TENSOR BASED REPRESENTATION AND ANALYSIS OF DIFFUSION-WEIGHTED MAGNETIC RESONANCE IMAGES

Transcription

1 TENSOR BASED REPRESENTATION AND ANALYSIS OF DIFFUSION-WEIGHTED MAGNETIC RESONANCE IMAGES By ANGELOS BARMPOUTIS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA

2 c 2009 Angelos Barmpoutis 2

3 To my wife, Λ ɛνα 3

4 ACKNOWLEDGMENTS I would like to thank my supervisor Dr. Baba C. Vemuri, for his inexhaustible inspiration and intellectual support during my doctorate studies. His insightful guidance affected positively not only my academic personality but also my character in general. It is also very important for me to acknowledge the financial support that my supervisor generously offered me in the form of research assistantship and travel grants. Furthermore, I would like to express my sincere appreciation to Dr. John R. Forder who supported my education related to the biomedical aspect of my research. The continuous collaboration with his lab in the McKnight Brain Institute at the University of Florida played important role in the clinical application of the techniques presented in this dissertation. Moreover, I would like to thank Dr. Arunava Banerjee, Dr. Jeffrey Ho, and Dr. Anand Rangarajan, for serving as members of my PhD committee and for devoting several hours to discus with me details of my research. I should also acknowledge Dr. Timothy Shepherd, Dr. Dena Howland, Dr. Evren Özarslan, Dr. Bing Jian, and Ritwik Kumar for our productive collaboration and discussions. I would also like to thank Saurav B. Chandra, and MinSig Hwang, both PhD candidates at the Biomedical Engineering Department, for their continuous feedback and useful technical comments, which helped me improve the clinical applicability of the scientific tools that I developed during my studies. In addition, I would like to mention the friendly and productive environment that was formed by my lab-mates Ting Chen, Santhosh Kodipaka, Dr. Nicholas Andrew Lord, Dr. Adrian Peter, Ajit Rajwade, Oneil Smith, Ozlem Subakan, Dr. Fei Wang, Yuchen Xie, Seniha Esen Yuksel and more others who shared with me unforgettable moments both in and out of lab. Moreover, I would like to thank Mrs. Rory Jean De Simone and Dr. Gerhard X. Ritter who, although are not directly related to my dissertation, both helped me to 4

5 integrate myself into the Computer and Information Science and Engineering Department during my first year of studies. Finally but not least, I would like to express my infinite appreciation to my wife Λ ɛνα, for her patience and continuous support. She gave me inspiration and courage all these years, which played significant role not only to my personal life but also to my academic work. 5

6 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS LIST OF ABBREVIATIONS ABSTRACT CHAPTER 1 DIFFUSION-WEIGHTED MRI PROCESSING: INTRODUCTION, PROBLEMS, GOALS AND APPLICATIONS Introduction Diffusion Tensor Field Processing Tensor Field Interpolation Tensor Field Segmentation Tensor Field Registration Higher Order Diffusion Tensors Problems in Tensor Processing Goals and Applications Real and Synthetic DW-MRI Data ESTIMATION OF REGULARIZED 2ND-ORDER DIFFUSION TENSOR FIELDS USING TENSOR SPLINES Introduction Mathematical Preliminaries Tensor Splines B-Splines Tensor Spline Interpolation Tensor Spline Fitting Log-Euclidean Splines Robust Tensor Splines Experimental Results APPLICATION OF TENSOR SPLINES IN SEGMENTING 2ND-ORDER TENSOR FIELDS Introduction DTI Segmentation Using Tensor Splines Experimental Results

7 4 ESTIMATION OF REGULARIZED 4TH-ORDER TENSOR FIELDS Introduction Diffusion Tensors of 4th Order Estimation From DWI Distance Measure Experimental Results Synthetic Data Experiments Real Data Experiments EXPONENTIAL TENSORS: A FRAMEWORK FOR EFFICIENT HIGHER-ORDER DT-MRI COMPUTATIONS Introduction Exponential Diffusion Tensors Distance Measure Distance from the Closest Isotropic Case Estimation of EDT Field from DWI Displacement Probability Profile Experimental Results and Discussion ESTIMATION OF DISPLACEMENT PROBABILITY FIELDS USING 4TH-ORDER TENSORS Introduction Spherical Function Tensorial Approximation Higher-Order Bases for HARDI Approximation Fast Estimation from HARDI Datasets Experimental Results Synthetic Data Experiments Real Data Experiments Application of the Displacement Probability Field for Tractography Extracting Tractosemas by Diffusing Probability Iso-Surfaces Experimental Results REGISTRATION OF 4TH-ORDER TENSOR FIELDS Introduction Locally Affine Registration of 4 th -Order Tensor Fields Distance measure Registration D Affine Transformation of 4 th -Order Tensors Group-Wise Registration and Atlas Construction Riemannian Metric for Positive-Valued Real Functions Group-Wise Registration of 4 th -Order Tensor Fields Implementation Details Robust Atlas Construction

8 7.4 Experimental Results Experiments Using Synthetic Datasets Real Data Experiments CLINICAL APPLICATIONS Introduction Comparison of Control and Injured Spinal Cord Datasets CONCLUSIONS REFERENCES BIOGRAPHICAL SKETCH

9 Table LIST OF TABLES page 2-1 Tensor field approximation errors using various algorithms Tensor field approximation errors using robust function Approximation errors of robust splines using different metrics Formulas to compute the tensor coefficients D i,j,k Comparison of DTI frameworks Properties of DTI frameworks Proposed bases functions

10 Figure LIST OF FIGURES page 1-1 Illustration of the DW-MRI acquisition using various diffusion sensitizing gradient directions A DW-MRI dataset from a rat brain The diffusion tensor coefficients computed from a DW-MRI dataset Illustration of a tensor spline Comparison of tensor field approximation methods Tensor spline example in real human hippocampus data Tensor spline example in isolated rat hippocampus Tensor field segmentation experiment Tensor field segmentation example of synthetic smoothly varying tensor field Illustration of segmentation under partial voluming effects D segmentation of the DT-MRI field from an isolated rat hippocampus D segmentation of the DT-MRI field from an isolated rat hippocampus Quantitative comparison of fiber orientations errors from 4 th -order tensors Quantitative comparison of the generalized trace of 4 th -order tensors Fiber orientation errors for various noise levels Visual comparison of 4 th -order tensor fields Visualization of the 4 th -order tensor field from a rat spinal cord Comparison between FA and 4 th -order f iso map Comparison of Exponential tensors and other higher-order tensor frameworks Exponential tensor field from a rat optic chiasm Illustration of exponential tensor processing D plots of three of the bases functions Synthetic simulated DW-MRI data Comparison of fiber orientation errors for various methods Estimated probability profiles from real data of a rat s spinal cord

11 6-5 Estimated tractosemas using synthetic data Estimated tractosemas from read hippocampal data Example of fiber tracking using tractosemas Quantitative comparison of registration methods Comparison of tensor atlases computed by various metrics Locally affine registration of hippocampal data th-order tensor field atlas construction The S0 images from control and injured rat spinal cords Comparison of the fiber orientations estimated in the control and injured cord Quantitative comparisons of spinal cord data Comparison of spinal cord data using Riemannian distances Comparison of tractosemas from control and injured spinal cord datasets

12 LIST OF SYMBOLS Symbols: b d(g) D g G P (r) q S γ δ The diffusion weighting factor (b-value) The diffusivity function The diffusion tensor The direction of the magnetic gradient (unit vector) The magnetic gradient The water molecule displacement probability, given displacement r The reciprocal space vector The acquired DW-MRI signal The gyromagnetic ratio The duration of the applied diffusion sensitizing gradients 12

13 LIST OF ABBREVIATIONS Abbreviations: DTI DW-MRI EDT HARDI MRI NMRI PSD ROI SPD Diffusion Tensor Imaging or Image(s) Diffusion-Weighted Magnetic Resonance Imaging or Image(s) Exponential Diffusion Tensor(s) High Angular Resolution Diffusion Image(s) Magnetic Resonance Imaging Nuclear Magnetic Resonance Imaging Positive Semi-Definite Region Of Interest Symmetric Positive-Definite 13

14 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TENSOR BASED REPRESENTATION AND ANALYSIS OF DIFFUSION-WEIGHTED MAGNETIC RESONANCE IMAGES Chair: Baba C. Vemuri Major: Computer Engineering By Angelos Barmpoutis May 2009 Cartesian tensor bases have been widely used to model spherical functions. In medical imaging, tensors of various orders can approximate the diffusivity function at each voxel of a diffusion-weighted MRI data set. This approximation produces tensor-valued datasets that contain information about the underlying local structure of the scanned tissue. The goal in this dissertation is to use the information provided in the tensor field in an automated system that detects changes in the tensor coefficients and correlates them with different types and levels of injury in spinal cord. In order to achieve this, one has to follow several intermediate steps of tensor field processing. These steps include robust estimation of the various orders diffusion tensor coefficients, tensor field segmentation, registration, and atlas construction as well as extraction of various features, such as anisotropy measures and fiber orientations. Several methods are presented for solving these problems in this dissertation. The proposed algorithms are either the first ever presented in literature for solving a specific problem, or improved alternative solutions to existing techniques. All methods are validated using synthetic simulated diffusion-weighted MR data and their effectiveness is demonstrated in several real datasets. 14

15 CHAPTER 1 DIFFUSION-WEIGHTED MRI PROCESSING: INTRODUCTION, PROBLEMS, GOALS AND APPLICATIONS 1.1 Introduction Studying and understanding the anatomy of the human and animal body is almost as old as the history of humanity itself. Ancient Egyptians according to archaeological evidence, performed operations on the human body. Studies on the function of the brain and its connection to human intelligence are reported in several ancient Greek documents (i.e. Hippocrates, 4th century BC). Progress toward understanding the neuroanatomy of the brain was gradually done over the ages; the invention of the microscope however (around 1595), became the starting point of a new era in the methodic and detailed research of the brain. After modern technological developments in image acquisition techniques, the acquisition of high resolution details of the human tissue was made possible in vivo. Magnetic Resonance Imaging (MRI) allows capturing of high contrast images of the soft human tissues. More specifically, Diffusion-Weighted MRI (DW-MRI) is an MRI modality that is the only non-invasive method for capturing the diffusivity of water molecules in human tissue. Variations of micro-structural properties across the neural system in the brain correspond to variations of diffusivity patterns, which can be captured by the DW-MRI signal. Stejskal and Tanner (1965) have modeled the attenuation of the acquired signal by a mono-exponential model given by S/S 0 = exp( bd) (1 1) where b is the diffusion weighting factor depending on the strength as well as the effective time of diffusion, D is the apparent diffusion coefficient, S is the measured attenuated signal and S 0 is the signal acquired with zero diffusion gradient. 15

16 Figure 1-1. This figure illustrates the acquisition of a DW-MR dataset that consists of N images. The images were acquired by applying a set of N magnetic gradients that sensitize diffusion in different orientations. The orientations are given by the reciprocal space vectors q 1... q N. The displacement probability P (r) of the water molecules is related to the DW-MRI signal attenuation through the following Fourier integral relationship (Callaghan, 1991; Özarslan et al., 2006). P (r) = S(q) S 0 e 2πiqT r dq (1 2) In Equation 1 2, r is the displacement vector, q is the reciprocal space vector, and S(q) is the DW-MRI signal value associated with vector q. The vector q is defined as a function of the magnetic gradient G given by q = γδg/2π, where γ is the gyromagnetic ratio and δ is the duration of the applied diffusion sensitizing gradients. Hence, both vectors q and G are pointing to the same direction g = q/ q = G/ G. Furthermore, the relation between the vector q and the diffusion weighting factor b is given by the expression b = 4π q 2 t, where t is the effective diffusion time. If we assume that the diffusion propagator follows a multi-variate (3-dimensional) Gaussian diffusion model, then its inverse Fourier transform can be computed analytically and is also a multi-variate Gaussian (Basser et al., 1994). In this case, the signal 16

17 Figure 1-2. A DW-MRI dataset from an excised rat brain. The S 0 image is shown on the top left. The rest of the 45 images were acquired using a set of 45 diffusion gradient directions. attenuation model in Equation 1 1 can be generalized as follows S/S 0 = exp( bg T Dg). (1 3) If we acquire several diffusion-weighted images (at least 7 including S 0 ), by applying diffusion-weighted magnetic gradient fields in different directions g, we can measure the coefficients of the diffusion covariance matrix D, called Diffusion Tensor. Figure 1-1 illustrates the acquisition of DW-MR images using a set of magnetic field gradient directions. Figure 1-2 shows a real rat brain dataset that consists of 45 diffusion weighted images and the S 0 image (shown on the top left). Observation of the images shows that the contrast patterns change when we apply gradients that sensitize diffusion in different directions. 17

18 Figure 1-3. A) The diffusion tensor coefficients computed from the DW-MRI dataset in Figure 1-2. The lower three coefficients are symmetric to the upper coefficients of the matrix and hence omitted. B) Each tensor can be visualized as an ellipsoid centered at the corresponding voxel. The corresponding diffusion tensor coefficients can be computed by fitting the model in Equation 1 3 to the acquired images using non-linear minimization techniques (Wang et al., 2004) and are shown in Figure 1-3. This figure also shows another more descriptive way of presenting the tensor field data, by plotting the corresponding ellipsoidal surfaces at each voxel. The ellipsoids can be computed by evaluating gg T Dg for g S 2, where S 2 denotes the space of unit vectors, i.e. unit sphere. This visualization of the matrix-valued field gives a very detailed insight of the fiber structures and connectivity in the scanned tissue. Further processing and study of such datasets helps us relate the various brain functions and disorders with specific patterns in changes of the diffusivity and finally develop mechanisms for automatic detection and classification of those neurological conditions. Such a framework for diffusion tensor field processing will include algorithms for interpolation, segmentation, registration of tensor fields as well as methods for atlas construction. Section 1.2 presents a brief introduction on the space of diffusion tensors and a review of several existing tensor field processing techniques. 18

19 1.2 Diffusion Tensor Field Processing The diffusion tensors are elements of the space of the (3 3) positive-definite matrices denoted by P (3). The space of SPD rank 2 tensors describes the space of all SPD matrices and in general, we use the notation P (n) to denote the space of SPD matrices of size (n, n). Mathematically, these positive definite diffusion tensors belong to a Riemannian symmetric space (Terras, 1988), where the Riemannian metric is defined as the inner product assigned to each point of this space. By using this metric, one can compute geodesic (shortest) distances between the points (diffusion tensors) of this space and can be employed in tensor field interpolation (Fletcher and Joshi, 2004; Wang and Vemuri, 2004, 2005; Lenglet et al., 2004; Pennec et al., 2005; San-Jose Estepar et al., 2005). Interpolation is required in several algorithms for tensor field registration, segmentation, atlas construction and fiber tacking. Below we briefly review some of the existing tensor interpolation techniques Tensor Field Interpolation Directly performing smooth interpolation of the individual components of the diffusion tensor matrices (Pajevic et al., 2002), does not preserve most of the properties, e.g., the value of the determinant of the diffusion tensors. Moreover, any processing of these SPD tensors by treating them component wise and ignoring the positive definiteness property will lead to estimation of unnatural diffusivity values. Smooth interpolation of orientation fields has been proposed by Barr et al. (1992). Although 2 nd order tensors contain the notion of orientation (e.g the orientation of its eigenvectors), their structure is much more complicated. By interpolating smoothly the orientations of a tensor field, we ignore other information provided by the tensors, (i.e., eigen values, determinant etc.). Wang and Vemuri (2004, 2005) used the symmetrized Kullback-Liebler (KL) divergence as a distance measure between two SPD tensors. The symmetrized KL, denoted here by KL s was shown to be a second order approximation to the true 19

20 Riemannian metric between SPD tensors and further was shown to be affine invariant. They also derived a closed form solution for computing the mean of two or more SPD tensors. A Riemannian metric for geodesic distance computation between two tensors was proposed by Fletcher and Joshi (2004); Batchelor et al. (2005); Lenglet et al. (2006). Having a tensor field, e.g., a volumetric DT-MRI, we can use the geodesic curves between spatially consecutive tensors in order to interpolate and approximate the dataset. However, none of the earlier reported methods on geodesic curve computation between tensors (Fletcher and Joshi, 2004; Batchelor et al., 2005; Lenglet et al., 2006) use higher order smoothness constraints in achieving the interpolation/approximation. Thus, although there is continuity of the interpolated dataset, higher order continuity and hence smoothness is lacking. Recently, a Log-Euclidean metric was proposed by Arsigny et al. (2005) for computing with tensors. In this work, the elements from the space of positive definite diffusion tensors, P (3), are mapped to their tangent space, denoted by Sym(3), using the matrix logarithm map. The tangent space of P (3) forms a vector space of dimension 6. Therefore, one can use the Euclidean norm for computations in this tangent space and finally by using the inverse mapping, the interpolated data are mapped back to the space of positive definite matrices P (3). Approximation of matrix-valued images can be achieved via various regularization methods. For instance, a PDE-based approach as was proposed by Weickert and Welk (2005). Although this is a feasible approach for denoising/smoothing and interpolation of matrix-valued images, the lack of use of an appropriate metric defined on the space of SPD matrices could lead to undesirable limitations in the solution, e.g., lack of affine invariance property etc. Another tensor field regularization method was proposed by Westin et al. (2006) using normalized convolution and Markov Random Fields (MRF) in a Bayesian framework. The SPD tensors are treated as vectors in 6D and their components 20

21 are treated independently. Once again, this disregards the special geometry of the space of SPD diffusion tensors in the regularization, which in turn may lead to inaccurate predictions (e.g., wrong determinants, lack of affine invariance etc.). Another important processing step is the segmentation of distinct regions in a tensor field; various techniques for achieving this goal are discussed in the Section Tensor Field Segmentation Several diffusion tensor field segmentation methods have been presented to date in literature, and we will briefly review them here. Zhukov et al. (2003), proposed a level set segmentation method that segments the scalar anisotropic property computed from the diffusion tensor. By using such a scalar field, the direction information contained in the tensor field has been ignored. Thus, such a method will fail to correctly segment two homogeneous regions of a tensor field that have the same scalar anisotropy property but are oriented in different directions. Feddern et al. (2003) extended the mean curvature flow and self-snakes models to matrix-valued data. However, their method employs the Euclidean metric to measure distance between tensors and not the Riemannian metric discussed earlier. Thus, it does not possess some of the interesting properties that accrue from the use of a Riemannian framework (e.g., the affine invariance property etc.). Wang and Vemuri (2004), developed a region-based active contour model for tensor field segmentation. They generalized the well known region-based active contour model for scalar field segmentation to that of tensor fields, and developed a variational principle using the Forbenius norm of the difference of tensors as a discriminant in the data term. In a later article by them, (Wang and Vemuri, 2005), an affine invariant tensor dissimilarity measure was proposed by using the J-divergence between oriented Gaussians representing the SPD tensors in the tensor field. This was incorporated in a region-based active contour model for the segmentation of DT-MRI data into either piece-wise constant or piece-wise smooth regions. 21

22 More recently, Lenglet et al. (2006) developed a statistical surface evolution framework using the Fisher-Rao metric. They employed the principle that within a region the diffusion tensors can be modeled by using statistics and distributions of diffusion tensors. The surface evolution framework and the active contour models can be extended to cope with multiple types of regions. Segmentation can also be achieved by means of registration. In this approach we estimate the unknown deformation that transforms a dataset into another given dataset whose segmentation is known. There are several methods for registering diffusion tensor fields and are briefly reviewed in Section Tensor Field Registration Registration of diffusion weighted MRI datasets by using 2 nd -order tensors was first proposed by Alexander et al. (2001). In this work a tensor re-orientation operation was defined as a significant part of the diffusion tensor field transformation procedure. This re-orientation operation rotates appropriately the diffusion tensors during the registration procedure, in order to maintain the local topology of the diffusivity patterns. A framework for non-rigid registration of multi-modal medical images was proposed by Ruiz-Azola et al. (2002). This technique performs registration based on extraction of highly structured features from the diffusion tensor field datasets. Registration of DTI using quantities which are invariant to rigid transformations and computed from the diffusion tensors was proposed by Guimond et al. (2002). By registering the rigid-tranformation invariant maps, one avoids the re-orientation step and thus can reduce the time complexity. The locally affine multi-resolution scalar image registration proposed by Ju et al. (1996) was extended to DTI images by Zhang et al. (2004). In this method the image domain of the image being registered is subdivided (using a multi-resolution framework) into smaller regions, and each region is registered using affine transformation. The affine transformation is parametrized using a transformation vector, a rotation, and an SPD 22

23 matrix. By using this parametrization one can avoid the polar decomposition step which is required in order to extract the rotation component for re-orientation purposes. This method has been also used for unbiased atlas construction using diffusion tensor images Zhang et al. (2007). Recently, various methods have been proposed for non-rigid registration of two tensor images employing continuous models of deformation (in contrast to the discontinuous locally affine models) and tensor metrics that make use of the geometry of symmetric positive definite tensors. The method proposed by Cao et al. (2006) registers the tensor fields using large deformation diffeomorphic metric mapping of tensor fields, resulting in optimizing for geodesics on the space of diffeomorphisms connecting two diffusion tensor images. A deformable diffusion tensor registration algorithm was proposed by Irfanoglu et al. (2008) which computes a B-spline defformation between two tensor field by minimizing the Geodesic-Loxodromes distances between the tensors. However, in all these approaches one of the registered images is fixed, which is biasing the obtained result. All the above methods (in Sections 1.2.1,1.2.2, and 1.2.3) perform processing (interpolation, segmentation, and registration respectively) of DW-MRI datasets based on scalar images or diffusion tensor images. However, the diffusion tensors are 2 nd -order tensorial approximations of the local diffusivity, which fails to represent complex local tissue structures, such as fiber crossings, and therefore DTI processing of dataset containing such crossings leads to inaccurate results. We can overcome this limitation by using algorithms that employ higher order approximations of the diffusivity function, and they are briefly reviewed in the Section Higher Order Diffusion Tensors In the previous sections we discussed methods for processing 2 nd order diffusion tensors. Several useful scalar quantities can be computed from the 2 nd -order tensorial approximation of the diffusivity function, such as the mean diffusivity (computed as the trace of matrix D i ) and various anisotropic measures of the tensor D i (fractional 23

24 anisotropy and relative anisotropy defined by Basser and Pierpaoli (1996)). The significance of these measures is that they provide an easy tool, since they are scalar quantities, for monitoring changes in the anisotropic properties of white matter fiber bundles caused either naturally by neurological disorders or artificially post-surgery or in response to a treatment. Fractional anisotropy has been successfully used clinically, and a reduction of its value has been reported in patients with trauma Ptak et al. (2003), multiple sclerosis Cercignania et al. (2001); Hasan et al. (2005), Hypoxic-Ischemic Encephalopathy Ward et al. (2006), multiple system atrophy Ito et al. (2007), meningitis Nath et al. (2007) and other pathologies. Similarly, reduction of the mean diffusivity, an other scalar quantity, has been also reported in the case of ischemic tissue van Gelderen et al. (1994). However, the above scalar measures are computed using the 2 nd -order tensorial approximation of the diffusivity, which although works well for simple tissue structures, fails to approximate more complex tissue geometry with multi-lobed diffusivity profiles. For instance, the value of fractional anisotropy drops significantly in areas of fiber crossings, although these locations are anisotropic. In order to overcome the above limitation, higher order tensors were introduced by Ozarslan and Mareci (2003) to represent more complex diffusivity profiles which better approximate the local diffusivity function. Generalized scalar quantities such as the variance of diffusivity and the generalized anisotropy were derived as functions of the higher order tensor coefficients (Evren Ozarslan, 2005). However, in all of these works the higher-order tensors are estimated without imposing the positivity of the diffusivity function approximation, which is significantly important since negative diffusivity values are non physical. The use of a 4 th -order covariance tensor was proposed by Basser and Pajevic (2003). This covariance tensor is employed in defining a Normal distribution of 2 nd order diffusion tensors. This distribution function has been employed by Basser and Pajevic (2007) 24

25 for higher-order multivariate statistical analysis of DT-MRI datasets using spectral decomposition of the 4 th -order covariance matrix into eigenvalues and eigentensors (2 nd order). However, 2 nd order tensors are used to approximate the diffusivity of each lattice point of a MR data set, failing to capture complex local tissue geometries, such as fiber crossings. Moakher and Norris (2006) presented a way to minimize the distance between a given higher-order tensor and the closest elasticity tensor of higher symmetry, using different metrics. This work was recently extended by Moakher (2008) where in he developed an interesting framework to describe the geometry of 4 th -order tensors. A 4 th -order symmetric positive definite tensor in three dimensions is represented by a 2 nd -order symmetric positive definite tensor in six dimensions and therefore, one can use the Riemannian metric of the space of 6 6 SPD matrices for the SPD 4 th -order tensor computations. However, this framework fails to parameterize the full space of SPD 4 th -order tensors. For example the groups of SPD tensors T (g) = ag1 4 + bg2 4 + cg3 4 and T (g) = (ag1 2 + bg2) cg3, 4 where a, b, c are positive valued coefficients, can not be written in the form of 6 6 SPD matrices. Furthermore, the Riemannian metric of symmetric positive-definite 2 nd -order tensors in six dimensions assigns infinite distance between 6 6 SPD and 6 6 semi-definite matrices. The SPD tensors in the aforementioned examples correspond to 6 6 semi-definite matrices and therefore, the Riemannian metric assigns infinite distance between them although their corresponding coefficients can be arbitrarily close. This leads to arbitrarily large errors in tensor processing and one should therefore strive to seek a more appropriate framework to characterize the full space of symmetric positive definite 4 th -order tensors. More than one distinct fiber tract structure within a voxel can be also estimated by employing other non-tensor-based models for reconstruction of the diffusion-weighted MR signal. Although the main topic of this dissertation is the tensor based representation of DW-MRI datasets, we briefly review here the major non-tensor-based techniques. Some of the models that have been proposed in literature include discrete and continuous mixture 25

26 of Gaussians by Tuch et al. (2003) and Jian and Vemuri (2007b) respectively, and the spherical harmonic transformation by Frank (2002). After reconstruction of the signal, one has to compute its Fourier transform in order to obtain the displacement probability whose peaks correspond to distinct fiber orientations. The displacement probability profiles can also be computed by transforming the diffusivity profiles using the diffusion orientation transform (DOT) (Özarslan et al., 2006). Multiple fiber orientations can also be estimated by reconstructing the orientation distribution function (ODF) (Descoteaux et al., 2007) using the so called Q-ball imaging (Tuch, 2004). Most of the above techniques (Basser et al., 1994; Tuch et al., 2003; Descoteaux et al., 2007; Jian and Vemuri, 2007b) can be expressed as a special case of a more generalized method in which the DW-MR signal can be expressed as the convolution over the sphere of a fiber bundle response function with the ODF (Tournier et al., 2004; Jian and Vemuri, 2007c). In this spherical deconvolution approach there is no limitation regarding the number of the estimated distinct fiber populations. 1.3 Problems in Tensor Processing Having acquired high angular and spatial resolution diffusion-weighted MRI datasets, we proceed with the development of algorithms that estimate useful quantities in order to explore and understand the connectivity of the various brain structures. Having this as the final goal, we should first develop a set of methods for solving smaller intermediate problems involved in the data processing and analysis. Initially, we need a robust method for estimating a field of diffusion tensors. This method should produce a denoised field of tensors in order to observe clearly the dominant directions of diffusion in the neural tissues. The use of the principal diffusion directions enables us to study the connectivity of the brain structures by segmenting it into regions with different diffusional characteristics. This segmentation process is another problem whose solution will contribute to the understanding of the underlying neural structures. 26

27 In several anatomical regions of the brain, the underlying structures involve multi-fiber crossings that can be recovered by using various models such as the higher-order diffusion tensor model. The estimation of positive higher-order diffusion tensors from diffusion-weighted MR images is an open problem. The difficulty lies in that positivity cannot be easily imposed on higher-order tensors. Another problem in multi-fiber reconstruction is the computation of the intravoxel distinct fiber orientations, which correspond to the maxima of the water molecule displacement probability. Therefore, computing accurately and efficiently the displacement probability from the acquired images is also an intermediate task that needs to be performed in order to track complex fiber structures. The tracking of fiber crossings as well as splaying fibers is another problem that needs to be addressed for the understanding of the brain connectivity. More specifically, splaying fibers correspond to asymmetric diffusivity patterns, the estimation of which is an open problem. These structures can be recovered either by using information from a single dataset, or can be estimated by combining statistical information taken from several datasets of the same subject. The latter involves construction of a statistical atlas for a type of subjects (e.g. normal spinal cords) which can be used later in an automated classification system for detecting type and degree of injury. At this point, we should emphasize that the existing methods for atlas construction involve scalar images or diffusion tensor fields of 2 nd -order, none of which account for multiple intravoxel fiber orientations. 1.4 Goals and Applications The goal of this dissertation is to propose solutions to the previously stated theoretical problems and present detailed demonstrations of their performance in synthetic and real DW-MRI datasets. The methods proposed in the following chapters are either the first ever presented in literature for solving a specific problem, or improved alternative solutions to already existing techniques. 27

28 More specifically in Chapter 2, Tensor Splines are proposed as a method for estimating a continuous and smoothly varying diffusion tensor field from a given diffusion-weighted MRI dataset. This method performs high-order continuous interpolation of a tensor field in contrast to the existing methods that employ linear interpolation using the Riemannian geometry of the space of SPD matrices. Tensor Splines are employed in Chapter 3 as a module in a probabilistic diffusion tensor field segmentation algorithm. In Chapter 4 I present the first ever method for estimating positive 4 th -order diffusion tensors from diffusion-weighted MR images. This method guarantees the positivity of the estimated diffusivity values, which is a physical property that is not being imposed by the existing methods. As a result our method outperforms the existing techniques in terms of accuracy. Another higher-order parametrization of the diffusivity is presented in Chapter 5 using Exponential Tensors. In this method we model the diffusivity function using the exponential of higher order polynomials. This framework guarantees the estimation of positive values and also allows fast tensor-field computations due to the simple geometry of the parametric space. To our knowledge this is the fastest framework for higher-order tensor field processing that imposes the positivity constraint on the estimates. Tensors of 4 th order are also used in Chapter 6 for computing the water molecule displacement probability, which is then employed for estimating symmetric as well as asymmetric local fiber structures. We should emphasize that the existing methods estimate fiber orientation distributions that are naturally symmetric and consequently cannot model asymmetries such as splaying fibers. In Chapter 7 we present a method for groupwise registration and atlas construction of 4 th order tensor fields. This method extends existing techniques that employ scalar images or 2 nd -order matrix valued images which cannot handle fiber crossing diffusivity profiles. Finally, in Chapter 8 the tensor field framework is demonstrated in a real application for classifying different levels of spinal cord injuries. This method involves registration and 28

29 atlas construction of 4 th -order tensor fields as well as extraction of invariant features for comparison and classification of real datasets. 1.5 Real and Synthetic DW-MRI Data The proposed methods are demonstrated using several real DW-MR data from various anatomical regions of the rat and human neural system. The corresponding data description along with the acquisition protocols are presented at the experimental sections of each chapter. All the real datasets were scanned in the McKnight Brain Institute at the University of Florida and provided, along with the acquisition protocol parameters, by Dr. Evren Ozarslan, Dr. Timothy Shepherd, Dr. John Forder, Dr Steve Blackband, Saurav B. Chandra, and MinSig Hwang. The spinal cord samples were prepared by Dr. Dena Howland and her research assistants. Furthermore, in order to produce quantitative results, we tested all the methods on synthetic data that depict multiple fiber crossings. The data were generated by simulation using a simplified approximation of the diffusional processes within real neural tissue. The diffusion-weighted MR signal attenuation from molecules, with free diffusion coefficient given by D 0, restricted inside a cylinder of radius ρ and length L, when applied diffusion gradient makes an angle θ with the orientation of the cylinder, is modeled by the following equation (Söderman and Jönsson, 1995): E(q) = n=0 k=1 m=0 2K nm ρ 2 (2πqρ) 4 sin 2 (2θ)γ km 2 [ (nπρ/l) 2 (2πqρcosθ) 2] 2 [ [1 ( 1) n cos(2πqlcosθ)] Jm (2πqρsinθ) L 2[ γ km2 (2πqρsinθ) 2] 2 (γkm2 m 2 ) exp ( [ (γkm ρ ) ] ) 2 ( nπ ) 2 + D 0 L ] 2 (1 4) In this expression, J m is the m th order Bessel function, γ km is the k th solution of the equation J m (γ) = 0 with the convention γ 10 = 0, and K nm = δ n0 δ m0 +2[(1 δ n0 )+(1 δ m0 )]. 29

30 In the presence of more than one cylinder, the signal attenuations from these cylinders become additive. Therefore, we can use Equation 1 4 to simulate multiple fiber crossings. Although this system is simpler than the diffusional processes within real neural tissue, it is a suitable test bed for the procedures developed for fiber orientation estimation and it has been used by Özarslan et al. (2006). Similar to that in von dem Hagen and Henkelman (2002), in our simulations we used the following parameters: L = 5 mm, ρ = 5 µm, D 0 = mm 2 /s, = 20.8 ms, δ = 2.4 ms, b = 1500 s/mm 2 and the infinite series in Equation 1 4 were terminated at n = 1000 and k, m =

31 CHAPTER 2 ESTIMATION OF REGULARIZED 2ND-ORDER DIFFUSION TENSOR FIELDS USING TENSOR SPLINES 2.1 Introduction Processing of DT-MRI data sets has scientific significance in clinical sciences. Most of these applications involve processing that more often than not involves interpolation of the diffusion tensor fields. For example, registration of DT-MRI data sets will require interpolation to be employed when a registration transformation is applied to a tensor field defined on a lattice. Other examples which require tensor field approximation as well as interpolation include, tensor field segmentation, atlas construction etc. In this chapter we present a novel diffusion tensor field approximation algorithm. Our algorithm approximates and interpolates the diffusion tensor fields by forming a higher order continuous tensor product of B-splines using the Riemannian metric on the space of SPD matrices. Our method involves a two step procedure wherein the first step uses Riemannian distances to evaluate a tensor spline by computing a weighted intrinsic average of tensors and the second step minimizes the Riemannian distance between the evaluated tensor spline curve and the given data. We present comparisons of our algorithms with existing methods applied to synthetically generated diffusion MRI data, and show significantly improved results in the presence of noise and outliers. We also present several 3D DT-MRI approximation results from an isolated rat hippocampus. The chapter is organized into the following sections: In Section 2.2, we present a brief note on the geometry of the space of diffusion tensors and the related mathematical background. This is followed by the presentation of a novel and robust algorithm for computing tensor splines in Section 2.3. Finally, in Section 2.4, comparisons of our tensor spline algorithm with existing methods of interpolation and approximation of DT-MRI data are presented. Validation results using synthetically generated noisy tensor field data with outliers are also presented. 31

32 2.2 Mathematical Preliminaries In this section we briefly review the geometry of the space of diffusion tensors. More detailed expositions on some of this material maybe found in the articles by Pennec et al. (2005); Lenglet et al. (2004); Fletcher and Joshi (2004). The space of rank two diffusion tensors can be viewed as a Riemannian Symmetric space (Helgason, 2001), where a Riemannian metric assigns an inner product to each point of this space. By using this metric we can compute geodesic distances between diffusions tensors and calculate statistics on this space (Fletcher and Joshi, 2004; Lenglet et al., 2004; Pennec et al., 2005). For example, the mean tensor of a set of diffusion tensors, can now be computed as that tensor which minimizes the sum of squared Riemannian distances between itself and the given set of tensors. The mean tensor maybe employed as, an interpolant, for performing principal geodesic analysis etc. In the aforementioned Riemannian framework, the distance between two tensors D 1 and D 2 is given by dist 2 (D 1, D 2 ) = trace(log(d 1/2 1 D 2 D 1/2 1 ) 2 ), where log is the matrix logarithm operation. By using this distance measure, the geodesic curve (shortest path) between D 1 and D 2 is defined uniquely. The tangent, specified at the first tensor with respect to the other one along the unique geodesic between them,. is a 3 3 symmetric matrix and is given by the Riemannian-log map, Log D1 (D 2 ) = D 1/2 1 log(d 1/2 1 D 2 D 1/2 1 )D 1/2 1. The inverse operation is given by the Riemannian-exp map, Exp D1 (D 2 ) = D 1/2 1 exp(d 1/2 1 DD 1/2 1 )D 1/2 1, where exp is the matrix exponential operation and D is a 3 3 symmetric matrix. We will use this Riemannian distance between SPD tensors in computing the distance between the given data and the tensor spline approximation of the data as well as in computing the weighted average for defining the tensor spline. In the following section the Riemannian exponential and logarithmic maps, and the expression for a geodesic between two diffusion tensors will be used in order to define and compute the tensor splines. 32

33 2.3 Tensor Splines In this section we present a novel and robust spline approximation algorithm given a noisy SPD tensor field. Our algorithm involves the use of the Riemannian distance between SPD tensors in order to evaluate a tensor spline by computing a weighted intrinsic average of SPD tensors. This module (the intrinsic weighted average calculator) is then used in a robust tensor product B-spline fitting method involving the minimization of the Riemannian distance between the tensor spline function and the SPD tensor valued data. The tensor valued data are obtained from the diffusion MRI data sets by employing the mono-exponential signal attenuation model in Equation 1 1 by Stejskal and Tanner (1965), and using a linear least squares model to estimate the diffusion tensors over the 3D lattice. This section has three subsections. First we provide a brief review of B-splines. Next, we present a novel algorithm for computing splines on a given SPD tensor field. After that we present tensor splines using the Log-Euclidean metric as an improvement over recent work by Arsigny et al. (2005) (described earlier in Section 1.2.1). Then, we present our robust tensor spline approximation algorithm. Finally a robust tensor spline approximation (fitting) technique is presented B-Splines The equation for a (k 1) th degree B-spline with (n + 1) control points (c 0, c 1,..., c n ) and n+k+1 numbers called knots (t k+1, t k+2,..., t n+1 ), is S(t) = n N i,k (t)c i (2 1) i=0 where t 0 t t n+1 (k 1). Each control point is associated with a basis function N i,k where 1 if t i t < t i+1 N i,1 = (2 2) 0 otherwise 33

34 and t t i N i,k (t) = N i,k 1 (t) + N i+1,k 1 (t) t i+k t (2 3) t i+k 1 t i t i+k t i+1 N i,k (t) functions are polynomials of degree k-1. Cubic basis functions N i,4 can be used for a 3rd degree B-spline. Knots must be series of monotonically increasing numbers. A more detailed discussion on B-splines can be found in the article by de Boor (1972). One useful property of the functions N i,k (t) is that N i,k (t) 0, for all i and N i,k (t) = 1. Considering the above properties, functions N i,k (t) behave as blending i=0 functions and Equation 2 1 is a weighted average of the control points c i Tensor Spline Interpolation Given two SPD tensors D 1 and D 2, we can use the tangent direction specified at D 1 with respect to D 2 by the unique geodesic (obtained using the Riemannian structure) between them. However, a geodesic curve does not contain information about the neighborhood of the two points D 1 and D 2 between which it is defined. Thus, although there is continuity of the interpolated tensors, there is lack of higher order continuity. It is more natural to have a higher order continuity in the interpolant when used to represent smoothly varying regions of tensors. Recent work by Pajevic et al. (2002), on continuous tensor field approximation achieves smoothness, however, a Riemannian framework is not employed for tensor calculations. In this section we define tensor splines which are curves interpolating or approximating matrix valued functions, constructed using the geometry of the space of SPD tensors. Note that we are defining tensor-splines by doing weighted intrinsic averages on P (n) and choosing the weight functions to be B-splines. As an illustration of interpolation on a 1D grid of tensors, figure (2-1B) depicts the idea of using weight functions (B-splines here) to perform weighted average of tensors using the Riemannian metric. This weighted averaging leads to the desired degree spline interpolant (approximant when used in a fitting problem) of the diffusion tensor data. 34

35 A N 0,4 N 1,4 N 2,4 N 3,4 N 4,4 N 5,4 N 6,4 t -3 t -2 t -1 t 0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 p 1 p 0 p 3 p 2 M S(t) p 4 B Figure 2-1. A) Tangent space of the manifold M of diffusion tensors at point p 1. The tangent vector X points to the direction of geodesic γ(t) between the points p 1 and p 2. B) A cubic tensor spline S(t), that approximates p i s of a 1-D tensor field. The given points p i and the points of the tensor spline S(t) are SPD matrices, elements of the Riemannian manifold M. However, the tensor spline is in P (n) R since the tensors lie on a 1-D lattice. 7 control points c i are required and 11 knots t i. The association between basis functions N i,4 (t), knots t i and given data points p i are displayed in this figure. Let us assume that we have a set of N diffusion tensors (D 0, D 1,..., D N 1 ) on a one dimensional grid, and we need to interpolate between them. Linear (1st degree) interpolation on the tensor space can be achieved by simply computing points on the geodesics connecting two consecutive diffusion tensors. Higher degree continuous interpolation can be achieved by using a set of control points and a knot vector. A k 1 degree tensor spline that fits to our data requires N + k 2 control points (c 0, c 1,..., c N+k 3 ) which are also tensors and N + 2(k 1) monotonically increasing knots (t k+1, t k+2,..., t N+k 2 ). A tensor S(t), where t [t j, t j+1 ), which is a point on 35

36 a tensor spline, can now be computed by generalizing the Equation 2 1 to the space of tensors. We can compute the value S(t) of the k-1 degree B-spline of tensors for a particular t value, by calculating a weighted intrinsic average,, of the control tensors c i, where the weights equal to the basis functions w i = N i,k (t), discussed earlier. S(t) = n i=0 w ic i (2 4) The intrinsic weighted average (Equation 2 4) of tensors is defined using the Riemannian distance instead of the Euclidean distance, and it is the minimizer of the function 1 min ρ(µ) = min µ P (n) µ P (n) 2 n w i dist(µ, c i ) 2 (2 5) where dist(.,.) is the Riemannian geodesic distance. The weighted average can be computed using a gradient descent algorithm which is an extension of the algorithm described by Fletcher and Joshi (2004) for computing the mean of tensors. The gradient of i=0 ρ(µ) is given by µ ρ = n i=0 w ilog µ (c i ) (2 6) Thus the intrinsic weighted average of a set of diffusion tensors can be computed by the following procedure: input : c 1,..., c N P (n) ; w 1,..., w N weights output: µ P (n), the weighted mean µ 0 I ; i 0 ; while X i > e do X i µi ρ; µ i+1 Exp µi (X i ) ; end Algorithm 1: Intrinsic Weighted Mean of Tensors Tensor Spline Fitting In order to fit a tensor spline to the diffusion tensor data, we have to approximate the control tensors of such a spline. A tensor spline that fits to our data, minimizes the 36

37 Riemannian distance of the given tensors from the tensor spline curve. E = 1 N 1 dist(s(t i ), D i ) 2 (2 7) 2N i=0 In Equation (2 7), the Riemannian metric should be used for the distance calculation, since the tensor space, where the data and control points live, is a curved manifold (convex cone). We need to find a set of control points (c 0, c 1,..., c N 1+k 2 ) that form the spline S(t) which minimizes the energy E. The gradient of E with respect to c j is then given by, Dj E = 1 N 1 2N S(t i )dist(s(t i ), D i ) 2 cj S(t i ) (2 8) i=0 The gradient of the square distance between S(t i ) and D i with respect to S(t i ) equals, S(ti )dist(s(t i ), D i ) 2 = 2Log S(ti )(D i ) (2 9) where Log S(ti )(D i ) is the Riemannian logarithmic map which is a tangent vector at S(t i ). Since, the gradient of the energy (see Equation 2 8) is with respect to c j, we need to express the gradient in Equation 2 9 by using tangent vectors at point c j. Taking this into consideration, Equation 2 9 can be approximated by the formula Λ cj (D i, S(t i )) = Log cj (D i ) Log cj (S(t i )), so we obtain S(ti )dist(s(t i ), D i ) 2 2Λ cj (D i, S(t i )) (2 10) Furthermore, the gradient of S(t i ) with respect to c j in the Equation 2 8 is cj S(t i ) = N j,k (t i ) (2 11) Using Equations 2 11 and 2 10 in Equation 2 8 we obtain, cj E = 1 N N 1 i=0 Λ c j (D i, S(t i ))N j,k (t i ). (2 12) 37

38 Starting with an initial guess of the control tensors, we can update them by using the gradient descent technique. The new values c j of control tensors will be c j = Exp cj ( 1 N ) Λ c j (D i, S(t i ))N j,k (t i ) i=0 N 1 (2 13) where Exp. (.) is the Riemannian exponential map. The initial guess of the control tensors can be either the given data or the average tensor of the given tensors. The gradient descent algorithm is summarized below: input : N tensors (D 0,..., D N 1 ), ; N+2(k-1) monotonically increasing knots (t k+1,..., t N 1+k 1 ) ; k, and a small value e output: N-1+k-1 control tensors (c 0,..., c N 1+k 2 ) X 0 e+1; while X j > e do j Evaluate S(t i ) for i=0... N-1 ; for j=0 to N-1+k-2 do X j zero matrix ; for i=0 to N-1 do X j X j + Λ cj (D i, S(t i ))N j,k (t i ) ; end c j Exp cj X j ; end c c ; end Algorithm 2: Control Tensors estimation The time complexity for a single iteration of algorithm 2 is of order O((k d c)n) where k is the degree of the spline, d is the dimensionality of the dataset (for 3D data d = 3) and c is the number of iterations of algorithm 1 and N is the given input data size (number of tensors to be approximated). In the experiments that we performed we found that algorithm 1 converges in at most in 5 iterations (c 5), a CPU time of 9.37 sec per iteration on a Pentium 2.4GHz processor for fitting a cubic (k = 3) tensor spline in a dataset of size As expected, the time complexity of algorithm 2 increases as we increase the degree of the spline k or the dimensionality of the dataset d. Note that we 38

39 chose to provide this machine independent measure of time complexity because execution time will depend on the machine architecture and therefore is not a preferred measure. The error introduced by the approximation of Equation 2 10 can be large, if the tensor spline approximation S(t i ) is far from the target D i. When S(t i ) tends (moves closer) to D i during the spline fitting procedure, the error introduced by the approximation of Equation 2 10 tends toward zero. By setting a small threshold e on the difference between consecutive iterates, the outer loop of algorithm 2 will be iterated sufficient times in order for the error of Equation 2 10 to be as small as needed. Thus, the control tensors c j, which are obtained as the output of algorithm 2, are estimated by taking the true geometry of P (n) into account and not doing approximations that undermine it. Tensor Splines can be easily extended to higher dimensional tensor fields. For example consider the case of a 2D N M tensor field. In this case we are doing an approximation on P (n) R 2. A (k 1) degree tensor spline that fits to our data requires (N + k 2)x(M + k 2) control tensors and (N + 2(k 1))x(M + 2(k 1)) monotonically increasing (in both the dimensions) knots (t k+1, k+1,..., t N+k 2,M+k 2 ). Note that in this case the knots are vectors with 2 elements, one for each parametric dimension. Finally the new basis functions are formed by the tensor product of 1-dimensional basis functions N i,j,k ([t 1 t 2 ]) = N i,k (t 1 )N j,k (t 2 ) Log-Euclidean Splines Recently, Arsigny et al. (2005) proposed a new Log-Euclidean metric for tensor calculations. In this framework, the diffusion tensors are first mapped using the matrix logarithmic map to the space of the symmetric matrices Sym(n). Thereafter, the Euclidean norm is used in all calculations in this space. Finally by using the matrix exponential mapping, the computed values are mapped back to the manifold P (n). Using the Log-Euclidean metric we can also fit a spline to the logarithmically mapped data in the space of symmetric tensors, and after that we can map the interpolated/approximated 39

40 symmetric tensors back to the space of SPD tensors using the exponential mapping. In Section 2.4, we provide a quantitative comparison between tensor splines and splines using the Log-Euclidean metric that we will call Log-Euclidean Splines Robust Tensor Splines The presence of outliers is common in DT-MRI data due to noise in the original data obtained from the MR scanners (Wang et al., 2003). A robust algorithm should reject these outliers from further consideration in any processing algorithms applied to the dataset. A robust function can be applied to the energy function, in order to weight the given data D i appropriately. We can use a robust function that assigns weights in the interval [0, 1], where weights which are almost zero imply rejection of the corresponding data point. Furthermore, high weights should be assigned to the data points whose distance from the unknown spline curve is small and on the other hand lower weights should be assigned to the data points whose distance from the unknown spline is larger. Let us consider the following function φ(x) = e x2 σ 2, whose derivative ψ(x) has the aforementioned properties. By using the above function φ, the energy function that we want to minimize can be written as E = 1 N 1 2N φ(dist(s(t i), D i )) (2 14) i=0 The gradient of this energy with respect to the control tensors now becomes cj E = 1 N N 1 i=0 ψ(dist(s(t i), D i )) Λ cj (D i, S(t i ))N j,k (t i ) In the above equation the quantity ψ(dist(s(t i ), D i )), weights the given data points D i, leading to a spline approximation that is robust to outliers. The distance function dist(.,.), as it was previously mentioned, measures the Riemannian distance between the tensors. 40

41 2.4 Experimental Results In this section, we present several approximation experiments with noisy synthetic as well as real DT-MRI data. We also present comparisons with four other existing methods to demonstrate the performance of our proposed Tensor Spline approximation algorithm. For visualization purposes in the tensor field plots we have assigned to the red-green-blue channel (RGB) of each tensor the values of the X-Y-Z components of its dominant eigenvector. We synthesized a tensor field on a 2D lattice of size (see Figure 2-2 ). For the generation of this field, we simulated the diffusion-weighted MR signal using the equation presented by Söderman and Jönsson (1995) (see Section 1.5 for details on formulas for generating the MR signal). Using this process at each voxel the MR signal was simulated as a function of the angle θ between the applied diffusion gradient and the orientation of the fiber. At each voxel of the 2D synthetic field the orientation of the fiber was assumed to be tangent to circles centered at the lower left corner of the field (Figure 2-2 ). Using this fiber structure the diffusion-weighted MR signal attenuation was simulated for 21 orientations that correspond to the second-order tessellation of the icosahedron on a unit hemisphere, using b-value=1500 s/mm 2. For more details on other simulation parameters, see Appendix B. The diffusion tensor field was estimated from the 21 diffusion-weighted images using a linear least squares technique applied to the log linearized Stejskal-Tanner Equation 1 1. The principal eigenvectors of the estimated diffusion tensor field are shown in Figure 2-2. This diffusion tensor field will be considered as the ground truth field for the experiment described below. Gaussian noise was added to the real and imaginary parts of the simulated diffusion MR signal and then the magnitude signal computed from this noisy complex-valued data. From this signal, we estimated the diffusion tensors as before and then subsampled it by a factor of 4. This process was repeated for different amounts of signal to noise 41

42 ratios. The first column of figure 2-2 depicts the primary eigenvector field of the 9 9 subsampled noisy tensor fields corresponding to signal to noise ratios of 5.0 (top) and 3.0 (bottom). Our goal now is to compare our tensor spline approximation and interpolation method against existing methods in literature, as well as our own modifications of these techniques. Figure 2-2. Comparison of approximation methods using a SNR=5.0 (top) and a SNR=3.0 (bottom). A) Primary eigenvectors of the noisy tensor fields. The rest of the columns shows the error in robust approximation using B) Euclidean spline, C) PDE interpolation, D) Log-Euclidean spline, E) Tensor spline. The Riemannian metric was employed for computing these errors. We first approximated (fitted) the noisy tensor fields by using four different techniques including ours and then interpolated the approximation (fitted) results by a factor of 4. The four methods that we employed were, A) Linear approximation of the elements of the SPD tensors, B) Log-Euclidean geodesic approximation (Arsigny et al., 2005), C) Riemannian geodesic approximation (Fletcher and Joshi, 2004), D) PDE-based anisotropic non-linear diffusion (Weickert and Welk, 2005) (table 2-1). Following this, we present a table of results comparing our method with statistically robust implementation of all the methods (except the PDE-based diffusion filter as its not a simple matter to implement this filter in a robust framework). After this, we present another table which depicts results of comparison of a spline version of others work with our own, all in a 42

43 Table 2-1. Tensor field approximation errors using various algorithms Riem. µ Riem. σ Fro. µ Fro. σ Euc. Geodesic Appr Log-Euc Geodesic Appr Riem. Geodesic Appr PDE interpolation Non-Robust Tensor Spline Table 2-2. Tensor field approximation errors using robust function Riem. µ Riem. σ Fro. µ Fro. σ Robust Euc. Geodesic Appr Robust Log-Euc Geodesic Appr Robust Riem. Geodesic Appr Robust Tensor Spline Appr robust framework. In all these comparisons, as expected, our tensor splines algorithms out-performed the competing methods. We use two methods to measure the distance of the estimated tensor fields from the ground truth tensor field of figure 2-2 ; a) the Riemannian metric and b) the Frobenius norm defined as trace((a B)(A B) T ) where A and B are two SPD matrices. These errors are computed at each voxel and the mean and standard deviation of these errors are reported in tables 2-1, 2-2 and 2-3 for two noisy data sets corresponding to a signal to noise ratio of 3.0. As evident, the error is much smaller for our algorithm in comparison to the others. These results demonstrate the superior performance of our algorithm over other existing methods. Table 2-3. Approximation errors of robust splines using different metrics Riem. µ Riem. σ Fro. µ Fro. σ Robust Euc. Spline Appr Robust Log-Euc Spline Appr Robust Tensor Spline Appr

44 A B Figure 2-3. Zero-gradient image of a slice from 3D volume of a human hippocampus autopsy specimen. Primary eigenvector field before (A) and after (B) cubic tensor spline approximation from the marked region of image. For the experiment presented in Figure 2-3, an 8-mm coronal segment of the human hippocampus was dissected from an autopsy specimen with short postmortem interval to fixation (21.2 hrs) and no histological evidence of neuropathology. Diffusion tensor microscopy data were collected from the hippocampal specimen using a 14.1-T magnet with a protocol that included 21 unique diffusion gradient orientations, diffusion time =17 ms, and b=1250 s/mm 2 (Shepherd et al., 2007). A diffusion tensor field was estimated from the diffusion-weighted images using linear fitting. Figure 2-3A shows a the diffusion-weighted MR image obtained using a zero gradient field of a slice from the 3D volume. A red box has been used to identify the region of interest (ROI) containing a reasonably noisy section in the DT-MRI volume. Figures 2-3A and 2-3B show the primary eigenvectors of the diffusion tensor field in the 3D ROI before and after cubic tensor spline approximation respectively. Notice that all of the outliers have been rejected and the field has been approximated smoothly. 44

45 Finally figure 2-4 shows another real data example from an isolated rat hippocampus. The diffusion weighted MR images for this example were acquired using the following protocol. This protocol included acquisition of 22 images using a pulsed gradient spin echo pulse sequence with repetition time (TR) = 1.5 s, echo time (TE) = 28.3 ms, bandwidth = 35 khz, field-of-view (FOV) = mm, matrix = with continuous 200-µm-thick axial slices (oriented transverse to the septo-temporal axis of the isolated hippocampus). After the first image set was collected without diffusion weighting (b 0 s/mm 2 ), 21 diffusion-weighted image sets with gradient strength (G) = 415 mt/m, gradient duration (δ) = 2.4 ms, gradient separation ( ) = 17.8 ms and diffusion time (T δ ) = 17 ms were collected. Each of these image sets used different diffusion gradients (with approximate b values of 1250 s/mm 2 ) whose orientations were determined from the 2 nd order tessellation of an icosahedron projected onto the surface of a unit hemisphere. The image without diffusion weighting had 36 signal averages (time = 81 min), and each diffusion-weighted image had 12 averages (time = 27 min per diffusion gradient orientation) to give a total imaging time of 10.8 h per hippocampus. Temperature was maintained at 20 ± 0.2 o C throughout the experiments using the temperature control unit of the magnet previously calibrated by methanol spectroscopy. In figure 2-4, the proposed tensor spline approximation algorithm is compared with the recently proposed Log-Euclidean metric based approximation algorithm by Arsigny et al. (2005), discussed earlier. Figure 2-4B shows a 3D view of the results using Log-Euclidean geodesic approximation and non-robust Tensor Spline algorithms. Notice that in the Tensor Spline approximation results the noise has been considerably reduced. This maybe attributed to the higher-order smoothness imposed by the Tensor Spline developed in this work. Figure 2-4A depicts the FA map of the original (noisy) and the approximated data. Note that the FA map after the approximation is much smoother that the FA map prior the approximation (see 2-4A). 45

46 A B Figure 2-4. Real DTI from an isolated rat hippocampus. A) FA maps before (top) and after (bottom) tensor spline approximation. B) The principal eigenvector field after Log-Euclidean geodesic approximation, and non-robust tensor spline approximation. 46

47 CHAPTER 3 APPLICATION OF TENSOR SPLINES IN SEGMENTING 2ND-ORDER TENSOR FIELDS 3.1 Introduction In this chapter we present a novel DT-MRI multimodal (multi-class regions) segmentation algorithm using the tensor splines (presented in Chapter 2) as an approximation module. The segmentation is achieved by jointly estimating the label (assigned to each tensor residing at a voxel) field and the parameters (control points) of each smooth tensor spline model representing the labeled regions. The label field is modeled by a Gauss Markov Measure Field (GMMF) and the segmentation algorithm very efficiently computes the posterior marginal probability distribution of the label field (given the parameters describing the region model) as the global minimizer of a linearly constrained quadratic energy. Furthermore we present several 3D DT-MRI segmentation results using synthetic and real dataset from an isolated rat hippocampus. The motivation for segmenting and analyzing the hippocampus is due to its importance in semantic and episodic formation that is particularly vulnerable to acute or chronic injury (Amaral and Witter, 1995; Squire et al., 2004). In current clinical practice, we only look at the whole hippocampus and describe atrophy for epilepsy (hippocampal sclerosis), schizophrenia, depression, hypoxia-ischemia, trauma and Alzheimer s disease and other dementias. Obviously these cannot be distinguished from each other and hippocampal atrophy is a late imaging sign of pathology. However, we know from more than 100-years of literature that the hippocampus is made of many different cytoarchitectural regions and that these regions are selectively vulnerable to the aforementioned diseases (e.g. the CA1 and subicular regions are affected by Alzheimer s disease while the dentate gyrus is affected by medial temporal lobe epilepsy). These regions can be distinguished by diffusion tensor MRI. Thus, the segmenting techniques being developed here could prove useful to improving the sensitivity and specificity 47

48 of diffusion MRI for detecting and monitoring hippocampal diseases. We can also use these methods for studies in animal models of hippocampal disease. Structural insights from high-resolution DT-MRI imaging of the isolated rat hippocampus were presented previously by Shepherd et al. (2006). In our experiments we use these structural results (Shepherd et al., 2006) for validation of the obtained segmentation. 3.2 DTI Segmentation Using Tensor Splines In this section, we pose the DT-MRI segmentation problem in a Bayesian estimation framework which has many advantages over the deterministic segmentation schemes in that it naturally allows for incorporation of any specific domain knowledge in the form of priors. Moreover, estimates are provided along with the uncertainty in the estimate. Also, one can get a soft segmentation i.e., a probabilistic segmentation when necessary. Our formulation of the segmentation problem drawn on recent work by Rivera et al. (2005). We assume that the given DT-MRI dataset D i consists of K regions, where i is the lattice index. Furthermore we assume that each region is represented by a model S k (t i ) where k = 1..K. There are different choices for the model, which can be either piece-wise constant or a smoothly varying tensor field model. In the space of diffusion tensors the piece-wise constant model has a parameter θ k which is a 3 3 SPD matrix. Therefore for this model the equation S k (t i ) = θ k holds for all t i. In the case of smoothly varying tensor fields we chose a tensor product of tensor splines as our model S k (t i ), whose parameters are the control tensors c i defined in Section The relation between the control tensors and the tensor spline is given by the Equation 2 4. Let b ki be the label map, where b ki = 1 indicates that the diffusion tensor at the i-th lattice point belongs to the k-th region class and b ki = 0 otherwise. Considering the above notation, a diffusion tensor dataset can be modeled as being generated using a generative model given in the following equation: D i = K S k(t i )b ki + ɛ i (3 1) k=1 48

49 where, ɛ i is assumed to be independent identically distributed noise process. In this framework, the parameters of the models and the labels b ki are assumed unknown and must be estimated given a diffusion tensor dataset D i. The goal then is to estimate these unknowns given the tensor field. A Bayesian framework has been popular in literature for solving such problems and an efficient solution for the scalar field segmentation was previously presented by Rivera et al. (2005). Our formulation here significantly extends their formulation to cope with tensor fields. Let v ki be the probability that the diffusion tensor D i was generated by the k-th model, the likelihood of the label field is then given by P D b,c = k i (v ki) b ki (3 2) where c are the control tensors (parameters of the tensor splines). In the case of tensor-valued images we can define the probability v ki as v ki = 1 e dist(d i,s k (t i )) 2 2σ 2 (3 3) 2πσ where dist(.,.) is the Riemannian geodesic distance between two SPD tensors. If we can estimate the marginal probabilities p ki, the label field for a hard segmentation (assigns labels in a yes/no fashion and not with a probability)) can be estimated by the maximum posterior marginals (MPM) estimator (Marroquín et al., 2001) which is defined as 1 if p ki p li, for k l b ki = 0 otherwise By using the above label field estimator the unknown variables of our problem are the parameters of the models and the marginal probabilities p ki. There are two ways to estimate the marginals: a)the mean field approximation (Zhang, 1992) and b) Gauss Markov Measure Field (GMMF) model (Marroquín et al., 2001). The mean field approximation leads to algorithms that are rather slow and sensitive to noise and the 49

50 GMMF approach by Marroquín et al. (2001) leads to significantly different (in the sense of entropy) distributions from the true ones. Rivera et al. (2005) developed a clever technique that controls the entropy of the solution distribution and constraints it to be closer to the true distribution. Using the formulation presented by Rivera et al. (2005), we can efficiently estimate the unknown parameters of the above mentioned GMMF model by minimizing the following energy function E(p c) = k i (p ki 2 ( logv ki µ) +λ s N i (p ki p ks ) 2 ) i γ i(1 k p ki) where to µ controls the entropy of the marginals, λ controls the smoothness of the label field and γ i are Lagrange multipliers that were introduced in order to enforce the condition K p ki = 1, i. (3 4) k The purpose of the entropy control is to bias the posterior marginals estimates toward distributions that have low entropy. Also, log v ki is introduced instead of v ki to make the data term quadratically depend on the model parameters (tensor spline control vertices) c and for the energy function to remain a quadratic positive definite function of the marginal probability v and µ < 2λ. For a detailed discussion on the nuances of this energy function (for scalar fields that are also applicable here), we refer the reader to the article by Rivera et al. (2005). The energy can be minimized using the expectation maximization (EM) procedure or a generalized EM (Neal and Barry, 1998). In this segmentation algorithm the number of labels K is not a hidden variable and is predefined. This number can be set equal to or greater than the number of regions that a neuroanatomist expects to find in a particular dataset. 3.3 Experimental Results In this section, several segmentation experiments with noisy synthetic tensor fields as well as real diffusion tensor data of an isolated rat hippocampus are presented. Validation 50

51 Figure 3-1. A) FA map of a synthetic tensor field similar to the synthetic tensor field in 3-2D but with smoothly varying tensors within each region and different amounts of noise. From left to right the SNR is: 1) no noise, 2) 8.8, 3) 3.8, and 4) 2.8. B) The corresponding segmentation using piece-wise constant representation of the regions, C) using smoothly varying representations. results are also presented to demonstrate the performance of our proposed algorithm for diffusion tensor field segmentation under different amounts of noise in the data. We synthesized several 2D synthetic tensor fields of size 32x32, with different shapes of regions and different anatomy of the diffusion tensor field in each region. All synthetic tensor fields were generated by simulating the diffusion-weighted MR signal (Equation 1 4 by Söderman and Jönsson (1995)) as described earlier in Section 1.5. Figure 3-2B presents a tensor field consisting of two piece-wise constant regions: 1) a small square region in the center of the tensor field and 2) the region forming the rest of the tensor field. The FA is 0.6 in both regions. The principal eigenvector of the diffusion tensors in the first region has horizontal direction and in the second region has vertical direction. The regions of the tensor field in figure 3-2C are the same as in figure 3-2C. However, now the tensor field within the regions is smoothly varying instead of piece-wise constant. The FA is again 0.6 in both regions. The principal eigenvectors of the diffusion tensors in the first region (small square) are tangent to circles with center in the upper left corner of the image. Similarly, the principal eigenvectors of the other region are tangent to circles with 51

52 A B C D Figure 3-2. A)The primary eigenvectors of a smoothly varying synthetic tensor field. B,C,D) Segmentation of different synthetic tensor fields. Figure 3-3. Illustration of segmentation under partial voluming effects. The top row shows averaging kernels at different locations of the diffusion-weighted images. The bottom row depicts the corresponding estimated tensor fields. center in the upper right corner of the image. Finally, figure 3-2D consists of the following regions: 1) a ring with principal eigenvectors tangent to circles centered in the lower left corner of the image, 2) two triangular regions with horizontal principal eigenvectors and 3) two triangular regions with vertical principal eigen vectors. The last two regions have the same FA equal to 0.6, while within the ring the FA is 0.8. We segmented the three tensor fields of figure 3-2 using the proposed entropy controlled segmentation algorithm described in Section 3.2. The parameters of the algorithm that we used are λ = 1 and µ = 0.1. The segmentation results are presented in the background of the vector fields of figure 3-2, in different gray levels for each segmented region. In order to demonstrate the segmentation performance of our proposed algorithm under partial voluming effects we synthesized a 2D field consisting of two rectangular 52

53 regions; the region on the left consists of fibers with orientations parallel to x-axis, and the region on the right is composed of fibers with orientations parallel to y-axis (Figure 3-3). Diffusion-weighted images for this 2D field were obtained by simulating the MR signal, similarly to previous experiments. Then the diffusion-weighted images were averaged using a kernel, which produces a single average measurement per image for every pixels. This process was repeated for different locations of the averaging kernel and each time the corresponding tensor field was estimated from the averaged images (see illustration in Figure 3-3). The estimated tensor fields were then segmented using our proposed algorithm. The segmentation performance defined as the ratio of the correctly classified area over the total area was computed for the pixels on the boundary between the two regions, and found to be For the rest of the pixels (non-boundary pixels) this ratio was 1.0. Finally we synthesized a tensor field similar to the one in figure 3-2D but with a piece-wise smooth variation of tensors within each region and a resolution of , and then by following the above averaging process we filtered the tensor field down to a) and b) After that the two obtained tensor fields were segmented using our algorithm and the segmentation performance found to be 0.92 and 0.96 respectively. As expected, in the case of the higher resolution (64 64) we observed better segmentation performance. In the following experiments we used the real diffusion tensor data set from an isolated rat hippocampus (Shepherd et al., 2006), shown earlier in figure 2-4. Figure 3-4 presents segmentation results on a 2D slice selected arbitrarily from the 3D data set. Figure 3-4A depicts the FA (fractional anisotropy) map segmentation obtained using the scalar field entropy-controlled segmentation algorithm presented by Rivera et al. (2005). Since the FA map does not contain any information about the orientation of diffusion, as expected, its segmentation yields erroneous results when compared to the expert s hand segmentation in figure 3-4D, taken as the ground truth. This example demonstrates that the tensor field segmentation obtained by segmenting a scalar-valued function 53

54 computed from the tensors would not suffice in achieving the desired results. Figure 3-4B depicts the segmentation result using a piece-wise constant model in our tensor field segmentation algorithm described earlier and figure 3-4C depicts the segmentation result using a piece-wise smooth segmentation model. Finally figure 3-4D is a manually labeled segmentation, based on expert knowledge of hippocampal anatomy. In figure 3-4E we compare our results (shown in the background) with the manual segmentation (shown as an overlay). There were some differences between expert manual segmentation (Figure 3-4D) and the automatic segmentations of the rat hippocampus, although automatic segmentation based on the smoothly varying tensor spline model better matched the expert manual segmentation than segmentation based on the piecewise constant model. Expert manual segmentation of the rat hippocampus was based on previous knowledge of hippocampal cytoarchitecture obtained from 2-dimensional visualization of the hippocampus with contrast generated by immunochemistry, various histological staining methods and tracer studies (Shepherd et al., 2006). The differences appreciated and boundaries denoted by these older techniques offer only indirect or inferential information about the orientations of neurons and glia within the hippocampus. Thus, manual segmentation may differ significantly in certain hippocampal regions from automatic segmentation based on the 6-dimensional tensor model of fiber coherences in the rat hippocampus. It must be further considered that study of the hippocampus using these new methods may produce novel insight into the cytoarchitecture of different hippocampal regions not previously known based on previous techniques. Several examples are shown in Figure 3-4C where the automatic segmentation using the smoothly varying tensor spline model recognized significant tensor differences that subdivide hippocampal regions into regions that make sense in the context of to hippocampal connectivity (please see the work by Shepherd et al. (2006) for in depth details), although those regions are usually not distinct in histological stains. In Figure 54

55 A B C D E Figure D segmentation of an isolated rat hippocampus DT-MRI. A) FA map segmentation using algorithm by Rivera et al. (2005). B) tensor field using piece-wise constant models. C) tensor field using a smoothly varying representation of the regions. D) Manually labeled image based on knowledge of hippocampal anatomy. The index of the labels corresponds to: 1) dorsal hippocampal commissure, 2) subiculum, 3) alveus, 4) stratum oriens, 5) stratum radiatum, 6) stratum lacunosum-moleculare, 7) molecular layer, 8) hilus, X) mixture of CA3 stratum pyramidale and stratum lucidum, Y) stratum oriens but ambiguous, 12) fimbria. E) comparison of our results (shown in the background) with the manual segmentation (shown as an overlay). 55

56 3-4C, the molecular layer was further divided into infra- and suprapyramidal blades ( e end i respectively) as well as the crest ( h ) that contains perforant fibers entering the dentate gyrus from the entorrhinal cortex. These regions do not differ in histological staining contrast, but are often divided based on spatial information because they do have different functional connectivities. Automatic segmentation using the smoothly varying tensor spline model also divided the subicular complex labeled 5 and 2 in Figure 3-4D into some of its component parts including presubiculum ( k ), prosubiculum ( j ) and subiculum proper ( l ). The tensor spline model also divided the stratum radiatum into proximal and distal CA1 regions ( c and f respectively) that reflect differences in septo-temporal connectivity between proximal and distal CA1 pyramidal neurons, as well as distinguishing a region of ipsilateral CA3-CA3 associational connections ( a ). In contrast, the segmentation based on the 6-dimensional tensor has difficulty distinguishing regions sometimes recognized by histology stains that are connected by a common fiber pathway (and thus, water diffusion coherence). For example, the perforant pathway continues from the presubiculum ( k ) to the stratum lacunosum-moleculare ( g ) to synapse on the terminal dendrites of pyramidal neurons. The automatic segmentation does not distinguish these areas because they share the same fiber coherence as the perforant fibers. As described above, this perforant pathway also branches into the crest of the molecular layer ( h ), but there is better contrast in the tensor spline model representation of the orthogonally-oriented basal dendrites of granule cells. Another example of this merging phenomena is created by mossy fibers extending through the hilus ( d ) onto the proximal apical dendrites of CA3 pyramidal neurons in the stratum lucidum ( b ). Similar regional mergence based on anatomical connectivity is noted between the prosubiculum ( j ) and the distal CA1 region of stratum radiatum ( f ) which are functionally connected. These interesting observations serve as areas for future research more related to diffusion tensor contrast generation and are not problems inherent to the smoothly varying tensor spline model or automatic segmentation employed herein. 56

57 A B Figure 3-5. A) A view of the 3D segmentation of an isolated rat hippocampus. B) Different views of the molecular layer from the segmentation in A. Finally, we present results of segmenting the whole 3D volume of the isolated rat hippocampus using our segmentation algorithm with the smoothly varying tensor spline representation of the regions. Figure 3-5A presents a 3D view of most of the regions detected by the proposed algorithm. Finally figure 3-5B shows different views of the molecular layer contained in the segmentation result shown in figure 3-5A. To the best of our knowledge this is the first report on automatic segmentation of an isolated hippocapmus. The clinical significance of an algorithm for automatic segmentation of an isolated hippocapmus has already been discussed earlier. 57

58 CHAPTER 4 ESTIMATION OF REGULARIZED 4TH-ORDER TENSOR FIELDS 4.1 Introduction As it was pointed out in Chapters 1, 2 and 3, a 2 nd order tensor has commonly been used to approximate the diffusivity profile at each image lattice point in a DW-MRI (Basser et al., 1994). The approximated diffusivity function is given by d(g) = g T Dg (4 1) where g = [g 1 g 2 g 3 ] T is the magnetic field gradient direction and D is the estimated 2 nd -order diffusion tensor. However, it is known that 2 nd -order polynomials fail to approximate more complex fiber structures such as fiber crossings (Basser et al., 2000). In order to overcome the above limitation, higher order tensors were introduced by Ozarslan and Mareci (2003) to represent more complex diffusivity profiles which better approximate the local diffusivity function. Generalized scalar quantities such as the variance of diffusivity and the generalized anisotropy were derived as functions of the higher order tensor coefficients (Evren Ozarslan, 2005). However, in all of these works the higher-order tensors are estimated without imposing the positivity of the diffusivity function approximation, which is significantly important since negative diffusivity values are non physical. In this chapter we propose a novel parametrization of the 4 th -order tensors for imposing the positivity of the estimated diffusivity. In this parametrization we express the 6 6 matrix G = CC T using its Iwasawa coordinates (Iwasawa, 1949). By combining the Iwasawa formulation with the theorem on ternary quartics by Hilbert (1888), we are interested in estimating Iwasawa decomposable 6 6 symmetric positive semi-definite matrices of at most rank-3. We derive formulas for uniquely computing the tensor coefficients as functions of the parameters of our model. The unknown parameters are 58

59 estimated from a given DW-MRI data set by minimizing an energy function in which we add a regularization term for estimating smooth 4 th -order tensor fields. The key contribution inf this chapter is the use of a parametrization which can decompose any 6 6 symmetric positive semi-definite matrix of at most rank-3. This parametrization has two main advantages: a) it minimizes the solution space by overcoming the non-uniqueness problems studied by Powers and Reznick (2000) and b) allows the simultaneous estimation and regularization of the 4 th -order tensor field. The motivation for using an estimated positive higher-order tensor model is that one can derive the generalized higher-order extensions of the commonly used 2 nd -order scalar measures for monitoring various brain diseases. In our experimental results we show that such scalar measures can be more accurately estimated by using the proposed framework compared to other methods. We show that our model is robust to noise in estimating fiber orientations as well as scalar measures derived from the 4 th -order tensor coefficients such as the generalized trace defined by Evren Ozarslan (2005). This demonstrates that the proposed framework is a directly applicable tool that extends efficiently the clinically used measures by overcoming the limitations of the current models in literature. The rest of the chapter is organized as follows: In Section 4.2, we present a novel parametrization of the 4 th -order tensors that is used to enforce the positivity of the estimated tensors using the Iwasawa decomposition of the Gram matrix. In Section 4.2.1, we present a functional minimization method to estimate 4 th -order tensors from diffusion-weighted MR images. Furthermore, in Section we propose a distance measure for the space of 4 th -order tensors, and employ it for estimating a smoothly varying tensor field as well as estimating the generalized variance of the tensors. Section 4.3 contains the experimental results and comparisons with other methods using simulated diffusion MRI data and real MR database of excised rat spinal cords. 59

60 4.2 Diffusion Tensors of 4th Order Studies (for instance Özarslan et al. (2006)) have shown that 2nd -order tensors fail to model complex local structures of the diffusivity in real tissues and higher-order tensors or other methods for multi-fiber approximation must instead be employed (see discussion in Section 1.2.4). In the case of 4 th -order tensors, the diffusivity function can be expressed using the standard notation of quartic forms (also employed by Powers and Reznick (2000)) as follows: d(g) = D i,j,k g1g i 2g j 3 k (4 2) i+j+k=4 where g = [g 1 g 2 g 3 ] T is the magnetic field gradient direction. It should be noted that in the case of 4 th -order symmetric tensors there are 15 unique coefficients D i,j,k, while in the case of 2 nd -order tensors we only have 6. In DW-MRI the diffusivity of the water is a positive quantity. This property is essential since negative diffusion coefficients are non-physical. However, in the parametrization of Equation 4 2 there is no guarantee that the estimated coefficients D i,j,k form a positive tensor. Therefore, there is need for the development of a new parametrization of the 4 th -order tensor, which guarantees the positivity of the estimated tensor. Letting g 1, g 2 and g 3 in 4 2 be the variables, the equivalence between symmetric tensors and homogeneous polynomials is straightforward to establish. Moreover, if a symmetric tensor is positive, then its corresponding polynomial must be positive for all real-valued variables. Hence, here we are concerned with the positive definiteness of homogenous polynomials of degree 4 in 3 variables, or the so called ternary quartics. Hilbert in 1888 proved the following theorem on ternary quartics (see Rudin (2000), and Powers et al. (2004) for a modern exposition) that we will employ in this work: Theorem 1. There exists an identity d = p p p 3 3 in which d is a real ternary form d = d(g 1, g 2, g 3 ) of degree four which is positive semi-definite and the p i are quadratic forms with real coefficients. 60

61 By using this theorem, Equation 4 2 can be expressed as a sum of 3 squares of quadratic forms as d(g) = (v T c 1 ) 2 + (v T c 2 ) 2 + (v T c 3 ) 2 = v T CC T v = v T Gv (4 3) where, v is a properly chosen vector of monomials, (e.g. [g1 2 g2 2 g3 2 g 1 g 2 g 1 g 3 g 2 g 3 ] T ), C = [c 1 c 2 c 3 ] is a 6 N matrix by stacking the 6 coefficient vectors c i and G = CC T is the so called Gram matrix. The Gram matrix G in Equation 4 3 is positive semi-definite and has at most rank 3. Here we should emphasize that G being semi-definite does not imply that the 4 th -order tensor d is also semi-definite, since the vector space of the former is 6-dimensional (i.e. vectors v) while the corresponding vector space of the latter is 3-dimensional (i.e. vectors g). In fact, using the parametrization of Equation 4 3 the whole space of the strictly positive definite ternary quartics as well as those which are semi-definite is spanned. Given a Gram matrix, the CC T parametrization in Equation 4 3 is not unique, i.e. there exist different matrices C which parameterize the same Gram matrix. For example C parameterizes the same Gram matrix as its antipodal pair C. Furthermore, there are infinitely many orthogonal matrices that yield the same G when appropriately applied in the aforementioned parametrization. This is due to the orthogonality property (OO T = I) of the rotation matrices O, where I is the identity matrix. Thus, in the case that C is of size 6 3, for any 3 3 orthogonal matrix O we have (CO)(CO) T = CC T. This produces an infinite solution space, which theoretically can not be handled by the optimization techniques. We should also note that in Equation 4 3 there are in total 18 parameters in C, which are 3 more coefficients than the number of unique tensor coefficients in Equation 4 2. The non uniqueness issues of the Gram matrix were also discussed by Powers and Reznick (2000), investigating how many fundamentally differnent Gram matrices parametrize the same ternary quartic. 61

62 In order to overcome the above issues, we use the Iwasawa decomposition which represents the components of a positive definite or semi-definite matrix in the Iwasawa coordinates (Iwasawa, 1949; Jian and Vemuri, 2007a). Every n n positive definite matrix G can be uniquely expressed using its Iwasawa components as follows. G = W 0 0 V I k X 0 I m (4 4) where W and V are SPD matrices of size k k and m m respectively, X R k m and A[B] denotes B T AB. In the case of n n positive semi-definite matrices of at most rank-k, the Iwasawa coordinates are also given by Equation 4 4 by setting rank(w) + rank(v) k. By computing the matrix multiplications in Equation 4 4 we derive the following parametrization of positive semi-definite matrices G = W WX X T W X T WX + V (4 5) where W and V are 3 3 positive definite or positive semi-definite matrices and their ranks sum up to k. Furthermore, by using the Cholesky decomposition of W = AA T, where A is a lower triangular matrix with non-negative diagonal elements, we can establish equivalence between the parametrization in Equation 4 5 and that in Equation 4 3 by defining B to be the matrix that satisfies the equation BA T = X T W. By using the above we derive the following parametrization for the Gram matrix G = W WX X T W X T WX + V = AAT BA T AB T BB T R where A = R R 0 0 and B R3 3. R R R 0 = A B A B T (4 6) 62

63 If W is positive definite, then the Cholesky factor A is unique, B = X T A is also unique, V = 0 and therefore, the parametrization in Equation 4 6 is unique. Note that in Equation 4 6 there are in total 15 parameters, 6 in A and 9 in B, which is equal to the number of unique tensor coefficients in Equation 4 2. According to Theorem 1, Equation 4 6 spans the whole space of positive semi-definite ternary quartics but can be used to define a parametrization for the case of strictly positive definite 4 th -order ternary quartics as follows: Lemma 1. For every real positive definite ternary quartic d there exists an arbitrarily small positive real number c such that d can be written as the sum of the ternary quartic c(x 2 + y 2 + z 2 ) 2 and three squares of quadratic forms. Proof. Let d(g) be a strictly positive definite ternary quartic. Therefore, d(g) > 0 g S 2, where, S 2 is the unit sphere, i.e. the space of unit vectors g. We define the following ternary quartic c(g1 2 + g2 2 + g3) 2 2, where c is any real number in the interval 0 < c min g S2 d(g). Since c > 0 this ternary quartic is also positive definite. Let us now define f(g) = d(g) c(g1 2 + g2 2 + g3) 2 2. Since min g S2 f(g) = min g S2 d(g) c 0, f(g) is a positive semi-definite ternary quartic and therefore can be expressed as a sum of three squares of quadratic forms (by Theorem 1). Thus, every positive definite ternary quartic d(g) can be written as d(g) = f(g)+c(g1 2 +g2 2 +g3) 2 2 where c is an arbitrarily small positive real number, which proves the lemma. The corresponding diffusivity function can be expressed using the Iwasawa coordinates as follows: d(g) = v T Gv = v T W WX X T W X T WX + V = v T A B A B T v + v T + C C v v (4 7) 63

64 where C is a 3 3 matrix whose elements equal to the same arbitrarily small positive real number c, v is a properly chosen vector of monomials, (e.g. [g1 2 g2 2 g3 2 g 1 g 2 g 1 g 3 g 2 g 3 ] T ), W, V and X are defined as in Equation 4 5, and A and B are defined as in Equation 4 6. Here we should note that in practice due to finite precision computations c can be set to the smallest possible value of a finite precision machine and therefore, is considered as a known variable. In the double-precision floating-point IEEE standard the smallest positive value is approximately c = Furthermore, we should emphasize that the semi-definite property of G does not necessarily imply semi-definiteness of d(g) since the former is semi-definite in R 6 while the latter may be semi-definite in R 3. In particular, using the proposed parametrization (Equation 4 7), we compute semi-definite matrices G that correspond to strictly positive-definite d(g). Given a Gram matrix, which in our case is parameterized using the matrices A and B in Equation 4 7, we can uniquely compute the tensor coefficients D i,j,k by using the formulas in table 4-1. In these formulas the tensor coefficients are expressed as functions of the components of A, B and the fixed parameter c. Therefore, we can employ the parametrization in Equation 4 7 for the estimation of the coefficients D i,j,k of the diffusion tensor from MR images using the following two steps: 1) first estimate the A and B matrices from the given images by using a functional minimization method (minimization of Equation 4 8 using the Lavenberg-Marquardt nonlinear least-squares method), and then 2) compute the unique coefficients D i,j,k of the 4 th -order tensor by using formulas in table 4-1. In the following section we will employ this method to enforce the positive definite property of the estimated fourth order diffusion tensors from the diffusion weighted MR images. 64

65 Table 4-1. Formulas to compute the tensor coefficients D i,j,k given the components a 11, a 22, a 33, a 21, a 31, a 32, b 11, b 12, b 13, b 21, b 22, b 23, b 31, b 32, b 33 and the fixed parameter c = D 400 = a c D 040 = a a c D 004 = a a a c D 220 = b b b a 11 a c D 202 = b b b a 11 a c D 022 = b b b a 21 a a 22 a c D 310 = 2 a 11 b 11 D 301 = 2 a 11 b 21 D 130 = 2(a 21 b 11 + a 22 b 12 ) D 031 = 2(a 21 b 31 + a 22 b 32 ) D 103 = 2(a 31 b 21 + a 32 b 22 + a 33 b 23 ) D 013 = 2(a 31 b 31 + a 32 b 32 + a 33 b 33 ) D 211 = 2(b 11 b 21 + b 12 b 22 + b 13 b 23 + a 11 b 31 ) D 121 = 2(b 11 b 31 + b 12 b 32 + b 13 b 33 + a 21 b 21 + a 22 b 22 ) D 112 = 2(b 21 b 31 + b 22 b 32 + b 23 b 33 + a 31 b 11 + a 32 b 12 + a 33 b 13 ) Estimation From DWI The coefficients D i,j,k of a 4 th order diffusion tensor can be estimated from diffusion-weighted MR images by minimizing the following cost function: E = M (S i S 0 e b ivi T Gv i ) 2 (4 8) i=1 where G is the Iwasawa parameterized Gram matrix given by Equation 4 7, M is the number of the diffusion weighted images associated with gradient vectors g i and b-values b i ; S i is the corresponding acquired signal and S 0 is the zero gradient signal. Using the magnetic field gradient directions g i we construct the 6-dimensional vectors v i = [gi1 2 gi2 2 gi3 2 g i1 g i2 g i1 g i3 g i2 g i3 ] T. In Equation 4 8, the 4 th order diffusion tensor is parameterized using the 3 3 matrices A and B which form together the Gram matrix G. Having estimated the matrices A and B that minimize Equation 4 8, the coefficients D i,j,k can be computed directly using the formulas in Table 4-1. S 0 can either be assumed to be known or estimated simultaneously with the coefficients D i,j,k by minimizing Equation

66 In the implementation of the proposed method, in order to enforce the diagonal elements a 1,1, a 2,2 and a 3,3 of matrix A to be non negative, we should use a mapping of R to the space of non-negative real numbers. In order for this mapping to be unique (one to one), the target space must be open, hence we seek a mapping to the positive part of R. This does not limit the solution space in our implementation, since it has been shown that in finite precision arithmetic, open spaces are equivalent to closed spaces (Wang et al., 2004). In our experiments we used the exponential mapping a 1,1 = exp(ã 1,1 ), a 2,2 = exp(ã 2,2 ) and a 3,3 = exp(ã 3,3 ) and therefore, in the minimization we solve for ã 1,1, ã 2,2 and ã 3,3 instead of a 1,1, a 2,2 and a 3,3. The total number of unknown parameters in G in Equation 4 8 are 15 and they are the following: ã 1,1, ã 2,2, ã 3,3, a 2,1, a 3,1, a 3,2 and the 9 elements of matrix B. Starting with an initial guess for S 0, A and B we can use any gradient based optimization method in order to minimize Equation 4 8. We should note here that the exponent in Equation 4 8 is in the form of a polynomial and therefore its gradient with respect to the unknown coefficients is easily derived analytically. Given A and B at each iteration of the optimization algorithm we can update S 0 by again minimizing Equation 4 8. The derivative of this equation with respect to the unknown S 0 is M S0 E = 2 (S i S 0 e b ivi T Gv i )e b ivi T Gv i (4 9) i=1 By setting Equation 4 9 equal to zero, we derive the following update formula for S 0 S 0 = M M S i e b ivi T Gv i / e 2b ivi T Gv i (4 10) i=1 i=1 In our experiments we used the well known Lavenberg-Marquardt (LM) nonlinear least-squares method, which has advantages over other optimization methods, in terms of stability and computational burden. The average execution time of our implementation on an AMD Athlon 2GHz was N M seconds where N is the number of voxels 66

67 per image and M is the number of the acquired DW-MR images used in the estimation of the 4 th -order tensor field Distance Measure In Section we discussed how to estimate the positive-definite 4 th -order tensors from DW-MRI data by minimizing Equation 4 8. The 4 th -order tensors are by definition (Equation 4 2) smoothly varying functions (within a voxel). In order to impose smoothness across the image lattice we can add to the energy function the following regularization term j i N j dist( D j, D i ) 2 (4 11) where N j is the set of lattice points in the neighborhood of j. In the regularization term defined in Equation 4 11 we need to employ an appropriate distance measure between the tensors D i and D j. Here we use the notation D to denote the set of 15 unique coefficients D i,j,k of a 4 th -order tensor. We can define a distance measure between the 4 th -order diffusion tensors D 1 and D 2 by computing the normalized L 2 distance between the corresponding diffusivity functions d 1 (g) and d 2 (g) leading to the equation, dist( D 1, D 2 ) 2 = 1 [d 1 (g) d 2 (g)] 2 dg (4 12) 4π S 2 67

68 = [( 4,0,0 + 0,4,0 + 0,0,4 + 2,2,0 + 0,2,2 + 2,0,2 ) 2 + 4[( 4,0,0 + 2,2,0 ) 2 + ( 4,0,0 + 2,0,2 ) 2 + ( 0,4,0 + 2,2,0 ) 2 + ( 0,4,0 + 0,2,2 ) 2 + ( 0,0,4 + 0,2,2 ) 2 + ( 0,0,4 + 2,0,2 ) 2 ] + 24( 2 4,0, ,4, ,0,4) 6( 2 2,2, ,2, ,0,2) + 2( 4,0,0 + 0,4,0 + 0,0,4 ) 2 + ( 2,1,1 + 0,3,1 + 0,1,3 ) 2 + ( 1,2,1 + 3,0,1 + 1,0,3 ) 2 + ( 1,1,2 + 3,1,0 + 1,3,0 ) [ ( 3,1,0 + 1,3,0 ) 2 + ( 3,0,1 + 1,0,3 ) 2 + ( 0,3,1 + 0,1,3 ) 2] + 2( 2 3,1, ,0, ,3, ,3, ,0, ,1,3)] where, the integral of Equation 4 12 is over all unit vectors g, i.e., the unit sphere S 2 and the coefficients i,j,k are computed by subtracting the coefficients D i,j,k of the tensor D 1 from the corresponding coefficients of the tensor D 2. As shown above, the integral of Equation 4 12 can be computed analytically and the result can be expressed as a sum of squares of the terms i,j,k. In this simple form, this distance measure between 4 th -order tensors can be implemented very efficiently. Note that this distance measure is invariant to rotations in 3-dimensional space because it is the L2 norm, which is well known for its invariance with respect to rigid motions. Furthermore, by using the formulas from Table 4-1 we can write the above distance as a function of the elements of the estimated matrices A and B. We should note that the obtained regularization term is in the form of polynomial and therefore its gradient with respect to the unknowns A and B of Equation 4 8 can be also computed analytically. Another property of the above distance measure is that the average element (mean tensor) ˆD of a set of N tensors D i, i = 1... N is defined as the Euclidean average of the corresponding D i,j,k coefficients of the tensors. This property can be proved by verifying that ˆD minimizes the sum of squares of distances dist( ˆD, D i ) 2. Similarly, it can be shown that geodesics (shortest paths) between 4 th -order tensors are defined as Euclidean geodesics. 68

69 Finally, the above distance measure can be used to compute the intra-voxel variance (defined by Evren Ozarslan (2005)) of a single displacement probability function parameterized as a 4 th -order tensor D. The variance is given by (dist( V = 1 D, 0) 2 ) 9 < D > 2 1 (4 13) where < D > is the generalized trace given by < D >= (D 4,0,0 + D 0,4,0 + D 0,0,4 + (D 2,2,0 + D 2,0,2 + D 0,2,2 )/3)/5. This variance has been used to define the generalized anisotropy in the article by Evren Ozarslan (2005) and we use it in our experimental results as well. 4.3 Experimental Results In this section we present experimental results on our method applied to simulated DW-MRI data as well as real DW-MRI data from excised rat s spinal cords Synthetic Data Experiments In order to motivate the need for the positive-definite constraint in the 4 th -order estimation process, we performed the following experiment using a synthetic data set. The synthetic data was generated by simulating the MR signal from a highly anisotropic single fiber (fractional anisotropy> 0.9) using the realistic diffusion MR simulation model in Section 1.5 by Söderman and Jönsson (1995) (b-value= 1250s/mm 2, 21 gradient directions). Then, we added different amounts of Riccian noise to the simulated data set and estimated the 4 th -order tensors from the noisy data by: a) minimizing M i=1 (S i S 0 exp( b i d(g i ))) 2 without using the proposed parametrization to enforce the SPD constraint, by employing the method employed by Özarslan et al. (2004) and b) our method, which guarantees the SPD property of the tensors. (S i is the MR signal of the i th image and S 0 is the zero-gradient signal). Studies on estimating fiber orientations from the diffusivity profile have shown that the peaks of the diffusivity profile do not necessarily yield the orientations of the distinct fiber bundles (Özarslan et al., 2006). One should instead employ the displacement probability functions given by Equation 1 2 for computing the fiber orientations. In our 69

70 Figure 4-1. Comparison of the fiber orientation errors for different amounts of noise in the data, obtained by using: a) our parametrization to enforce positivity and b) without enforcing positivity of the estimated tensors. experiments, we estimated the displacement probability profiles from the 4 th -order tensors using the method that is presented in Chapter 6. Then, we computed the displacement probability functions from the 4 th -order tensors estimated earlier using the two different methods, and after that we computed the fiber orientations from the maxima of these probability functions. The error angles (mean and standard deviation) between ground truth and estimated orientations from the two methods for different amount of noise in the data are plotted in Figure 4-1. Here, we should emphasize that the intensity of the simulated signal was significantly smaller when the diffusion gradient orientation formed a small angle with the fiber orientation than that corresponding to gradient orientations perpendicular to the fiber. These smaller intensity values are most likely to become negative when we corrupt the signal with noise, and as a result any method that does not impose the SPD property is affected drastically. As expected, our method yields smaller errors in comparison with the method that does not enforce the SPD property of the tensors. When we increase the amount of noise in the data, the errors observed by the latter method are significantly increased, while our method depicts less sensitivity. This demonstrates the need for enforcing the SPD property of the estimated tensors and validates the accuracy of our proposed method. 70

71 Figure 4-2. Comparison of the standard deviation of the generalized trace defined by Evren Ozarslan (2005), obtained by using: a) our parametrization that enforces positivity and b) without enforcing positivity of the estimated tensors. As was expected, similar results were observed by comparing the generalized trace (defined by Evren Ozarslan (2005)) of the 4 th -order diffusion tensors estimated in the previous experiment. In the case of estimating the 4 th -order tensors without imposing the SPD property, 44% of the estimated tensors yielded negative generalized trace values, while the values computed using the proposed method were all positive. In both cases we computed the standard deviation of the estimated generalized trace for different amounts of noise in the data (excluding from our calculations the negative values computed by the non SPD method). Figure 4-2 shows the standard deviation of the generalized trace computed using both methods. By observing the figure we conclude that the proposed method produces significantly more accurate results even in the higher noise cases. Finally, in order to compare our proposed method with other existing techniques that do not employ 4 th -order tensors, we performed another experiment using synthetic data. The data were generated for different amounts of noise by following the same method as previously using the simulated MR signal of a 2-fiber crossing. We estimated SPD 4 th -order tensors from the corrupted simulated MR signal using our method and then computed the fiber orientations from the corresponding probability functions. For 71

72 Figure 4-3. Fiber orientation errors for different SNR in the data using our method for the estimation of positive 4 th -order tensors and two other existing methods: 1) DOT and 2) ODF. comparison we also estimated the fiber orientations using the DOT method described by Özarslan et al. (2006) and the ODF method presented by Tuch (2004). For all three methods we computed the estimated fiber orientation errors for different amounts of noise in the data (shown in Figure 4.3.1). The results conclusively demonstrate the accuracy of our method, showing small fiber orientation errors ( 5 o ) for typical amount of noise with std. dev Furthermore, by observing the plot, we also conclude that the accuracy of our proposed method is very close to that of the DOT method and is significantly better than the ODF method. For larger amounts of noise our method yielded smaller errors than all the other methods Real Data Experiments In the following experiments, we used MR data from excised rat s spinal cord. The protocol that we used in this experiment included acquisition of 22 images using a pulsed gradient spin echo pulse sequence with repetition time (TR) = 1.5 s, echo time (TE) = 27.2 ms, bandwidth = 30 khz, field-of-view (FOV) = mm. After the first image set was collected without diffusion weighting (b 0 s/mm 2 ), 21 diffusion-weighted image sets with gradient strength (G) = 664 mt/m, gradient duration (δ) = 1.5 ms, gradient separation ( ) = 17.5 ms and diffusion time (T δ ) = 17 ms were collected. The image 72

73 Figure 4-4. The elements of matrices A and B estimated by the proposed method as well the estimated S 0 and generalized anisotropy defined by Evren Ozarslan (2005). without diffusion weighting had 8 signal averages, and each diffusion-weighted image had 2 averages. First, we estimated an SPD 4 th -order tensor field by applying the proposed method to the real DW-MRI data. Figure 4-4 shows the estimated elements of matrices A and B, which parameterize the Gram matrix discussed in Section 4.2. In the first row the estimated S 0 and generalized anisotropy defined by Evren Ozarslan (2005) are also shown. The red color in the images denotes negative values. By observing the images we can see that each coefficient shows different details of the underlying tissue. 73

74 Figure th -order tensors estimated A) without imposing the SPD property and B) by using the proposed method. On the top row the corresponding estimated S 0 images are shown colored by mapping the X, Y, Z coordinates of the largest diffusivity orientation to the R, G, B color components. In this region of interest we expected single-lobed diffusivities with peaks predominantly in the axial direction (shown in blue). Furthermore, we computed the 4 th -order tensor field first without imposing the positive-definite constraint. In order to compare the obtained results with that estimated by the proposed algorithm, in Figure 4-5 we plot the corresponding tensors from a region of interest in the white matter. In this region of interest we expected single-lobed diffusivities with peaks predominantly in the axial direction which is represented by the blue color. The X, Y, Z components of the dominant orientation of each probability profile are assigned to R, G, B (red, green, blue) components of the color of each tensor. By observing this figure, we can say that the tensor field is incoherent if we do not enforce the SPD constraint (Figure 4-5 left) and the expected single-lobed nature of this white 74

75 Figure 4-6. Visualization of the 4 th -order tensor field estimated by applying proposed method to a real DW-MRI dataset from an excised rat s spinal cord. matter region is lost. On the other hand the tensors obtained by our method (Figure 4-5 right) are more coherent. Note that this is a result of enforcing the SPD constraint, since in this experiment we did not use any regularization. Similarily to the simulated data examples (Section 4.3.1), the ROI in Figure 4-5 corresponds to highly anisotropic fibers in the white matter of the spinal cord, which are most likely to yield negative diffusivities when the SPD property is not imposed. This demonstrates the superior performance of our algorithm and motivates the use of the proposed SPD constraint. Figure 4-6 depicts a visualization of the tensor field estimated by the proposed technique. In this figure multi-lobed diffusivity profiles and other complex structures can be observed. Finally, a zoomed in region of interest shows smooth transitions from single-lobed diffusivities to multi-lobed structures in the estimated 4 th -order tensor field. 75

76 CHAPTER 5 EXPONENTIAL TENSORS: A FRAMEWORK FOR EFFICIENT HIGHER-ORDER DT-MRI COMPUTATIONS 5.1 Introduction In the Chapter 4 we parametrized 4 th -order diffusion tensors using the Iwasawa decomposition of positive definite matrices and Hilbert s theorem on positive ternary quartics. This method guarantees the estimation of positive definite 4 th -order diffusivity functions. The estimated coefficients of these higher-order tensors are directly related to usefully physical quantities such as partial derivatives of the diffusivity. Furthermore, the generalized trace of the tensors corresponds to the mean diffusivity value, which has been used in clinical applications (see discussion in Section 1.2.4). However, it is not trivial to extend this parametrization to orders higher than 4. To date, none of the reported methods in literature for the estimation of the coefficients of higher order tensors (e.g. 6 th, 8 th ) preserve the positive definiteness of the diffusivity function. In this chapter we present a novel parameterization of the diffusivity function that guarantees the positive definite property using the scalar exponential of a higher-order homogeneous polynomial. We present an efficient framework for computing distances and geodesics in the space of the coefficients of our proposed diffusivity function. The key contribution of this chapter is that we employ this framework for estimating higher (4 th, 6 th and 8 nd ) order diffusivity approximations from diffusion-weighted MR images. Note that we are only interested for symmetric tensors and therefore we consider only even orders. We compare our method with other existing DTI methods showing high efficiency of the proposed method. Finally, we validate our framework using real diffusion-weighted MR data from excised, perfusion-fixed rat optic chiasm. 5.2 Exponential Diffusion Tensors We define an Exponential Diffusion Tensor (EDT) of order 2 as a 3 3 symmetric matrix E, which will be used in the following diffusivity function 76

77 d(g E) = e gt Eg (5 1) The EDT matrix E is not necessarily an SPD matrix since the diffusivity function 5 1 is positive for any symmetric matrix. For example, in the case that E is the 3 3 zero matrix, we have d(g) = e 0 = 1 g. If we use the standard diffusivity function g T Dg (Equation 4 1), the previous example corresponds to the diffusion tensor D = I. In this case we have g T Dg = 1 g. In Equation 5 1 the diffusivity function was defined by using a 2 nd order exponential tensor E. This function d(g E) can be generalized by using higher order tensors. In the case of a 4 th order symmetric tensor we have 15 unique coefficients collected into a vector E =(E 4,0,0, E 0,4,0, E 0,4,0, E 2,2,0, E 0,2,2, E 2,0,2, E 2,1,1, E 1,2,1, E 1,1,2, E 3,1,0, E 3,0,1, E 1,3,0, E 0,3,1, E 1,0,3, E 0,1,3 ). In the case of higher order tensors we will use the notation E p1,p 2,p 3 to indicate that it is the coefficient of the term g 1 p 1 g 2 p 2 g 3 p 3. By using this notation Equation 5 1 can be generalized as ( N ) d(g E) p = exp g 1i p 1 g 2i p 2 g 3i 3 E p1i,p 2i,p 3i i=1 (5 2) where in the case of 2 nd, 4 th, 6 th and 8 th order N = 6, 15, 28 and 45 respectively Distance Measure We can define a distance measure between same order EDTs E 1 and E 2 by computing the normalized L-2 distance of the corresponding diffusivity functions d(g E 1 ) and d(g E 2 ), given by dist( E 1, E 2 ) 2 = 1 dg [d(g E1 ) d(g E 2 )] 2 dg where the integration is over the unit sphere (i.e. for all unit vectors g). As an example, the distance between the 2 nd order EDT matrices E 1 = 0 and E 2 = (lim x x)i is dist(e 1, E 2 ) 2 = 1. This is true, since d(g E 1 ) = 1 and d(g E 2 ) = 0 g. However, we need to define a metric that assigns infinite distance between the purely isotropic d(g E 1 ) and the degenerate case d(g E 2 ). Here we use the term degenerate in order to highlight the correspondence 77

78 between d(g E 2 ) and the standard diffusivity function g T Dg, where D = 0. A distance measure that satisfies this property is given by the following equation dist( E 1, E 2 ) 2 = 1 4π [log(d(g E 1 )) log(d(g E 2 ))] 2 dg (5 3) By analytically computing the integral, Equation 5 3 can be written in the from of sum of squares, which is very fast to compute. As an example in the case of 2 nd order EDTs Equation 5 3 can be evaluated using 8 additions and 10 multiplications, in the 4 th order case using 47 add. and 33 mul.. The analytic expression for the distance in the 4 th -order case is given in Equation 4 13, which has been also used in the experiments presented in Chapter 4. Note that the metric defined above is rotation invariant in the case of any order exponential tensors. Furthermore, by using this distance measure it is easy to prove that the mean element E µ is defined as the Euclidean average ( E E N )/N (or geometric mean of d 1,...,d N ) and the geodesic (shortest path) between two elements E 1 and E 2 is defined as Euclidean geodesic γ(t) = (1 t) E 1 + te 2, t [0, 1]. In the following section we employ this distance measure to define an anisotropy map of 2, 4, 6 & 8 th -order EDTs Distance from the Closest Isotropic Case In the isotropic case the quantity d(g E) is the same constant for every unit vector g, forming an isotropic sphere c = log(d(g)) = c(g g g 2 3 ) (K/2), where c R and K denotes the order of the symmetric tensor E and is even. The above equation is satisfied by: 2 nd order exponential diffusion tensors of the form E = ci, 4 th order EDTs of the form E 4,0,0 = E 0,4,0 = E 0,0,4 = c, E 2,2,0 = E 0,2,2 = E 2,0,2 = 2c, 6 th order EDTs of the form E 6,0,0 = E 0,6,0 = E 0,0,6 = c, E 4,2,0 = E 4,0,2 = E 2,4,0 = E 0,4,2 = E 2,0,4 = E 0,2,4 = 3c, E 2,2,2 = 6c, and 8 th order EDTs of the form E 8,0,0 = E 0,8,0 = E 0,0,8 = c, E 6,2,0 = E 6,0,2 = E 2,6,0 = E 0,6,2 = E 2,0,6 = E 0,2,6 = 4c, E 4,4,0 = E 0,4,4 = E 4,0,4 = 6c, E 4,2,2 = E 2,4,2 = E 2,2,4 = 12c, where c is a scalar and the rest of the elements of E are equal to zero. 78

79 Figure 5-1. Comparison between FA and 4 th -order f iso map using rat optic chiasm data. A) FA. B) f iso. C) f iso mapped to [0,1] using the fitted function shown in D. D) plot of f iso vs. FA. Given an arbitrary K th -order E, we can compute the closest isotropy tensor coefficients E iso by finding the scalar c that minimizes the distance of E from the isotropic case. In the 2 nd -order case c = (E 1,1 + E 2,2 + E 3,3 )/3, in the 4 th -order case c = (E 4,0,0 + E 0,4,0 + E 0,0,4 + E 2,2,0 + E 0,2,2 + E 2,0,2 )/9, in the 6 th -order case c = (E 6,0,0 + E 0,6,0 + E 0,0,6 + E 4,2,0 + E 4,0,2 + E 2,4,0 + E 0,4,2 + E 2,0,4 + E 0,2,4 + E 2,2,2 )/27 and in the 8 th -order case c = (E 8,0,0 + E 0,8,0 + E 0,0,8 + E 6,2,0 + E 6,0,2 + E 2,6,0 + E 0,6,2 + E 2,0,6 + E 0,2,6 + E 4,4,0 + E 0,4,4 + E 4,0,4 + E 4,2,2 + E 2,4,2 + E 2,2,4 )/81. The function f iso ( E) = dist( E, E iso ) maps the space of E to the space of non-negative real numbers. The smaller the value of the function, the closer is E to the E iso. The behavior of the function f iso is similar to that of the well-known fractional anisotropy (FA) map of 2 nd order diffusion tensors. To illustrate this, we estimated the DT field and the EDT field from a real dataset, and then we computed the FA and the f iso map respectively (Figure 5-1A and 5-1B). Furthermore, we plot the f iso as a function of FA in Figure 5-1D. The same figure also contains the plot (in red) of the fitted function ( c)log(1 F A) for an estimated c = The inverse of the above function is 1 exp( (1/c)f iso ( E)) and can be used to map f iso to values in the interval [0,1] as was done in Figure 5-1C. 79

80 5.2.3 Estimation of EDT Field from DWI The coefficients of any order exponential diffusion tensor can be estimated from diffusion weighted images (DWI) by minimizing the function E( E, M S 0 ) = (S i S 0 e b id(g i E) ) 2, where M is the number of diffusion weighted images associated with gradient vectors g i and b-values b i, S i is the corresponding acquired signal and S 0 is the zero gradient signal. S 0 can either be assumed to be known or estimated simultaneously with the coefficients of E. In our experiments we minimized the above equation using simple gradient descent algorithm, however any non-linear functional minimization method can be used. One can use also additional regularization terms in the above function in order to enforce smoothness across the lattice Displacement Probability Profile According to the discussion in Section 4.3.1, the peaks of the diffusivity profile does not necessarily yield the orientations of the distinct fiber populations. One should instead employ the displacement probability P (R) which is given by the Fourier integral in Equation 1 2. In the case of 2 nd order exponential diffusion tensors the peak of the diffusivity profile coincides with the peak of the displacement probability. Therefore the fiber orientation can be estimated by computing the eigenvector associated with the largest eigenvalue of matrix E. In higher order case, instead of finding the maxima of P (R) we N can compute the maxima of the expression: E(qq i ) exp( 2πiqq i R) 4πq 2 dq, which approximates the reciprocal space using the icosahedral tessellation of the unit hemisphere, where q is a scalar and q i are unit vectors corresponding to the tessellation. In the case of third-order tessellation we have N = 81, which is the approximation that we used in our experiments. By computing the integral analytically in the above i=1 i=1 approximation we have P (R) π 4N N i=1 ( exp α ) ( 2 4β β α ) 5/2 β 3/2 (5 4) 80

81 Table 5-1. Comparison of DTI frameworks Frameworks: Affine invar. Log-Euc. EDT framework Distance map 0.19 sec 0.45 sec 0.04 sec Smoothing 5.65 sec 0.64 sec 0.10 sec Table 5-2. Properties of DTI frameworks Properties / Frameworks Affine inv. Log-Euc. EDT Affine. Invariance X Rotation Invariance X X X Fast DTI processing X X Unconstrained estimation X Use of higher order tensors X where α = (2πq i R) 2, β = 4πtd(q i, E) and t is the effective diffusion time. 5.3 Experimental Results and Discussion In this section we present experimental results using synthetic and real data. All the synthetic data were generated by simulating the MR signal from single fibers or fiber crossings using the realistic diffusion MR simulation model in Equation 1 4 by Söderman and Jönsson (1995). In order to compare the time performance of the proposed framework with other existing frameworks (affine invariant by Pennec et al. (2005), and Log-Euclidean by Arsigny et al. (2005)) for processing tensor fields we synthesized a single row of a 2 nd -order DTI and EDT field of size and then we applied two simple calculations on every pair of tensors: a)computing their distance and b) smoothing by finding their average using the above mentioned frameworks. According to the times reported in Table 5-1, our framework is the fastest. In the case of smoothing, it is significantly faster than the Affine invariant framework and asymptotically faster than the Log-Euclidean. A comparison of their properties is presented in Table 5-2. Note that only the EDT framework can be used for higher-order approximations. For the special case of 4 th -order approximations, we compared a) the method introduced by Özarslan et al. (2004) for computing generalized diffusion tensors, b) the method presented in Chapter 4, and c) the exponential tensor framework described 81

82 Figure 5-2. Comparison of 4 th -order DTI using positivity constraint (see Chapter 4), without imposing the positivity (Özarslan et al., 2004) and EDT in estimating fiber orientations for different SNR in the data. in this chapter. Figure 5-2 presents comparison of these techniques in estimating fiber orientations using simulated MR signal (Equation 1 4) for different amounts of Riccian noise in the data. The errors observed by using the exponential tensor framework or the method in Chapter 4 are significantly smaller than those of 4 th -order generalized DTI (Özarslan et al., 2004), which conclusively validates the accuracy of our proposed methods. As expected, the two algorithms presented in this dissertation, have similar performance regarding the accuracy of the estimates, since they both guarantee the positive-definite property of the estimated diffusivity functions. The advantage of the method presented in Chapter 4 is that it computes the coefficients of a positive definite 4 th -order tensor, which are directly related to useful physical quantities, e.g. mean diffusivity, while the exponential tensor coefficients do not have such properties. On the other hand, the advantage of the exponential tensor framework is that it can be used in orders higher than 4, and its coefficients lie in the Euclidean space. Furthermore, we estimated the 4 th -order exponential tensor field of a dataset acquired from excised, perfusion-fixed rat optic chiasm Özarslan et al. (2006). Figure 82

83 Figure 5-3. Displacement probability profiles of a 4 th -order EDT field from a rat optic chiasm data set (Özarslan et al., 2006). In the background the distance from the closest isotropy f iso is shown. 5-3 shows the displacement probability profiles computed from the estimated field. The probability profiles demonstrate the distinct fiber orientations in the central region of the optic chiasm where myelinated axons from the two optic nerves cross one another to reach their respective contralateral optic tracts. These orientation maps are consistent with other studies on this anatomical region of the rat optic chiasm (Özarslan et al., 2006). Furthermore, the f iso map (Figure 5-1c) has slightly brighter intensities in the central region, compared to the FA map (Figure 5-1a). This is because FA uses 2 nd -order approximation, which fails in approximating the fiber crossings in this region and produces estimations close to the isotropy (lower intensities). In our framework, after having estimated the coefficient vectors E, we can use algorithms developed for vectorfield processing in order to compute statistics (average, principal components), interpolate EDT fields etc. Figure 5-4 shows some examples of processes for resolving fiber crossings, interpolating and computing principal components using the proposed framework, for the entire image of Figure

84 Figure 5-4. Applications of the proposed framework for: resolving fiber crossings (8 th -order example), geodesic interpolation and calculating PCA (4 th -order example) from dataset of Figure

85 CHAPTER 6 ESTIMATION OF DISPLACEMENT PROBABILITY FIELDS USING 4TH-ORDER TENSORS 6.1 Introduction In all the previous chapters the local diffusivity function has been modeled by a second or fourth-order tensor. However, in all cases the peaks of the estimated higher-order tensor do not necessarily yield the distinct orientations of the underlying distinct fiber bundles (Özarslan et al., 2006). Hence, one should instead employ the displacement probability profiles given by the Fourier integral (Equation 1 2). However, the computation of the integral cannot be performed analytically and hence is a task that involves numerical integration or approximations. In order to avoid the aforementioned computational effort and the possible inaccuracies introduced by this step, one can directly estimate the displacement probability from the given DW-MRI data. In this chapter we propose a novel representation of the displacement probability profile by using the 4 th -order Cartesian tensor bases. 4 th -order tensors have been studied in Chapter 4 showing their capability to model multi-lobed diffusivity functions. In the model that we will study in this chapter, the lobes of the probability profile, which is expressed in the form of a 4 th -order tensor, correspond directly to orientations of distinct fiber distributions and thus there is no need to evaluate the Fourier integral in Equation 1 2. We also present a novel method for efficiently estimating the 15 unknown tensor coefficients of the displacement probability from a given HARDI dataset. We compare the performance of our method in computing accurate fiber orientations with several other existing techniques, demonstrating the efficiency and accuracy of our model. Finally, we employ the estimated displacement probabilities in an asymmetric diffusion framework in order to estimate asymmetric complex fiber structures by inferring inter-voxel information. 85

86 6.2 Spherical Function Tensorial Approximation A spherical function can be approximated by a n th -order Cartesian tensor expressed in the following form T (v) = T k,l,m v1v k 2v l 3 m (6 1) k+l+m=n where v = [v 1 v 2 v 3 ] T is a unit vector. The spherical functions modeled by Equation 6 1 are antipodal symmetric (T (v) = T ( v)) for even orders. The ability of a Cartesian tensors to approximate the complex geometry of a spherical function with many lobes increases with the order. A 2 nd -order tensor (used in DT-MRI processing presented in Chapters 2 and 3) can only be used for approximating antipodal symmetric spherical functions with a single lobe. For approximating functions with more lobes, higher-order tensors are required (see 4 th -order approximation of the diffusivity function in Chapter 4). In the following sections, we employ higher-order tensors to model the displacement probability profile in HARDI datasets. 6.3 Higher-Order Bases for HARDI Approximation In this section we present a novel representation of the displacement probability profile as a higher-order tensor and a method for direct estimation from a given DW-MRI dataset. We model the probability profile by using the 4 th -order Cartesian tensor bases as follows P (r) = c i,j,k (r 0 )r1r i 2r j 3 k (6 2) i+j+k=4 where c i,j,k (r 0 ) are the tensor coefficients estimated for a given magnitude r 0 of the displacement vector. Given r 0, Equation 6 2 is a spherical function since the only argument is the unit vector r. In the case of 4 th -order tensors there are 15 unique unknown coefficients c i,j,k (r 0 ) that need to be estimated. However, since in our application, the given data is the DW-MRI signal and not the displacement probability, we need to define a set of functions that approximate the signal, and have the following properties: a) 86

87 Table 6-1. Proposed bases functions P (r) basis Corresponding S(q)/S 0 basis (12 48q q1)e 4 qt q r1 4 r2 4 r3 4 r1r r2r r1r r1r 2 2 r 3 r 1 r2r 2 3 r 1 r 2 r3 2 r1r 3 2 r1r 3 3 r 1 r2 3 r2r 3 3 r 1 r3 3 r 2 r3 3 r1r i 2r j 3 k (12 48q q2)e 4 qt q (12 48q q3)e 4 qt q ( 2 + 4q1)( q2)e 2 qt q ( 2 + 4q2)( q3)e 2 qt q ( 2 + 4q1)( q3)e 2 qt q 4q 2 q 3 ( 2 + 4q1)e 2 qt q 4q 1 q 3 ( 2 + 4q2)e 2 qt q 4q 1 q 2 ( 2 + 4q3)e 2 qt q 2q 2 (12q 1 8q1)e 3 qt q 2q 3 (12q 1 8q1)e 3 qt q 2q 1 (12q 2 8q2)e 3 qt q 2q 3 (12q 2 8q2)e 3 qt q 2q 1 (12q 3 8q3)e 3 qt q 2q 2 (12q 3 8q3)e 3 qt q ( / q 1 ) i ( / q 2 ) j ( / q 3 ) k B(q) their Fourier integral gives the 4 th -order tensorial bases functions, and b) can be computed analytically. A set of functions that have the above properties is presented in the right column of table 1. This set of functions was obtained by taking all the possible combinations of partial derivatives ( / q 1 ) i ( / q 2 ) j ( / q 3 ) k B(q), where i + j + k = 4 and B(q) is a real-valued function in R 3 defined as B(q) = e qt q (6 3) The Fourier integral of each partial derivative of Equation 6 3 is expressed as r1r i 2r j 3c k i j k = q1 q i 2 q j B(q)e 2πiqT rr 0 dq (6 4) 3 k where c is a constant whose value is not dependent on r. Note that the left side of Equation 6 4 is a Cartesian tensor basis function. Hence we can employ the functions 87

88 Figure D plots of three of the bases functions from Table 1. The functions were evaluated for varying q over a unit circle of directions q. The circle was defined by fixing the elevation spherical coordinate to π/3 and varying azimuth. presented in table 6-1 to model the displacement probability profile as P (r) = i+j+k=4 c i,j,k i j k q1 q i 2 q j B(q)e 2πiqT rr 0 dq (6 5) 3 k In the model proposed in Equation 6 5, the approximated DW-MRI signal has asymptotic behavior (when q ) similar to that of the commonly used Stejskal-Tanner Equation 1 1. Furthermore, by substituting Equation 6 4 into 6 5 the displacement probability profile is modeled as a 4 th -order tensor (Equation 6 2). By using known properties of the Fourier transform, the integral in Equation 6 5 can be re-written in the following more convenient form that we will use later in Section 6.4. P (r) = i+j+k=4 c i,j,k i j k q1 q i 2 q j B(q/r 3 k 0 )e 2πiqT r dq (6 6) where, c i,j,k = c i,j,k /r 0. Note the presence of r 0 in the argument of B(q/r 0 ). Figure 6-1 shows 3D plots of three of the bases functions ( / q 1 ) i ( / q 2 ) j ( / q 3 ) k B(q). By observing the figure, we can see the variability in the shape of the bases functions (e.g. crosses and peanut-like shapes). These characteristics of the functions provide the ability to our model to approximate complex geometries such as fiber crossings. In Section 6.4 we employ our proposed model (given by Equation 6 6) to approximate the displacement probability profiles from a given set of high angular resolution diffusion-weighted images. 88

89 6.4 Fast Estimation from HARDI Datasets Given a set of N diffusion-weighted MR images S n associated with diffusion gradient directions g n, n = 1... N, we seek to estimate the 15 unknown coefficients c i,j,k in Equation 6 6 by minimizing the following energy ( N E = n=1 i+j+k=4 c i,j,k ) 2 i j k q1 q i 2 q j B(q/r 3 k 0 ) S n /S 0 (6 7) where S 0 is the zero gradient image. If N > 15 Equation 6 7 can be minimized by solving an over-determined linear system. This system is formed by constructing an N-dimensional vector S that consists of the signal values S n and a N 15 matrix A whose entries are the values of our 15 dimensional basis ( / q 1 ) i ( / q 2 ) j ( / q 3 ) k B(αg n ) n = 1... N, where α = q n /r 0. We note that α depends only on r 0 if the diffusion-weighted images were acquired with a fixed b-value since in this case q n is the same constant n = 1... N. In our experiments we used α = 0.5. The estimated coefficients c i,j,k are the components of the vector x in the following over-determined linear system Ax = S, which can be solved very efficiently. After solving the above system, we can compute the directions of the distinct fiber populations by finding the peaks of Equation 6 2. The probability profile can be visualized by plotting Equation 6 2 as a spherical function (i.e. for all unit vectors r) as shown in the experimental result section. 6.5 Experimental Results In this section we present experimental results on our method applied to simulated DW-MRI data as well as real HARDI data from a fixed rat spinal cord Synthetic Data Experiments In order to test our method in approximating displacement probability profiles from single fibers as well as from fiber crossings, we synthesized a HARDI dataset of size by simulating two fiber bundles crossing each other using the realistic diffusion MR simulation model given by Equation 1 4 (b-value=1250 s/mm 2, 81 gradient 89

90 Figure 6-2. Probability profiles estimated by applying our method to simulated data of: A) 2-fiber crossing bundle and B) corrupted crossings for different amounts of Riccian noise. directions).the probability profiles that were estimated from this dataset by using our method are presented in Figure 6-2A showing correct fiber orientations. Furthermore, in order to compare our proposed method with other existing techniques, we performed another experiment using simulated noisy MR signal of 2-fiber crossings with different amounts of Riccian noise (Figure 6-2B). We estimated the displacement probability profiles from the corrupted signal using our proposed method and the following existing methods: a)the DOT method described by Özarslan et al. (2006), b)the ODF method presented by Tuch (2004) and c) the positive 4 th -order diffusion tensor model in Chapter 4. For all methods we computed the estimated fiber orientation errors for different amount of noise in the data (shown in Figure 6-3). The results conclusively demonstrate the accuracy of our method, showing small fiber orientation errors ( 6 o ) for typical amount of noise with signal to noise ratios (SNR): Furthermore, by observing the plot, we also conclude that the accuracy of our proposed method is very close to that of the DOT method and is better than the ODF method for higher noise cases. Here, we should note that our method estimated each probability profile in approximately 2ms (on a Pentium 2.4GHz) which demonstrates the efficiency of our method. 90

91 Figure 6-3. Fiber orientation errors for different SNR in the data using our method (P4) and three other existing methods: 1) DOT, 2) ODF and 3) 4 th -order DT. In the experiment we used simulated MR signal of a 2-fiber crossing, whose probability profile is shown in Figure 6-2(right) Real Data Experiments Figure 6-4 shows a real data example from a fixed rat spinal cord. The protocol that used in this experiment included acquisition of 22 images using a pulsed gradient spin echo pulse sequence with repetition time (TR) = 1.5 s, echo time (TE) = 27.2 ms, bandwidth = 30 khz, field-of-view (FOV) = mm. After the first image set was collected without diffusion weighting (b 0 s/mm 2 ), 21 diffusion-weighted image sets with gradient strength (G) = 664 mt/m, gradient duration (δ) = 1.5 ms, gradient separation ( ) = 17.5 ms and diffusion time (T δ ) = 17 ms were collected. The image without diffusion weighting had 8 signal averages, and each diffusion-weighted image had 2 averages. By observing figure 6-4 it is clear that the white matter probability profiles shows peaks that correspond to fiber tracts which, as expected, are predominantly in the axial direction which is represented by the blue color. This indicates that our proposed method estimates correct fiber orientations in real HARDI datasets. The same figure also presents (in the 91

92 Figure 6-4. Estimated probability profiles from real data of a rat s fixed spinal cord. The zoomed ROI shows single fiber distributions in white matter and other more complex tissue structures. zoomed plate) regions where more complex probability profiles were estimated that show the underlying complexity of the tissue structures. 6.6 Application of the Displacement Probability Field for Tractography The estimated displacement probabilities are spherical functions which characterize the intra-voxel fiber structure without taking into consideration any inter-voxel information. The computed probabilities are always anti-podally symmetric and therefore they can only model either single fiber tracts or symmetric crossings of multiple fiber tracts. However, it is well known that neural fiber tracts can also form asymmetric local structures such as in sprouting fibers (Basser et al., 2000). To date there are no existing methods in literature 92

93 for estimating locally asymmetric fiber orientation functions and one has to resort to an existing fiber tracking procedure that can accommodate for multiple fibers at a voxel (Melonakos et al., 2008; Deriche and Descoteaux, 2007; Basser et al., 2000), in order to infer the presence of a sprouting or anti-symmetric crossing structures. In this section we present a novel method for estimating an intra-voxel asymmetric spherical function that can model complex local fiber structures using inter-voxel information. The peaks of the estimated spherical function correspond to directions that point to distinct local fiber tracts and are appropriately dubbed tractosemas. Tractosema is a pointer/sign of used here for neural tracts and has its roots in the Greek word sēma (sign). In our work here, we extract a field of tractosemas from a given field of ODFs or displacement probabilities by following asymmetric and orientation depended diffusion of spherical functions. The kernel that controls the diffusion process between two elements (in our case spherical functions) is defined as a function over the spatial location (R 3 ) and the domain (S 2 unit sphere) of the two elements, which leads us to the space (R 3 S 2 ) (R 3 S 2 ). We construct the diffusion kernel as a tensor product of the von Mises and Gaussian probability distributions and by using it we derive an update formula for the field of tractosemas which is expressed in the form of a discrete kernel convolution. The main contribution of tractosemas is that they can depict complex asymmetric fiber structures without the need for fiber tracking. To the best of our knowledge, it is the first method that estimates a field of asymmetric spherical functions for modeling splaying fibers and other asymmetric as well as symmetric structures. Furthermore, the estimated field of tractosemas can be used as input by any existing fiber tracking algorithm for finding fiber junctions and branches without the need for multiple seeds (a common requirement in many existing methods (Melonakos et al., 2008; Friman et al., 2006; McGraw et al., 2004; Basser et al., 2000)). Finally, the experimental results demonstrate the robustness and accuracy of our model in estimating fiber orientations in the presence 93

94 of varying amount of noise as demonstrated via simulation experiments with realistic MR data synthesis (Equation 1 4) Extracting Tractosemas by Diffusing Probability Iso-Surfaces After having estimated the displacement probability p x (r) x R 3, where x is the lattice index, we use the obtained spherical function field in the following diffusion process. In this process the spherical functions are updated iteratively by diffusing the displacement probability field. In general, diffusion can be seen as a smoothing process which can be performed by minimizing a smoothness measure. In our case, we minimize the following function with respect to p x (r). E(p x (r)) = K(x, y, r, v)dist(p x (r), p y (v))dvdy (6 8) R 3 S 2 Equation 6 8 is expressed in the form of a kernel integration, where dist(.) can be any norm or edge-stopping function Black et al. (1998), the kernel K(.) is a function of x, y, r, v, and the integration is over all vectors y and unit vectors v. In our particular application, the kernel is a probability function expressing the probability of diffusion between the elements p x (r) and p y (v). The kernel we seek should exhibit the following properties: a) the probability of diffusion between locations x and y decreases with their distance, b) the probability of diffusion between orientations r and v decreases with the angle between them, and c) the probability of diffusion is larger at the locations along the maxima of p x (r). These properties are satisfied by single peaked distributions. One such function used here is, K(x, y, r, v) = K dist ( y x )K orient (r v)k fiber (r (y x)/ y x )). (6 9) The first property mentioned above is imposed by defining K dist using a multivariate Gaussian distribution. K dist ( y x ) = 1 y x 2 e 2σ (2πσ) 3/2 3 (6 10) 94

95 Figure 6-5. Synthetic data example. A) Simulated data. B) The field of computed tractosemas. C) Tractosemas in ROI under varying noise. D) Plot of fiber orientation errors. The most natural way to impose the last two properties is to employ the single peaked von Mises distribution for both K orient and K fiber, given by, K orient (cos(φ)) = K fiber (cos(φ)) = κeκcos(φ) 4πsinh(κ) (6 11) where φ is the angle between r and v, and the angle between r and (y x) in K orient and K fiber respectively. The distribution parameters σ and κ in Equation 6 10 and 6 11 respectively control the sharpness of the kernel. Having a discrete lattice of probabilities p x (r) the integral over R 3 in Equation 6 8 becomes summation over the lattice. Furthermore, since the Gaussian part of the kernel takes its largest values in the region around its center (at location x), we can define a set N(x) that contains the lattice indices in the neighborhood of x. Furthermore, we discretize the space of unit vectors by using a 4 th order subdivision of the icosahedral tessellation of the unit sphere. By using the above discretization, Equation 6 8 can be written in the following form E(p x (r)) = y N(x) v S K(x, y, r, v)dist(p x(r), p y (v)) (6 12) By setting for simplicity dist(a, b) = (a b) 2 and taking the derivative of Equation 6 12 with respect to p x (r) and setting it equal to zero, we derive the following update 95

96 Figure 6-6. Real hippocampal data. A) The data set shown in 3D (top) and the region of interest shown enlarged (bottom). The rest of the plates depict the displacement probability profiles (bottom) and the orientations corresponding to their maxima shown as tubes (top) obtained by using: B) DTI, C) fourth order tensors, and D) tractosemas. formula for the field of spherical functions (tractosemas) p new x (r) = K(x, y, r, v)p y(v) (6 13) y N(x) v S Equation 6 13 is expressed in the form of a discrete kernel convolution and it is applied iteratively to all indices x and vectors r on the discretized S 2. This method produces very efficient implementations since only kernel multiplications are involved in the evaluation of Equation 6 13, which is a fully parallelizable process. Furthermore, only few iterations (2 to 3) are required to observe visually the diffused asymmetric tractosemas. Finally, choosing a different dist (e.g. L1 norm), would lead to more anisotropic solutions Experimental Results In the experiments presented in this section, we tested the performance of our method using simulated diffusion-weighted MR signal and real HARDI data sets from an isolated rat hippocampus and an excised rat spinal cord. 96

97 For the validation of tractosemas we synthesized a dataset representing splaying fiber bundles, whose orientations were taken to be tangent to two ellipsoids centered at the two lower corners of the image. The data set was of size and was generated by simulating the diffusion-weighted MR signal using the realistic simulation model in Equation 1 4 (b-value=1250s/mm 2, 81 gradient directions). After that, we estimated the displacement probability field (Figure 6-5A) from the simulated signal by using the method in Barmpoutis et al. (2008) (one can also use any other method). The above obtained field of probability functions was then input to our proposed method for extracting tractosemas (σ = 1, κ = 10, 3 iterations). Figure 6-5B shows the field of tractosemas computed by our technique. By observing the figure, we can see that our method estimated correctly single fiber distributions in the lower part of the image and splaying fibers in the central region of the field, which demonstrates the effectiveness of our technique. Note the smooth transition from single fiber to splaying structure in the ROI, and the expected anti-aliasing effect observed in the voxels close to the splaying fibers. Furthermore, to quantitatively test the performance of our method in estimating fiber orientations we added varying amounts of Riccian noise (SNR between 20:1 and 3.3:1) to the data. We applied our method to these noise corrupted data sets and then computed the estimated fiber orientation errors. Figure 6-5D depicts a plot of the mean and the standard deviation of the angle error between computed and ground truth orientations (in degrees). These results validate the accuracy of our model and demonstrate its robustness to noise. The proposed method was also applied to a real DW-MRI from an isolated rat hippocampus (Figure 6-6A). The dataset consists of 22 images acquired using a pulsed gradient spin echo pulse sequence with TR=1.5 s, TE= 28.3 ms, G= 415 mt/m, δ= 2.4 ms, = 17.8 ms, T δ = 17 ms and b 1250s/mm 2. 97

98 Figure 6-7. The field of tractosemas estimated from the hippocampal data set. A) Three zoomed voxels depicting the variability in the estimated structures. B) The fiber sprouting with the estimated tractosemas superimposed. Figure 6-6 shows a region of interest (ROI) in the hippocampus containing mixture of CA3 stratum pyramidale, stratum lucidum and part of the hilus. The rest of the images in this figure show a comparison of the estimated local fiber structures using Diffusion tensors (order-2 DTs), fourth order tensors, and tractosemas. In the DT field we can observe two dominant orientations one pointing to the upper left and the other to the upper right corner of the ROI, however, the structure at the junction is lost. The junction was recovered using the fourth order tensors however, they depict the two aforementioned fiber orientations as symmetric structures. The complicated junction structure is correctly captured in the estimated field of tractosemas with asymmetric structures that depict splaying fibers. Figure 6-7 depicts fiber tracks estimated from the hippocampal data set by following the peaks of tractosemas. The capability of tractosemas in capturing various structures is demonstrated on the plate B of this figure. 98

99 CHAPTER 7 REGISTRATION OF 4TH-ORDER TENSOR FIELDS 7.1 Introduction In Chapters 4, 5 and 6 we employed symmetric positive 4 th -order Cartesian tensors to approximate the local diffusivity function (see Chapters 4 and 5) or the local water molecule displacement probability (see Chapter 6). All these methods produce 4 th -order tensor fields from given DW-MRI datasets. Hence, given two different DW-MRI datasets depicting the same or different subjects, one can register them by using the information provided by the coefficients D i,j,k of the corresponding 4 th -order tensor fields. In this chapter we present two different methods for 4 th -order tensor field registration. The first method computes a locally affine transform that registers one dataset with a fixed target dataset. The advantage of a locally affine transformation is that it stores the local translation vector and rotation matrix of the transformation. The rotation matrices are used to reorient appropriately the 4 th -order tensor coefficients in order to preserve the local geometry of the microstructural properties in the transformed image. The second method performs group-wise registration of several datasets and simultaneously estimates the average tensor field (atlas). This algorithm computes diffeomorphic deformations that map each dataset to the estimated atlas using a functional minimization method. The advantage of this method is that the computed result is unbiased, i.e. there is no fixed target dataset and therefore the result is not depended on the order of the given datasets. The chapter is structured as follows. Section 7.2 presents an algorithm for locally affine registration of 4 th -order tensor fields. In Section 7.3 we review the Riemannian geometry of positive-valued functions and we use it for group-wise registration and atlas construction of higher order tensor fields. Both methods are demonstrated in synthetic and real datasets and the results are discussed in Section

100 7.2 Locally Affine Registration of 4 th -Order Tensor Fields In order to compute the locally affine transformation that maps one deforming tensor field to another fixed dataset we need to minimize an energy functional which measures the distance between the two tensor fields. For this purpose, we need to define the appropriate metric between higher-order tensors, which will be later employed by the registration algorithm Distance measure In this section we define a distance measure between symmetric positive definite 4 th -order tensors using their corresponding normalized representations which are angular distributions. A family of angular distributions for modeling antipodal symmetric directional data is the angular central Gaussian distribution family, which has a simple formula and a number of properties discussed by Watson (1983). The family of angular central Gaussian distributions on the q-dimensional sphere S q with radius one is given by p(g) = 1 Z q(t) (gt T 1 g) q+1 2 where g is a (q+1)-dimensional unit vector, T is a symmetric positive-definite matrix and Z q (T) is a normalizing factor. In the S 2 case, g is a 3 dimensional unit vector, Z 2 (T) = 4π T, and T is a 3 3 symmetric positive-definite matrix similar to the 2 nd -order tensor used in DTI. A generalization of this distribution family for the case of higher-order tensors should involve an appropriate generalized normalizing factor as a function of higher-order tensors and a generalization of the tensor inversion operation, which may not lead to a closed-form expression. In order to get closed-form expressions we define a new higher-order angular distribution as p(g) = 1 S 2 d(g) 2 d(g)2 (7 1) where in the case of 4 th -order tensors d(g) is given by Equation 4 2 and the integral is over S 2 (i.e. over all unit vectors g). The integral in Equation (7 1) can be analytically computed and it can be written in a sum-of-squares form as it has been shown in Section

101 Given two angular distributions we need to define a scale and rotation invariant metric in order to make true shape (obtained after removing scale and orientation) comparison between them. This can be efficiently done by the Hellinger distance between 4 th -order tensors D 1 and D 2 : dist 2 (D 1, D 2 ) = ( p 1 (g) p 2 (g)) 2 = S 2 S 2 ( d 1 (g) d 2 (g) ) 2 (7 2) (d S2 1(g)) 2 (d S2 2(g)) 2 Here we use the notation D to denote the 15-dimensional vector consisting of the unique coefficients D i,j,k of a 4 th -order tensor. Equation 7 2 can also be analytically expressed in a sum-of-squares form and it is invariant to scale and rotations of the 3D space, i.e. the distance between p 1 (g) and p 2 (g) is equal to the distance between p 1 (srg) and p 2 (srg), where R is a 3 3 rotation matrix and s is a scale parameter. In the following section we use the above distance measure for registering a pair of misaligned 4 th -order tensor fields Registration In this section we present an algorithm for 4 th -order tensor field registration. Given two 4 th -order tensor fields I 1 (x) and I 2 (x), where x is the 3D lattice index, we need to estimate the unknown transformation F (x), which transforms the dataset I 1 (F (x)) in order to better match I 2 (x). In the case of an affine transformation we have F (x) = Ax + T, where A is a 3 3 transformation matrix and T is the translational component of the transformation. The estimation of the unknown transformation parameters can be done by minimizing the following energy function E(A, T) = dist 2 (I 1 (Ax + T), A 1 I 2 (x))dx R 3 (7 3) where dist(.,.) is the distance measuere between 4 th -order tensors defined in section 7.2.1, and the integral is over the 3D image domain. A 1 I 2 (x) denotes some higher-order tensor re-transformation operation. This operation applies the inverse transformation to 101

102 the tensors of the dataset I 2 in order to compare them with the corresponding tensors of the transformed image I 1. In the case of registering 2 nd -order tensor fields, it has been shown by Alexander et al. (2001) that the unknown transformation parameters can be successfully estimated by applying only the rotation component of the transformation to the dataset I 2. This happens because of the fact that 2 nd -order tensors can approximate only single fiber distributions, whose principal direction transformation can be adequately performed by applying rotations only to the tensors. In the case of 4 th -order tensors, multiple fiber distributions can be resolved by a single tensor, whose relative orientations can also be affected by the deformation part of the applied transformation. Therefore, tensor re-orientation is not meaningful for higher-order tensors and in this case a tensor re-transformation operation must be performed instead, using the full affine matrix A, which is defined in section Affine transformation has been also used in DTI; for more details and justification of the scheme, the reader is referred to the work by Wang and Vemuri (2005). Equation 7 3 can be extended for non-rigid registration of 4 th -order tensor fields by dividing the domain of image I 1 into N smaller regions and then registering each smaller region by using affine transformations. Similar method has been used for scalar image registration by Ju et al. (1996) and DTI registration by Zhang et al. (2004). The unknown transformation parameters can be estimated by minimizing E(A 1, T 1,..., A N, T N ) = N r=1 R 3 dist 2 (I 1,r (A r x + T r ), A 1 r I 2 (x))dx (7 4) Equation 7 4 can be efficiently minimized by a conjugate gradient algorithm used in a multi-resolution framework, similar to that used by Ju et al. (1996) and Zhang et al. (2004). 102

103 D Affine Transformation of 4 th -Order Tensors Assume that we have vectorized the coefficients D i,j,k into a 1 15 vector D in some specific order D n = D in,j n,k n, ( e.g. D 1 = D 4,0,0, D 2 = D 2,2,0, etc.). By using this vector, Equation 4 2 can be written as 15 n=1 D ng 1 i n g 2 j n g 3 k n. If we apply an affine transformation defined by the 3 3 matrix A to the 3D space, the previous equation becomes 15 n=1 D n(a 1 g) in (a 2 g) jn (a 3 g) kn,where (a 1 g) in (a 2 g) jn (a 3 g) kn is a polynomial of order 4 in 3 variables g 1, g 2, g 3, and a i is the i th raw of A. In this summation there are 15 such polynomials and each of them can be expanded as 15 m=1 C m,ng 1 i m g 2 j m g 3 k m, by computing the corresponding coefficients C m,n as functions of matrix A. For example if we use the same vectorization as we did in the previous example, we have C 1,1 = (A 1,1 ) 4, etc. Therefore, we can construct the matrix C, whose elements C m,n are simple functions of A, and use it to define the operation of transforming a 4 th -order tensor D by a 3D affine transformation A as A D = C(A) D (7 5) 7.3 Group-Wise Registration and Atlas Construction In Section 7.2 we presented an algorithm for registering two datasets by fixing one of them and deforming the other one by minimizing an appropriately defined dissimilarity measure. Here we present a method for unbiased group-wise registration of several 4 th -order tensor fields and simultaneously estimation of the atlas field. The calculations in this algorithm are being performed in the Riemannian space of real positive-valued functions (since symmetric positive 4 th -order tensor belong to this space) which is studied in Section Riemannian Metric for Positive-Valued Real Functions Assume a, b R +, i.e. are elements of the space of positive real numbers. The Logarithmic map at location a is given by Log a (x) = log(x/a) and corresponds to the local tangent vector toward x. Its inverse function is the Exponential map, which is 103

104 given by Exp a (t) = exp(t)a and projects the tangent t R back to the space R +. The corresponding Riemannian distance between a and b R + is given by the length of the tangent dist(a, b) = log a (7 6) b which satisfies scale invariance, i.e. dist(sa, sb) = dist(a, b) a, b, s R +, additionally to the properties of distance measures. The Riemannian metric in R + can also be used to define distances between n-tuplets whose elements are positive real numbers. The distance between A = {a 1, a 2,..., a n } and B = {b 1, b 2,..., b n } a i, b i R + can be defined as dist 2 (A, B) = N i=1 dist2 (a i, b i ). Similarly, the distance between positive-valued functions f a (x) and f b (x) x Ω can be defined using the Riemannian metric in R + as dist 2 (f a, f b ) = Ω dist2 (f a (x), f b (x))dx. In the particular case of parametric spherical functions d(g D 1 ) and d(g D 2 ), where g S 2 and D 1 and D 2 are the corresponding parameter vectors, the distance is given by dist 2 (D 1, D 2 ) = S 2 log d(g D 1) d(g D 2 ) 2 dg. (7 7) Note that the integral in Equation 7 7 is over S 2, i.e. the space of unit vectors g. The above distance function is invariant with respect to 3D rotations and scale, i.e. dist(sr D 1, sr D 2 ) = dist(d 1, D 2 ) s R + and R SO 3. In Section we employ the distance measure in Equation 7 7 in order to achieve simultaneous group-wise registration and atlas construction of fields of spherical functions modeled using Cartesian tensor bases Group-Wise Registration of 4 th -Order Tensor Fields Cartesian tensor bases of various orders have been used for approximating physical quantities computed from DW-MRI datasets. 4 th -order tensors have been employed to approximate the water molecule displacement probability isosurfaces in Chapter 6 and the diffusivity function in Chapter 4 and the kurtosis component of the diffusion in diffusion kurtosis images (studied by Jensen et al. (2005)). 104

105 In the case of generalized diffusion tensors, the diffusivity is a positive-valued function and can be computed using the parametrization presented in Chapter 4. This produces fields of positive-valued spherical functions whose processing can be achieved using the Riemannian metric presented in Section The problem of group-wise registration of N tensor-fields and simultaneous atlas estimation can be formulated into a functional minimization problem. By using Equation 7 7 the energy function to be minimized is given by E(φ n, D µ ) = N n=1 Ω S 2 ( log d(g D ) 2 n φ n ) dgdx + d(g D µ ) N n=1 Ω cost(φ n )dx (7 8) where D µ is the 4 th -order tensor coefficients of the estimated atlas, φ n is the estimated deformation to be applied to the n th tensor field, and cost() is a cost function that adds constraints to the estimated deformations. Note that the tensor coefficients are depended on the local rotation of the coordinate system. Hence, given a deformation φ n the transformed spherical function field can be computed as d(g D n φ n ) = i,j,k D i,j,k n (x φ n )(R x g) i 1(R x g) j 2(R x g) k 3 (7 9) where R x is the rotation of deformation φ n at location x, and the notation (R x g) 1 represents the first component of the rotated vector g. The deformation can be parametrized as a time varying vector field such that φ n (x, t)/ t = v n (x, t), t [0, 1]. In this formulation the estimated deformation is given by φ n = φ n (x, 1) = 1 0 v n(x, t)dt. Furthermore, the cost() function in Equation 7 8 can be defined as 1 0 v n(x, t) 2 dt. We will minimize the energy function (Equation 7 8) by evolving the deformation fields φ n using a greedy iterative scheme which approximates the solution to the above minimization problem, similarly to the work by Joshi et al. (2004). For this purpose we will construct a field of forces by computing the variation of the first term in Equation

106 with respect to v n as follows F n = 2 log S 2 ( d(g Dn φ n ) d(g D µ ) ) [ trans + rot ]log(d(g D n φ n ))dg (7 10) where the variation trans is related with the local translation (i.e. variation of Dn i,j,k (x φ n ) in Equation 7 9) and rot is related with the local rotation (i.e. variation of (R x g) i 1(R x g) j 2(R x g) k 3 in Equation 7 9). The computation of these terms is discussed in Section After the estimation of the fields of forces F n, n = 1... N we compute the update vector fields v n = Ω K(x)F n(x)dx, where K is a kernel applied to the field of forces. In our experiments we employed the kernel K(x) = η(x)g(x), where G is a Gaussian kernel centered at x and η is a smooth function that takes zero value at the boundaries and therefore imposes zero boundary conditions to the kernel K similarly to the work by Cao et al. (2006). Note that the integration of K with F n is a convolution that becomes multiplication in the frequency domain, hence it can be efficiently computed using the discrete Fourier transform (see the work by Joshi et al. (1997)). Then, the deformation fields are updated as φ new n = φ old n (x + ɛv n ) using a small step ɛ. Finally, the tensor coefficients of the atlas can be updated by also minimizing the first term in Equation 7 8 with respect to the parameters of a positive definite 4 th -order tensor using the parametrization in Chapter Implementation Details In general, the integral over the sphere in Equation 7 10 cannot be computed analytically when the Cartesian tensor parametrization is used for modeling the diffusivity function. Therefore we approximate the integration over the sphere by using a sum over a set of unit vectors g m m = 1... M uniformly distributed on the sphere. This set of vectors can be constructed by tessellating the icosahedron and then projecting the vectors on the unit hemisphere (we consider only a hemisphere due to antipodal symmetry of diffusivity functions). We use this set of vectors in order to evaluate the spherical 106

107 functions log(d(g m D n φ n )), m = 1... M and n = 1... N. This creates N vector valued images, whose vectors contain M elements that are rotation depended, similarly to the D i,j,k tensor coefficients. The above discretization helps us also reducing the time complexity of atlas computation, which can now be efficiently computed by d µ (g m ) = exp( 1 N N log(d(g m D n φ n ))) (7 11) n=1 where d µ (g m ) is also in the form of a vector valued image, whose vectors contain M elements. Note that log(d(g m D n φ n )) is an already computed image, and therefore there is no need to re-deform the images and re-compute the log maps. Furthermore, there is no need to compute the exp in Equation 7 11 during the minimization procedure, since the Riemannian metric employed in Equation 7 10 involves only computations in the log-space. The corresponding driving forces in Equation 7 10 are now computed as follows F n = 2 M ( ) d(gm D n φ n ) log log(d(g m D n φ n ) + rot E (7 12) d µ (g m ) m=1 where log(d(g m D n φ n ) is simply the spatial gradient of a scalar valued image and the term rot E is approximated numerically as log(d(r δx g m D n φ n )/d(rg m D n φ n ))/δx for a small step δx. The rotation matrices R and R δx are efficiently computed by the Gram-Schidt algorithm, which orthogonalizes the Jacobians J and J δx that correspond to the deformation φ n (x) and the perturbed deformation φ n (x + δx) respectively. Finally, after the termination of the iterative minimization procedure the 4 th -order tensor coefficients can be computed by fitting the tensorial model to the estimated values d µ (g m ) using the positive-definite parametrization in Chapter Robust Atlas Construction The energy in Equation 7 8 is in the form of sum (and integral) of square distances (L 2 ). However it is well known that the sum of absolute distances (L 1 ), is more robust to 107

108 Figure 7-1. A) Synthetically generated dataset by simulating the MR signal using Equation 1 4. B) Dataset generated by applying a non-rigid transformation to A. C) Quantitative comparison of the registration errors. The errors were measured by Equation 7 2 for the whole field. variations in the data. In this case the field of forces is computed as follows: F n = 2 M ( ) d(gm D n φ n ) log(d(gm D n φ n ) + rot E log d µ (g m ) log(d(g m D n φ n )) log(d µ (g m )) m=1 (7 13) when d(g m D n φ n ) d µ (g m ), otherwise F n = 0. Furthermore, the atlas can also be computed by minimizing a sum of absolute distances and it corresponds to the median in the Riemannian space presented in Section as follows: d µ (g m ) = exp(median n=1:n {log(g m D n φ n )}). In Section 7.4 we compare the atlases computed using the Riemannian mean and median in terms of accuracy in estimating fiber orientations under different levels of outliers and variations in the data. 7.4 Experimental Results In the experiments presented in this section, we tested the performance of our methods using simulated diffusion-weighted MR data and real HARDI data sets. 108

109 7.4.1 Experiments Using Synthetic Datasets The synthetic data were generated by simulating the MR signal from single fibers and fiber crossings using the simulation model in Equation 1 4. A dataset of size was generated by simulating two fiber bundles crossing each other (Figure 7-1A). Then, a non-rigid deformation was randomly generated as a b-spline displacement field and then applied to the original dataset. The obtained dataset is shown in Figure 7-1B. In order to compare the accuracy of our 4 th -order tensor field registration methods with other methods that perform DTI regitration or registration of scalar quantities computed from tensors (e.g. GA), we registered the dataset of Figure 7-1A with that of Figure 7-1B by performing: a) General Anisotropy (GA) map registration using the method by Ju et al. (1996), b) DTI registration using the algorithm by Zhang et al. (2004) and c) 4 th -order tensor registration using our proposed methods. Figure 7-1C shows a quantitative comparison of the above results by measuring the distance between the corresponding misaligned tensors by using the measure defined in section The results conclusively validate the accuracy of our methods and demonstrate their superior performance compared to the other existing methods. We can also see that the unbiased registration method performs slightly better than the locally affine transformation method. Furthermore, in order to compare the Riemannian metrics presented in Section 7.3 in terms of fiber orientation accuracy of the atlas estimated by each metric, we performed the following experiment. We synthesized a 2-fiber crossing DW-MRI dataset (in a single voxel) using the same simulation method as above. Then, we computed a 4 th -order tensor from the synthetic dataset using the method in Chapter 6 and it is shown in Figure 7-2. Then we generated 100 more datasets by applying small rotations to the simulated crossing (few of them are shown in Figure 7-2). The arbitrary rotations were generated by computing the matrix exponential of 3 3 skew symmetric matrices whose elements were generated using a Gaussian distribution with zero mean and varying standard deviation. 109

110 Figure 7-2. Comparison of the 4 th -order tensor atlases computed by various metrics: a) Euclidean mean, b) Riemannian mean and c) Riemannian median (the last two were computed in the space presented in Section 7.3.1). Finally, we also added various quantities of 1-fiber outliers in the dataset in order to test the robustness of the proposed metrics. After that we computed the atlas tensor by minimizing a) the sum of Euclidean square distances, b) the sum of Riemannian square distances, and c) the sum of Riemannian distances. The results are shown in the bar chart of Figure 7-2. As expected, the Riemannian mean outperforms the Euclidean mean since the physical space of the data is the space of positive real-valued functions, where the Riemannian metric was constructed. Furthermore, as it is widely known, the median is more robust to the presence of outliers in the data compared to the mean Real Data Experiments In this section we present experiments using real DW-MRI datasets from 4 human hippocampi. In the data acquisition first an image without diffusion-weighting was collected, and then 21 diffusion-weighted images were collected with a 415 mt/m diffusion gradient (T d =17 ms, δ = 2.4ms, b = 1250 s/mm 2 ). Figure 7-3 shows two S 0 images from the 3D volumes of two hippocampi (A,B), and two checkers images showing the datasets before (C) and after locally affine registration (D). A checkers image is a way to display two images at the same time, presenting one image in the half boxes, and the other in 110

111 Figure 7-3. A and B) S 0 images from two HARDI volumes of human hippocampi. C,D) datasets before and after locally affine registration. Tensors from the ROI in D showing crossings. the rest of the boxes. Based on knowledge of hippocampal anatomy, fiber crossings are observed in several hippocampal regions such as CA3 stratum pyramidale and stratum lucidum. Therefore, one should employ one of our 4th-order tensor methods instead of DTI registration. By observing Figure 7-3d all the hippocampal regions were successfully alligned by our method, transforming appropriately the fiber crossings Figure 7-3(right). Finally we applied our group-wise 4 th -order tensor field registration algorithm in order to co-register all the 4 datasets and compute the atlas using the Riemannian median. Figure 7-4 shows the initial misalignment of the S 0 images and the direction of largest diffusion before (top row) and after group-wise registration using our method (bottom row). By observing the images we can see that all datasets were correctly aligned and the primary directions of diffusions become more coherent. 111

112 Figure 7-4. Overlayed S 0 images and the corresponding Riemannian median atlas (direction of largest diffusion is shown) before (top row) and after (bottom row) group-wise 4 th -order tensor field registration. 112

113 CHAPTER 8 CLINICAL APPLICATIONS 8.1 Introduction In this chapter we demonstrate the methods presented in this dissertation, by showing a clinical application using real DW-MRI data from rat spinal cords. The experiments were performed using a set of 4 controls and 3 injured rat spinal cords. Sample S 0 images of the data sets are shown in Figure 8-1. The goal in this clinical application is to perform unsupervised clustering of the datasets and compare quantitative comparisons. 8.2 Comparison of Control and Injured Spinal Cord Datasets First we estimated the 4 th -order diffusion tensors by applying the technique proposed in Chapter 4 to the DW-MR images. Then, we registered the estimated tensor fields using the 4 th -order tensor-field group-wise registration method presented in Chapter 7. Figure 8-2 shows two corresponding slices of the control and an injured cord data set. In this figure, the orientation of the peaks in the estimated 4 th -order tensors are shown as tubes. By comparing the corresponding S 0 images (shown on the top of Figure 8-2) one cannot easily observe changes in the underlying fiber structures due to the injury. However, by comparing the corresponding fiber orientations we can observe significant changes in the white matter region between the horns (marked with a red circle). Figure 8-3(left) depicts a 3D visualization of the region of interest (ROI) in the white matter region between the horns (shown in pink). In order to perfrom various quantitative comparisons in the ROI, we computed the average normalized diffusivity as the minimizer of the Helinger s distance between the 4 th -order tensors in the ROI. In this region, we Figure 8-1. The acquired S 0 image of a control (left) and three injured rat spinal cords. 113

114 Figure 8-2. Comparison of the fiber orientations estimated in the control and the corresponding registered injured cord dataset. The S 0 images are shown on the top of the figure. expected to find anisotropic diffusivities with fiber orientations predominant in the axial direction. By observing the orientations in Figure 8-2, one can see that the estimated diffusivities from the injured dataset are less coherent than those in the control spinal cord. In order to compare these orientation plots we computed the angle between the peaks of the average diffusivity in the ROI and the results are shown in Figure 8-3. In this plot, there is a clear reduction of the fiber orientation angle in all injured spinal cord cases due to the changes in the underlying structures caused by the injury. The intra-voxel variances were also estimated in the ROI and the histograms of the obtained values are shown on the right plot of the same figure. This plot also shows a reduction of the variance after the injury which corresponds to a drop of the anisotropy caused by the injury. In all the above experiments we derived either a scalar or an orientational quantity in order to compare the spinal cord data sets. However, one can use all the information included in the 15 real-valued parameters of our model, which fully characterize the 114

115 Figure 8-3. Visualization of the ROI (shown in pink) in 3D. The plots show comparisons between the fiber orientation angle of the average diffusivity in the ROI and the histogram of variances in the ROI. corresponding 4 th -order tensors. In the ROI, we treated the 15 coefficients as elements of R 15 and we constructed a covariance matrix for each data set. These matrices are symmetric and positive-definite and therefore, we can employ the Riemannian metric of P 15 Pennec et al. (2005) in order to compute distances between the datasets and various statistics (e.g. Principal Geodesic analysis Fletcher and Joshi (2004)). Figure 8-4 shows the table of Riemannian distances between the datasets. This table was then used in the Aglomerative clustering technique Duda et al. (2001); Gil-Garcia et al. (2006) to construct the hierarhical tree shown on the right of the same figure. In this plot, the distance between the branches shows the affinities between the given datasets. In this case the three injured datasets were clustered together whose Riemannian distance from the control data set is significantly larger than the distance between the injured data sets. Finally, we extracted tractosemas from the control and injured rat spinal cord datasets using the technique presented in Chapter 6. Figure 8-5 shows the Cornu Posterius region in one of the control and one of the injured spinal cords. A variety of different fiber structures are shown (single bundles, crossings, branchings). In order to compare the estimated structures we plotted the average percentage of tractosemas with 1,2,3... peaks found in all control and injured sets. By observing the plot, one can see quantitative differences in control and injured cords in terms of the distribution of the number 115

116 Figure 8-4. Quantitative comparison of the rat spinal cord dataset using the Riemannian metric of the positive definite matrices. The table shows the Riemannian distances between the covariance matrices. The corresponding hierarchical dendrogram was computed using the Riemannian distances. Figure 8-5. Tractosemas extracted from a control (A) and an injured (B) cat s spinal cord dataset. C) comparison of the number of peaks in the estimated tractosemas. of peaks. This result is consistent with the findings in the previous experiments and demonstrate that there is clear reduction in the diffusion observed in the datasets form injured spinal cords. 116