NEURONAL FIBER TRACKING IN DT-MRI

Transcription

1 NEURONAL FIBER TRACKING IN DT-MRI By TIM E. MCGRAW A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2002

2 Copyright 2002 by Tim E. McGraw

3 For my wife, Jo, and my mother, Patti.

4 ACKNOWLEDGMENTS I would like to express thanks to everyone who made this research possible. First, thanks go to Dr. Baba C. Vemuri for serving as the chairman of my thesis committee, and being available for consultation and advice. Thanks go, also, to Dr. Jörg Peters for serving on my committee and imparting the graphics knowledge required for the visualization aspect of this project. Also, many thanks go to the members of the Computer Vision, Graphics and Medical Imaging (CVGMI) group, especially Zhizhou Wang, Jundong Liu and Fei Wang, for their assistance in preparing this work. I am grateful to the faculty members of the UF Mathematics Department, Dr. Yunmei Chen, for serving on my committee, and Dr. Murali Rao, for advisement and encouragement. I would also like to acknowledge the McKnight Brain Institute for providing data, analysis and validation of results. Thanks go to Dr. Tom Mareci, Dr. Steve Blackband, Dr. Paul Reier, and Everen Ozarslan. This research was funded in part by the NIH grant RO1-NS iv

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS iv LIST OF FIGURES vii KEY TO ABBREVIATIONS viii ABSTRACT ix CHAPTERS 1 INTRODUCTION Overview Contributions Outline of thesis BACKGROUND OVERVIEW DT-MRI Data Acquisition Overview of Diffusion Overview of MR Imaging DT-MRI Acquisition Stejskal-Tanner Equation PDE Based Scalar-Valued Image Restoration Linear Filtering Nonlinear Filtering Perona-Malik Tensor anisotropic diffusion ALM Variational Formulation Membrane spline Thin-Plate spline Total Variation Image Restoration PDE Based Vector-Valued Image Restoration Vector-Valued Diffusion Riemannian Metric Based Anisotropic Diffusion Beltrami Flow Color Total Variation v

6 3 DT-MRI IMAGE RESTORATION Noise Model Restoration Formulation NEURONAL FIBER TRACKING Overview of Neuronal Fiber Tracking Formulation NUMERICAL METHODS Data Denoising Fixed-Point Lagged-Diffusivity Discretized Equations Fiber Regularization Lagged-Diffusivity Crank-Nicholson Method NEURONAL FIBER VISUALIZATION Rendered Ellipsoids Streamtubes Line Integral Convolution Particles EXPERIMENTAL RESULTS Data Denoising Fiber Tracking LIC Streamtubes Particles CONCLUSIONS AND FUTURE WORK Conclusions Future Work Noise Model For Measured Data Quantitative Validation Robust Regression Non-Tensor Model of Diffusion Automatic Region-Of-Interest Extraction APPENDIX DERIVATION OF EULER-LAGRANGE CONDITIONS REFERENCES BIOGRAPHICAL SKETCH vi

7 LIST OF FIGURES Figure page 2.1 Diffusion Ellipsoid TV(f1) >TV(f2) =TV(f3) Noisy image (left) and restored (right) by TV norm minimization LIC visualization of synthetic field FAimage of coronal slice of raw rat brain data FAresults for smoothed data LIC fiber tracts in coronal slice of smoothed rat brain data LIC fiber tracts in axial slice of smoothed rat brain data Comparison of fluoroscopic image (left) with extracted streamtubes (right) Axial view of streamtubes in corpus callosum Details of corpus callosum Sequence from Fiber visualization (2 seconds between images) vii

8 KEY TO ABBREVIATIONS CNS: central nervous system DT: diffusion tensor DTI: diffusion tensor imaging FA: fractional anisotropy LIC: line integral convolution MRI: magnetic resonance imaging PDE: partial differential equation RF: radio frequency ROI: region of interest TV: total variation viii

9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science NEURONAL FIBER TRACKING IN DT-MRI By Tim E. McGraw December 2002 Chair: Baba C. Vemuri Major Department: Computer and Information Sciences and Engineering Diffusion tensor imaging (DTI) can provide the fundamental information required for viewing structural connectivity. However, robust and accurate acquisition and processing algorithms are needed to accurately map the nerve connectivity. In this thesis, we present a novel algorithm for extracting and visualizing the fiber tracts in the CNS, specifically the spinal cord. The automatic fiber tract mapping problem will be solved in two phases, namely a data smoothing phase and a fiber tract mapping phase. In the former, smoothing is achieved via a weighted TV-norm minimization, which strives to smooth while retaining all relevant detail. For the fiber tract mapping, a smooth 3D vector field indicating the dominant anisotropic direction at each spatial location is computed from the smoothed data. Neuronal fibers are traced by calculating the integral curves of this vector field. ix

10 Results are expressed using three modes of visualization. Line integral convolution (LIC) produces an oriented texture which shows fiber pathways in a planar slice of the data. A streamtube map is generated to present a three-dimensional view of fiber tracts. Additional information, such as degree of anisotropy, can be encoded in the tube radius, or by using color. A particle system form of visualization is also presented. This mode of display allows for interactive exploration of fiber connectivity with no additional preprocessing. x

11 CHAPTER 1 INTRODUCTION 1.1 Overview The understanding of neurological function, and medical diagnosis of disease and injury require knowledge of the structural organization of the brain. Magnetic Resonance Imaging provides a non-invasive means of studying anatomical connectivity. This connectivity, in the form of axonal nerve fiber bundles, can only be measured indirectly. The presence of these fibers must be inferred from the behavior of water molecules near the tissue. Diffusion tensor magnetic resonance imaging (DT-MRI) is a method of measuring the rate of water diffusion in biological structures. A diffusion tensor at each location on a regular lattice describes a volumetric average of the directional properties of diffusion within each voxel. The observation that diffusion is anisotropic in areas of white-matter fiber bundles allows tracking of fibers through the lattice. However, acquisition noise corrupts the data measurements. Voltage variations in the receiving coil of the MRI machine due to thermal noise are a major source of signal degradation. The noise is accepted to be zero-mean, and additive in nature. Integrating a fiber path over a noisy field is an ill-posed problem. The goals of the work presented in this thesis are to trace fibers through an ideal, unknown field given only noisy observations, and to visualize the fibers in a useful way. 1

12 2 The fiber tracts are estimated in a two stage process, (a) smoothing of the raw directional images acquired for varying magnetic field strengths and estimating the diffusion tensor, D, field over the 3D image lattice, (b) computing the dominant eigenvector field from this regularized D and estimating regularized streamlines/integral curves as the desired fiber tracts. Computed fibers must be visualized in a way that is useful for anatomical study or medical diagnosis. Existing vector field visualization techniques, such as streamlines/streamtubes can be used to visualize fibers. These techniques can also be modified to convey more information about diffusion by incorporating quantities derived from the tensor. Here, we adapt line integral convolution (LIC), streamtubes, and particles to the task of tensor field visualization. We will review the literature on the raw data smoothing, streamline generation, and visualization techniques where appropriate. 1.2 Contributions Although is well known that the DT-MRI data requires denoising prior to fiber tracking, previous work has concentrated on smoothing the vector field of dominant diffusion directions, or even the diffusion tensor itself. Employing these methods requires that the diffusion tensor, and possibly eigensystems of the tensor be computed from noisy data. Results of these calculations may be meaningless due to properties of the noise, and the diffusion tensor calculation. We propose to smooth the data before these calculation to provide for more physically meaningful results.

13 3 A new application of particle-based visualization is presented. This technique has previously been used in fluid mechanics for visualization of velocity fields. Here we will adapt the technique to tensor field visualization. This technique has no associated preprocessing time, and allows real-time interactive visualization of the reconstructed data. Many other techniques require extensive computing time for streamline integration prior to visualization. 1.3 Outline of thesis The rest of this thesis is organized as follows. In Chapter 2 we will discuss the diffusion process and give a short introduction to the DT-MRI data acquisition process. An overview of PDE based image denoising will follow. Some linear and nonlinear techniques for scalar and vector-valued images will be reviewed. Variational techniques for formulating image denoising, and the TV norm will be presented. Chapter 3 will describe our vector-valued image restoration technique, a weighted TV norm minimization. The noise model used to formulate this technique is discussed, as well as the motivation for modifying the TV norm usually used for vector-valued images. In Chapter 4 we present the process of tracking neuronal fiber bundles through the restored DT-MRI data. Previous methods of fiber tracking will be discussed. A quantification of diffusion tensor anisotropy will be given. In this chapter streamline regularization will be presented as a way of enforcing a smoothness constraint on the fibers.

14 4 Chapter 5 will show the linearization and discretization of the PDEs involved in data denoising and fiber regularization. The numerical methods used to solve linear equations will be described. In Chapter 6 we will present visualization techniques for DT-MRI data. These include rendered ellipsoids, streamtubes, LIC and particles. In Chapter 7 experimental results will be reported. The data denoising results for spinal cord and brain data sets will be presented. Fiber tracking results using several visualization techniques will be presented. Conclusions and possible directions for future work will be discussed in Chapter 8.

15 CHAPTER 2 BACKGROUND OVERVIEW 2.1 DT-MRI Data Acquisition Fundamental advances in understanding living biological systems require detailed knowledge of structural and functional organization. This is particularly important in the nervous system where anatomical connections determine the information pathways and how this information is processed. Our current understanding of the nervous system is incomplete because of a lack of fundamental structural information [21] necessary to understand function. In addition, understanding fundamental structural relationships is essential to the development and application of therapies to treat pathological conditions (e.g. disease or injury). However, most imaging methods give only an anatomically isolated representation of living tissue because the images do not contain connectivity information. Such information would allow the identification and correlation of system elements responding during function. For example in brain trauma, the relationship between anatomy and behavior will only become apparent when we are able to discriminate the afferent nerve fiber pathways transmitting the sensation from a stimulus to the brain or the efferent pathways transmitting impulses from the brain area controlling behavior. 5

16 6 Research during the last few decades has shown that the central nervous system is able to adapt to challenges and recover some function. However, the structural basis for this adaptive ability is not well understood. For the entire central nervous system, understanding and treating evolving pathology, such as brain trauma, depends on a detailed understanding of the anatomical connectivity changes and how they relate to function. Recently MR measurements have been developed to measure the tensor of diffusion. This provides a complete characterization of the restricted motion of water through the tissue that can be used to infer tissue structure and hence fiber tracts. In a series of papers, Basser and colleagues [2, 3, 4, 5, 6, 36] have discussed in detail general methods of acquiring and processing the complete apparent-diffusion-tensor of MR measured translational self-diffusion. They showed that directly measured diffusion tensors could be recast in a rotationally invariant form and reduced to parametric images that represent the average rate of diffusion (tensor trace), diffusion anisotropy (relationship of eigenvalues), and how the diffusion ellipsoid (eigenvalues and eigenvectors) can be related to the laboratory reference frame. The parametric images [5, 35] of volume ratio, fractional anisotropy, and lattice anisotropy index represent scalar measures of diffusion that are independent of the lab reference frame and subject orientation. Therefore, these measures can be used to characterize the tissue pathology, e.g., ischemia, independent of the specific frame of reference used to acquire the images. The development of diffusion tensor acquisition, processing, and analysis methods provides the framework for creating fiber tract maps based on this complete

17 7 diffusion tensor analysis [19, 24, 29, 30]. This has been used to produce fiber tract maps in rat brains [30, 47] and to map fiber tracts in the human brain [24] then the first steps were taken to relate this structural connectivity to function [19]. The directional properties of diffusion can be characterized by a diffusion tensor, a 3 3 symmetric matrix of real values. In order to calculate the 6 independent components of the tensor, the subject is imaged in seven different directions with several magnetic field strengths. The relationship S = S 0 exp( ij b ijd ij ) allows the diffusion tensor, D, and the T-2 weighted image S 0 to be calculated given the samples S and field gradient strength, b. Previous work has concentrated on smoothing the field of eigenvectors of D. More recent work [14] has formulated regularization techniques for the tensor field itself, even constraining the resulting tensors to be positive-definite. Here we will take the approach of smoothing the observed vector-valued image S prior to calculating D. This decision to smooth in this manner will be justified in Chapter 3. In summary, the anisotropy of water translational diffusion can be used to visualize structure in the brain and provides the basis for a new method of visualizing nerve fiber tracts. Initial results have been very encouraging and suggest that this approach to fiber mapping may be applied to a wide range of studies in living subjects. However, it is essential to optimize the acquisition and processing algorithms for fiber tract mapping and validate the results relative to known measures of fiber tracts.

18 8 Figure 2.1: Diffusion Ellipsoid Overview of Diffusion Random molecular motion (Brownian motion) can cause transport of matter within a system. Within a volume of water, molecules freely diffuse in all directions. The water abundant in biological systems is also subject to such stochastic motion. The properties of the surrounding tissue can affect the magnitude of diffusion, and the directional properties as well. Tissue can form a barrier to diffusion, restricting molecular motion. Within an oriented structure, such as a bundle of axonal fibers, diffusion can be highly anisotropic. The white matter of the brain and spinal cord is characterized by many such bundles. The directional properties of diffusion can be characterized by a tensor. The diffusion tensor, D, is a symmetric, positive-definite 3 3 matrix. We will make use of the eigenvalues and eigenvectors of this tensor, sorting the eigenvalues (λ 1, λ 2, λ 3 ) from largest to smallest, and labelling the corresponding unit eigenvectors (e 1, e 2, e 3 ). The eigenvalues represent the magnitude of diffusion in the direction of their corresponding eigenvector. For isotropic diffusion λ 1 = λ 2 = λ 3. A

19 9 popular representation for describing anisotropic diffusion is the diffusion ellipsoid. This ellipsoid is the image of the unit sphere under the transformation defined by the tensor, D. The eigenvectors of D form an orthogonal basis, representing the orientation of the ellipsoid. The length of each axis of the ellipsoid is the corresponding eigenvector. For isotropic diffusion, the diffusion ellipsoid is a sphere Overview of MR Imaging In this section we will present a brief overview of the MRI acquisition process. A detailed treatment of the subject was done by Haacke et al. [23]. The protons in the nuclei of atoms align their axis of spin with the direction of an applied magnetic field. The magnetic field also induces a wobble, known as precession, in the spin of the protons. This frequency, the resonant frequency, is proportional to the strength of the applied field. For protons the resonance frequency lies in the RF range. In the MRI machine, a field B 0 is applied through out the imaging process. The direction of this field defines the axial direction of the image. Protons will absorb energy from an RF pulse of the resonance frequency, and tip away from the direction induced by B 0. The amount of tip is proportional to the pulse duration. The RF pulse also causes the protons to precess in phase with each other. This pulse is called the B 1 field. When the B 1 transmitter is turned off, the absorbed energy at the resonant frequency is re-emitted by the protons. This occurs as the spins, tipped by B 1 return to their previous B 0 alignment. The time constant associated with this

20 10 exponential process is known as the T 1 relaxation time. The protons precessions also dephase exponentially with time constant T 2. The final image contrast is influenced by strength, width and repetition time of the RF pulses in the B 1 signal. By spatially varying the intensities of B 0 and B 1, position information is encoded. For instance, specially designed magnets add a gradient field to B 0. This causes the proton resonance frequency to be a function of axial position. The frequency of B 1 can then be chosen to tip protons within a chosen slice. To encode x, y position within a slice, two additional additional gradient fields are employed. The first gradient, G y is pulsed, causing a phase variation, just as in T 2 relaxation. The phase variation is a function of position in the y direction. A perpendicular gradient, G x is then applied, changing resonance frequencies in the x direction. A 2D Fourier transform reconstructs the image of each slice from the data in the spatial frequency domain DT-MRI Acquisition By carefully designing gradient pulse sequences, the measured signal from protons in water molecules undergoing diffusion can be attenuated. The first gradient pulse induces a known phase shift in proton precession. After some delay, a second gradient pulse is applied, inducing the opposite phase shift. Protons which have not moved between the two gradient pulses are returned to their previous phase. Protons belonging to molecules which have changed location have some net change in phase, changing their T 2 relaxation time. Conventional MRI images of

21 11 white matter in the brain suggest a material of homogeneous composition. The fibrous nature of white matter is apparent in DT-MRI however Stejskal-Tanner Equation The intensity of the received signal, S, at each voxel depends on the properties of the diffusion-encoding gradient, and the apparent diffusion tensor, D at that location. The Stejskal-Tanner equation relates all of these quantities ln ( S 3 3 ) = b i,j D i,j (2.1) S 0 i=1 j=1 In 2.1, S 0 is the intensity with no diffusion-encoding gradient present, and b is a matrix characterizing the gradient pulse sequence. The physics behind 2.1 is beyond the scope of this thesis. The motivated reader may refer to the work of Haacke et al. [23] for more details. This imaging process must be performed with 7 noncoplanar gradient directions in order to fully generate a diffusion tensor image. Multiple samples, usually 3 or 4 are taken for each gradient direction. ln S 1. ln S 28 = 1 b 1 xx b 1 yy b 1 zz 2b 1 xy 2b 1 yz 2b 1 xz b 28 xx b 28 yy b 28 zz 2b 28 xy 2b 28 yz 2b 28 xz... ln S 0 D xx D yy D zz D xy D yz (2.2) D xz

22 12 The overconstrained linear system, is solved for S 0 and the elements of the symmetric tensor D by a least squares linear regression. 2.2 PDE Based Scalar-Valued Image Restoration Image data smoothing or denoising is a fundamental problem in image processing. Image denoising (noise removal) is a technique that enhances images by attempting to reverse the effects of degradations occurring during acquisition or transmission. Image noise makes it difficult to perform other processing tasks such as edge-detection, segmentation, or in our case, fiber tracking. For this reason, denoising is the first step in most image analyses. The goal of image denoising is to recover an unknown, ideal image given some observed image. In our case, we wish to recover the smooth S values from which the DT image is calculated. The most common degradation source is the noise from the image acquisition system and is commonly modeled by additive Gaussian random noise. In the following, we will briefly review representative schemes, specifically, partial differential equation (PDE) based methods that lend themselves to fast numerical implementations. Image denoising can be formulated using variational principles which in turn require solutions to PDEs. Recently, there has been a flurry of activity on the PDE-based smoothing schemes. For scalar valued image smoothing using nonlinear diffusion filters with scalar diffusivity coefficient, we refer the reader to the following articles and references therein [1, 11, 12, 15, 28, 31, 32, 34, 43, 45]. Anisotropic diffusion filters that use a tensorial diffusivity parameter

23 13 were introduced in Weickert [45]. These filters can be tailored to enhance image structures (edges, parallel lines, curves etc.) that occur in preferred directions Linear Filtering Linear filters are a simple and efficient means of removing noise from images. One such filter may be implemented by the process of isotropic diffusion. This diffusion process is governed by the heat equation I(x, t) t = 2 I(x, t) I(x, 0) = I 0 (x) (2.3) In the same way that a heated plate will seek an equilibrium state of a smooth temperature gradient, so will an image evolved according to 2.3 smooth out discontinuities in intensity. It can be shown that isotropic diffusion is equivalent to convolving the image with a gaussian kernel. Taking the Fourier transform of 2.3 with respect to x, and defining F(I(x, t)) = U(w, t). U(w, t) t = w 2 U(w, t) U(w, 0) = U 0 (w) (2.4) The solution to 2.4 is U(w, t) = U 0 (w)e w2 t (2.5)

24 14 The solution to 2.3 is then the inverse Fourier transform of I(x, t) = I 0 e x 2 2σ t 2 2πσt σ 2 t = 2t (2.6) Clearly, the linear diffusion process continues until a I(x, t) becomes a constantvalued image. By 2.6 we can consider the effect of 2.3 as t to equivalent to convolving the original image with gaussian kernels of ever-increasing variance. Since the gaussian kernel is separable, this type of filter is simple to implement for images of arbitrary dimension. Although the images produced by this simple filtering technique show a reduction in the high frequency noise, there is also the unwelcome effect of blurred edges and lost details. The linear filter still has applications in image resampling, and generating scale-space image representations, such as image pyramids Nonlinear Filtering The nonlinear filters described in this chapter were designed to overcome the inter-region blurring effect of linear filters. Nonlinear filters cannot be modelled with gaussian convolution. The median filter is a nonlinear filter with a simple implementation. A median filter replaces each intensity value in an image with the median of the neighboring values. The size of this neighborhood determines the amount of smoothing.

25 15 There are numerous models of nonlinear diffusion for image smoothing. To implement the filters presented in the rest of this section we must numerically solve a PDE Perona-Malik An anisotropic diffusion process which inhibits blurring at edges was formulated by Perona and Malik [34] by modifying 2.3. I(x, t) t = div(c(x) I(x, t)) I(x, 0) = I 0 (x) (2.7) When comparing 2.7 to 2.3 recall that 2 I I div( I). By using a diffusion coefficient which is a decreasing function of I(x, t), such as c(x) = I(x, t) (2.8) we can inhibit diffusion at locations of high image gradient, presumed to be edges. By adding a reactive term to 2.7 we can actually enhance edges. The presence of I(x, t) in 2.8 makes 2.7 nonlinear. The diffusion coefficient serves only to slow diffusion at edges. The steadystate (t ) solution of 2.7 is still a constant valued image. A constraint can be added to impose the condition that the smoothed image be close to the original image. By introducing a reaction term to the diffusion equation we can impose a

26 16 data fidelity requirement I(x, t) t = div(c(x) I(x, t)) + µ(i 0 (x) I(x, t)) I(x, 0) = I 0 (x) (2.9) we penalize solutions that differ from the initial image. The parameter µ controls the degree of smoothing in the final image. The steady-state solution of 2.9 is nontrivial, so we no longer need a stopping time Tensor anisotropic diffusion An alternative to merely slowing diffusion at edges is to align the direction of diffusion to be parallel with edges in the image. Weickert [45] controls diffusion direction by replacing the scalar c(x) in 2.7 with a diffusion tensor-valued function. I(x, t) t = div(d(x) I(x, t)) I(x, 0) = I 0 (x) (2.10) The function, D(x) produces tensors such that the unit eigenvector, e 1 is parallel with I and e 2 is perpendicular to I. The eigenvectors λ 1 = g( I ) and λ 2 = 1 are suggested to generate anisotropic tensors whose representative ellipsoid has the major axis parallel with image edges. In this way, diffusion near edges still occurs, mostly along the edge. By limiting smoothing across the edge, edge location and intensity can be preserved.

27 ALM Alvarez, Lions, and Morel [1] propose a nonlinear parabolic PDE of the form I(x, t) t I(x, t) = g( I) I div( I(x, t) ) I(x, 0) = I 0 (x) (2.11) The effect is that I is smoothed in a direction orthogonal to the gradient at each point. This is best considered in a level-set framework. Embedding I in R 3 by considering t to be the third dimension, and φ(t = 0) to be the isocontours of I we have φ(x(t), t) t + φ(x(t), t) x (t) = 0 (2.12) Recall the definitions of level-set normal,n = φ φ and curvature, κ = div( φ φ ). The normal component of the velocity will be v n = φ φ x (t) (2.13) So 2.12 becomes φ(x(t), t) t = v n φ (2.14) Letting v n be proportional to κ we obtain Curve-shortening flow of the isocontours of φ smoothes the image I Variational Formulation The problem of image denoising is ill-posed, as are many inverse problems. Since different pixel intensities could result in the same value when corrupted by noise, the solution to the denoising problem is not unique. The technique of

28 18 Tikhonov regularization involves incorporating some additional knowledge about the ideal image, similar to the prior distribution involved in a Bayesian analysis. In Tikhonov regularization, the solution space is restricted by posing the problem as a minimization problem. The function (image) which minimizes some stabilizing functional can be proven to be unique for appropriate choice of functional. We will consider problems of the form min I (µe s (I) + E d (I)) (2.15) The functionals, E involved in image smoothing often correspond to physical models, and usually represent energy. For instance, the energy associated with data fidelity is E d = 1 k i I 0 I 2 (2.16) 2 i A physical analogy for the data constraint is a number of springs between the initial image, I 0 and the ideal image, I. The second energy term, E s, represents the internal potential energy of the surface described by the image. The regularization parameter, µ, is introduced to control the amount of smoothing Membrane spline A first-order stabilizer, stretching energy (arc-length), is one possible stabilizer E ms = Ix 2 + I2 y dx dy (2.17) R 2

29 19 The Euler-Lagrange condition for the minimization of 2.17 is 2 I = 0, the steadystate solution to 2.3. In fact, most of the filters considered so far can be obtained from a variational principle. By setting E s = E ms in 2.15 we obtain a membrane spline solution. In this case, µ controls the tension of the spline surface Thin-Plate spline Using a second-order stabilizer bending energy (curvature) E tps = Ixx 2 + 2Ixy 2 + Iyy 2 dx dy (2.18) R 2 as a stabilizer we obtain a thin-plate spline solution. For this spline, µ controls the stiffness of the spline surface. The Euler-Lagrange condition for the minimization of 2.18 is 4 I = Total Variation Image Restoration Another side-constraint that can be used as an energy functional is called the total variation (TV) norm. This norm represents oscillation. In our case, image noise is considered to be wrinkling of the surface described by the image. Minimizing the TV norm produces very smooth images while permitting sharp discontinuities between regions [40]. The TV norm is formulated by TV n,1 (I(x)) = I(x) dx, Ω R n (2.19) Ω The TV norm is essentially the L 1 norm of the image gradient. At discontinuities, the weak derivative DI(x) to calculate the TV norm. For a piecewise continuous function, the TV norm is then the sum of the TV norm

30 20 of each continuous piece plus the sum of the absolute values of the jumps. For example, functions f1 and f2 in 2.2 have the same TV norm. The oscillatory function, f1, has a higher TV than the function with a discontinuity, f3. Since Figure 2.2: TV(f1) > TV(f2) = TV(f3) most images consist of piecewise smooth regions separated by discontinuities (edges), this is a useful model for image denoising. The Euler-Lagrange condition for the minimization of 2.19, written in gradient-descent form is I(x, t) t I(x, t) = div( I(x, t) ) I(x, 0) = I 0 (x) (2.20) An example of image denoising by TV norm minimization is shown in 2.3. The restored image on the right is characterized by a high degree of intra-region smoothing, and edges have also been preserved.

31 21 Figure 2.3: Noisy image (left) and restored (right) by TV norm minimization. 2.3 PDE Based Vector-Valued Image Restoration The main application of vector-valued image restoration has been the restoration of color images. The simplest implementation of vector-valued smoothing is uncoupled smoothing. By treating each component of the vector field as an independent scalar field, we can proceed by smoothing channel-by-channel using one of the methods presented in section 2.2 for scalar image smoothing. This can, however, result in a loss of correlation between channels as edges in each channel may move independently due to diffusion. To prevent this, there must be coupling between the channels. There are many other PDE-based image smoothing techniques which we will not cover here, but will refer the reader to Weickert [45] and Caselles et al.[10].

32 Vector-Valued Diffusion Whitaker and Gerig introduced anisotropic vector-valued diffusion which was a direct extension of the work of Perona and Malik [34]. The equations I n (x, t) t = div(c(i) I n (x, t)) I n (x, 0) = I n,0 (x) (2.21) are coupled through the function C, and can be written in vector form as I(x, t) t = (c(i) I(x, t)) I(x, 0) = I 0 (x) (2.22) Riemannian Metric Based Anisotropic Diffusion Sapiro et al. [41] introduced a selective smoothing technique where the selection term is not simply based on the gradient of the vector valued image but based on the eigen values of the Riemannian metric of the underlying manifold. In 2 dimensions, the underlying manifold can be thought of as the parametric surface described by the image I(x). By geometrically smoothing this surface, we also smooth the vector-valued image. The entries of the metric tensor, G, are defined by g i,j = I I (2.23) x i x j The largest and smallest eigenvalues of G, (λ +, λ ) and their corresponding eigenvectors (ξ +,ξ ) describe surface properties of the Riemannian manifold. The

33 23 degree and direction of maximal change are given by λ + and ξ +. Smoothing is achieved by evolution in the direction of minimal change, that is, along edges in the vector-valued image I(x, t) t = g(λ +, λ ) 2 I(x, t) ξ I(x, 0) = I 0 (x) (2.24) Beltrami Flow More recently, Kimmel et al. [25] presented a very general flow called the Beltrami flow as a general framework for image smoothing and show that most flow-based smoothing schemes may be viewed as special cases in their framework. I i (x, t) t = 1 G 2 µ=1 x µ ( G 2 ν=1 G I i x ν ) I(x, 0) = I 0 (x) (2.25) where G is the metric tensor, and G is the determinant of the metric tensor. The Beltrami flow can be thought of as nonlinear diffusion on a manifold, where G acts as an edge-detecting diffusion coefficient Color Total Variation Blomgren and Chan [8] introduced the T V n,m norm for vector valued images. TV n,m (I(x)) = m [TV n,1 (I i )] 2 (2.26) i=1

34 For m = 1, 2.26 reduces to the scalar TV norm The Euler-Lagrange condition for the minimization of 2.26, written in gradient descent form is 24 I i (x, t) t = TV n,1(i i ) TV n,m (I) ( I i I i ) I(x, 0) = I 0 (x) (2.27) This was shown to be quite effective for color images, preserving edges in the color space while attenuating noise. However, for much larger dimensional data sets (m=7) as in the work proposed here, the Color TV method becomes computationally very intensive and thus may not be the preferred method in such applications.

35 CHAPTER 3 DT-MRI IMAGE RESTORATION 3.1 Noise Model The degradation associated with the measurement of the S values is modelled by an additive gaussian process. Let Ŝ(X) be the vector valued image that we want to smooth where, X = (x, y, z) and let S(X) be the unknown smooth approximation of the data that we want to estimate. We have Ŝ(X) = S(X) + η. Although we can consider the noise to be on the components of the tensor, D, the form of this noise is no longer so simple. In fact, even computing D from S, using the Stejskal-Tanner relation may be meaningless. Substituting the noise model into 2.1 ln S + η = ln S b i,j D i,j (3.1) i=1 j=1 The physical principles governing diffusion do not allow for negative values of S, as evidenced by the dependence of 3.1 on the natural logarithms of S and S 0. Low-intensity S measurements may be overwhelmed by noise. However, since η is a zero-mean distribution, these measurements may become negative. Clearly, we cannot calculate D in this case. Nor should we propagate the noise model through the Skeskal-Tanner equation in order to write D as a function of η. We propose to instead smooth the vector of S measurements before any further analysis. 25

36 Restoration Formulation Smoothing the raw vector valued image data is posed as a variational principle involving a first order smoothness constraint on the solution to the smoothing problem. We propose a weighted TV-norm minimization for smoothing the vector valued image S. The variational principle for estimating a smooth S(X) is given by min E(S) = [g(λ +, λ ) S Ω 7 S i + µ 2 i=1 7 S i Ŝ i 2 ]dx (3.2) i=1 where, Ω is the image domain and µ is a regularization factor. The first term here is the regularization constraint on the solution to have a certain degree of smoothness. The second term in the variational principle makes the solution faithful to the data to a certain degree. The weighting term in this case g(s) = 1/(1 + s) where s = F A is the fractional anisotropy defined by Basser et al. [2]. This selection criteria preserves the dominant anisotropic direction while smoothing the rest of the data. Note that since we are only interested in the fiber tracts corresponding to the streamlines of the dominant anisotropic direction, it is apt to choose such a selective term. Here we have used a different TV norm than the one used by Blomgren and Chan [8]. The T V n,m norm is an L 2 norm of the vector of T V n,1 norms for each channel. We use the L 1 norm.

37 27 The gradient descent of the above minimization is given by S i t = div ( ) g(λ+, λ ) S i µ(s i S i Ŝi) i = 1,..., 7 S i n + Ω R = 0 and S(x, t = 0) = Ŝ(x) (3.3) The derivation of equation 3.3 from 3.2 is presented in section The use of a modified TV norm in 3.2 results in a looser coupling between channels than the use of the true T V n,m norm would have. This reduces the numerical complexity of 3.3 and makes solution for large data set feasible. Note that the TV n,m norm appears in the gradient descent solution 2.27 of the minimization problem. Consider that our data sets consist of 7 images, corresponding to gradient directions. Each of these images consists of several samples (usually 3 or 4) corresponding to different gradient strengths. Calculating the TV n,m norm by numerically integrating over the 3-dimensional data set at each step of an iterative process would have been prohibitively expensive.

38 CHAPTER 4 NEURONAL FIBER TRACKING 4.1 Overview of Neuronal Fiber Tracking Water in the brain preferentially diffuses along white matter fibers. By tracking the direction of fastest diffusion, as measured by MRI, non-invasive fiber tracking of the brain can be accomplished. Fibers tracks maybe constructed by repeatedly stepping in the direction of fastest diffusion. The direction along which the diffusion is dominant corresponds to the direction of eigen vector corresponding to the largest eigenvalue of the tensor D. In Conturo et al., [18], fiber tracks were constructed by following the dominant eigenvector in 0.5 mm steps until a predefined measure of anisotropy fell below some threshold. This usually occurred in grey matter. The tensor, D, was calculated at each step from interpolated DT-MRI data. This tracking scheme is primarily based on heuristics and is not grounded in well founded mathematical principles. Mori et al. [30] achieved fiber tracking by using several heuristics. The tracking algorithm starts from a voxel center and proceeds in the direction of the major axis of the diffusion ellipsoid. When the edge of the voxel is reached, the direction is changed to that of the neighboring voxel. Tracking stops when a measure of adjacent fiber alignment crosses a given threshold. The scheme however is resolution dependent since the MRI data only reflects average axonal orientation 28

39 29 within a voxel. Small fibers adjacent to each other may not be distinguished. Another problem occurs with branching fibers. This method will only track one path. In this situation, multiple points within a bundle may be independently tracked. Westin et al. [46] reported techniques for processing DT-MRI data using tensor averaging. Diffusion tensor averaging is an interesting concept but does not address the issue of estimating a smooth tensor from the given noisy vectorvalued data. More recently, Poupon et al. [38] developed a Bayesian formulation of the fiber tract mapping problem. Prior to mapping the fibers, they use robust regression to estimate the diffusion tensor from the raw vector valued image data. No image selective smoothing is performed in their work prior to application of the robust regression for estimating the diffusion tensors. Coulon et al. [20] determined fibers tracts after smoothing the eigen values and vectors. Once again, this scheme is faced with the same problem i.e., the eigen vector and the eigen values are computed from a noisy tensor field and hence may not be meaningful at several locations in the field. Parker et al. [33] also presented a follow up article on fiber tract mapping wherein, they use the idea of the fast marching method from Sethian et al. [42] for growing seeds initialized in the smoothed dominant eigen vector field. Batchelor et al. [7] reported an interesting fiber tract mapping scheme wherein, they produce a map indicating the probability of a fiber passing through each location in the field. However, they do not address the issue of computing the diffusion tensor from a noisy set of vector valued images. This is a very important issue and should not be overlooked as is demonstrated in the preliminary results of this proposal.

40 30 Given the dominant eigen vector field of the diffusion tensor in 3D, tracking the fibers (space curves) tangential to this vector field is equivalent to finding the stream lines/integral curves of this vector field. Finding integral curves of vector fields is a well researched problem in the field of Fluid Mechanics [17]. The simplest solution would be to numerically integrate the given vector field using a stable numerical integration scheme such as a fourth order Runge-Kutta integrator [39]. However, this may not yield a regularized integral curve. In the context of the fiber tract mapping, two sub-tasks are involved namely, (1) estimating a denoised diffusion tensor from noisy vector-valued image measurements and (2) estimating the streamlines of the dominant eigen vectors of the diffusion tensor. The denoising involves a weighted TV-norm minimization of the vector valued data S followed by a linear least squares estimation of D from the smoothed S and then an estimation of regularized stream lines of the dominant eigen vectors of D. 4.2 Formulation Diffusion at each point can be characterized by a diffusion ellipsoid. The ellipsoid axes are the eigenvectors of the diffusion tensor. The radii are the corresponding eigenvalues. The shape of the ellipsoid reflects the isotropy of diffusion. Nearly spherical diffusion ellipsoid represent areas of free water, where diffusion is unimpeded. Areas of white-matter fiber bundles have elongated ellipsoids, since water diffusion is restricted in directions perpendicular to fiber

41 31 direction. The phenomenon was quantified by Basser and Pierpaoli [2]. F A = 3 (λ 1 λ) 2 + (λ 2 λ) 2 + (λ 3 λ) 2 2 λ λ2 2 + λ2 3 (4.1) For the stream line estimation problem, we pose the problem in a variational framework incorporating smoothness constraints which regularize the stream lines/integral curve. The variational principle formulation leads to a PDE which can be solved using efficient numerical techniques. The variational principle 1 min E(p) = min c (p) + β c c 0 2 c (p) v(c(p)) 2 dp (4.2) where, c(p) = (x(p), y(p), z(p)) T is the integral curve we want to estimate and p [0, 1] is the parameterization of the curve, v(c(p)) is the vector field v restricted to the curve c(p). The first term of 4.2 is a smoothness constraint. By minimizing arc-length we penalize spurious oscillation in the curve. The second term provides data fidelity : we wish for the fiber to be tangent to the vector field at every point along the curve. The parameter, β controls the degree of smoothness of the solution. The gradient descent of (4.2) c t = c (p) kn + β[c (p) V(c(p))c (p)] + βv T (c(p))(c (p) v(c(p))) (4.3)

42 32 where k is the curvature of the space curve, β is a regularization parameter and V T = The transpose of V v 1x v 1y v 1z V = Jv = v 2x v 2y v 2z v 3x v 3y v 3z (4.4) The initial condition, c(p, t = 0) is provided by an ordinary streamline integration routine.

43 CHAPTER 5 NUMERICAL METHODS 5.1 Data Denoising The nonlinear PDE for denoising the raw data is div(g s s ) µ(s s0 ) = 0 (5.1) where g : R 3 R is the selective smoothing and µ is the constant regularization parameter Fixed-Point Lagged-Diffusivity Equation 5.1 is nonlinear due to the presence of s in the denominator of the first term. We linearize 5.1 by using the method of lagged-diffusivity presented by Chan and Mulet [13]. By considering s to be a constant for each iteration, and using the value from the previous iteration we can instead solve 1 s t ( g st + g 2 s t+1 ) + µ(s t+1 s 0 ) = 0 (5.2) Here the superscript denotes iteration number. First, rewrite 5.2 with all of the s t terms on the right-hand side 2 s t+1 + µ st s t+1 = µ st s 0 + g s t g g (5.3) 33

44 Discretized Equations To write 5.3 as a linear system (As t + 1 = y), discretize the laplacian and gradient terms. Using central differences for the laplacian we have 2 s t+1 = s t+1 x 1,y,z + st+1 x,y 1,z + st+1 x,y,z 1 6st+1 x,y,z + st+1 x+1,y,z + st+1 x,y+1,z + st+1 x,y,z+1 (5.4) Define the standard forward, backward and central differences to be + x s = s x+1,y,z s x,y,z + y s = s x,y+1,z s x,y,z + z s = s x,y,z+1 s x,y,z x s = s x,y,z s x 1,y,z y s = s x,y,z s x,y 1,z z s = s x,y,z s x,y,z 1 x s = 1 2 (s x+1,y,z s x 1,y,z ) y s = 1 2 (s x,y+1,z s x 1,y,z ) z s = 1 2 (s x,y,z+1 s x 1,y,z ) (5.5) We can rewrite 5.3 in discrete form using the definitions in 5.5 s x 1,y,z s x,y 1,z s x,y,z 1 + (6 + µ ( xs t ) 2 +( ys t ) 2 +( zs t ) 2 g )s x,y,z s x+1,y,z s x,y+1,z s x,y,z+1 = 1 g (µs0 ( x s t ) 2 + ( y s t ) 2 + ( z s t ) 2 + x g x s t + y g y s t + z g z s t ) (5.6)

45 This results in a banded-diagonal linear system with 7 nonzero coefficients per row. 6 + µ st 0 g µ st 1 g µ st 3 g s t+1 0 s t+1 1. s t+1 n 3 35 = f0 t f1 t. f t n 3 (5.7) where the right-hand side of 5.6 has been replaced with fn. t The matrix in equation 5.7 is symmetric and diagonally dominant. We have successfully used conjugate gradient descent to solve this system. The solution of 5.7 represents one fixed-point iteration. This iteration is continued until s t s t+1 < c, where c is a small constant. 5.2 Fiber Regularization The PDE for fiber regularization is c t = c (p) kn + µ[c (p) V(c(p))c (p)] + µv T (c(p))(c (p) v(c(p))) (5.8) where k is the curvature of the space curve, µ is a regularization parameter and V T = The transpose of V v 1x v 1y v 1z V = Jv = v 2x v 2y v 2z v 3x v 3y v 3z (5.9)

46 36 Using the definitions of tangent, T, curvature, k, and normal, n T = c c, k = T c, n = T T (5.10) we have c t = ( c c ) + µ(c + V T (c v) Vc ) (5.11) Lagged-Diffusivity We linearize 5.11 using the concept of lagged-diffusivity, as we did in the data denoising case. c ) = ( c t+ t ) = c t+ t c t c t ( c (5.12) We can simplify notation by calling the coefficients of c t+ t and c t+ t respectively α t (p) and β t (p) and calling the additive constant K t. c t = α t (p)c t+ t + β t(p)c t+ t + K t (5.13) Crank-Nicholson Method We solve 5.13 using the Crank-Nicholson method. This method achieves stability by using averaged differences to estimate derivatives. For the second derivative term we use 2 c p 2 = 1 2 p 2 (c t(p + p) 2c t (p) + c t (p p) + c t+ t (p + p) 2c t+ t (p) + c t+ t (p p)) (5.14)

47 37 The averaged difference for the first derivative is c p = 1 2 p (c t(p + p) c t (p p) + c t+ t (p + p) c t+ t (p p)) (5.15) In finite-difference form c t+ t (p) c t(p) t = αt(p) 2 p 2 (c t+ t (p + p) 2c t+ t (p) + c t+ t (p p)) + β t(p) 2 p (c t(p + p) c t (p p) + α t(p) 2 p 2 (c t (p + p) 2c t (p) + c t (p p)) + β t(p) 2 p (c t+ t(p + p) c t+ t (p p) + K t (p) (5.16) Equation 5.16 can be expressed as a linear system to be solved for c t+ t. A t c t+ t = M t c t + K t (5.17) This tridiagonal linear system can be solved by Crout s factorization, an optimized LU factorization for this type of matrix.

48 CHAPTER 6 NEURONAL FIBER VISUALIZATION 6.1 Rendered Ellipsoids A very simple visualization strategy is to simply render the diffusion ellipsoid at a subset of data points. Since a 3D field of ellipsoids would occlude each other, this visualization is usually done for 2D slices of data. Additionally, only ellipsoids on a sparse grid can be rendered in order for each ellipsoid to be discerned. This type of visualization can easily become visually cluttered and convey so little information as to be useless. Laidlaw[26], however, successfully incorporated ellipsoids in a layered visualization approach. 6.2 Streamtubes Streamtubes are a three-dimensional analogue to streamlines. In fact, the streamtube is computed by using a streamline as the centerline of the tube. We can use the streamtube diameter to encode some additional information about the tensor field being visualized, such as FA value. Previously, Laidlaw [27] has applied the streamtube visualization approach to DT-MRI. 6.3 Line Integral Convolution It is also possible to visualize the 3D vector field corresponding to the dominant eigenvalues of the diffusion tensor using other visualization methods such as the line integral convolution technique introduced by Cabral et al. [9] a concept 38

49 39 explored in this work as well. The advantage of this visualization technique is that it is well suited for visualizing high density vector fields and does not depend on the resolution of the vector field moreover, it also has the advantage of being able to deal with branching structures that cause singularities in the vector field. Since Figure 6.1: LIC visualization of synthetic field. the fiber direction is parallel to the dominant eigenvector of the diffusion tensor, we can calculate fiber paths as integral curves of the dominant eigenvector field. The stopping criterion is based on FA value. When FA falls below 0.17 we consider the diffusion to be nearly isotropic and stop tracking the fiber at this point. Once the diffusion tensor has been robustly estimated, the principal diffusion direction can be calculated by finding the eigen vector corresponding to the dominant eigen value of this tensor. The fiber tracts may be mapped by visualizing the streamlines through the field of eigen vectors. LIC is a texture-based vector field visualization method. The technique generates intensity values by convolving a noise texture with a curvilinear kernel

50 40 aligned with the streamline through each pixel, such as by I(x 0 ) = s0 +L s 0 L T (σ(s))k(s 0 s)ds (6.1) where I(x 0 ) is the intensity of the LIC texture at pixel x 0, k is a filter kernel of width 2L, T is the input noise texture, and σ is the streamline through point x 0. The streamline, σ can be found by numerical integration, given the discrete field of eigen vectors. The result is a texture with highly correlated values between nearby pixels on the same streamline, and contrasting values for pixels not sharing a streamline. In our case, an FA value below a certain threshold can be a stopping criterion for the integration since the diffusion field ceases to have a principal direction for low FA values. Stalling and Hege [44] achieve significant computational savings by leveraging the correlation between adjacent points on the same streamline. For a constant valued kernel, k, the intensity value at I(σ(s + ds)) can be quickly estimated by I(σ(s)) + ɛ, where ɛ is a small error term which can be quickly computed. Previously, Chaing et al. [16] have used LIC to visualize fibers from diffusion tensor images of the myocardium. 6.4 Particles The LIC and streamtube techniques presented in the previous sections are time-consuming operation. All of the streamline are completely traced before an image can be displayed. For interactive display of fibers, we use a particle

51 41 based visualization technique. The particles are analogous to smoke introduced into a wind-tunnel to visualize streamlines. Rather than simulating the actual diffusion process, the particles simply advect through a velocity field described by the dominant eigenvector of the diffusion tensor at each point. By seeding a few streamlines within a region of interest, and performing a single step of numerical integration at a time, interactive frame-rates can be achieved. Each particle is implemented as a small textured quadrilateral which is always oriented to face the viewer. We vary the size and color of this quad as a means of visualizing the FA value at the particle s location in the tensor field. We adapt this technique to tensor field visualization by incorporating the FA value at each field location in to the LIC texture. By modulating the image intensity with an increasing function of FA, we highlight the areas of white matter, and are able to resolve where the fibers eventually track into grey matter. Unlike LIC and streamtube tracing, the particle visualization requires no preprocessing time. The other techniques require completely integrating many, perhaps thousands, of streamlines. Particle-based techniques allow immediate visualization.

52 CHAPTER 7 EXPERIMENTAL RESULTS 7.1 Data Denoising In all of the experiments, we first smooth the seven 3D directional images using the novel selective smoothing technique outlined in section 3. Following this, the diffusion tensor is estimated from the smoothed data using a standard least squares technique. The results of FA calculation from the smoothed data, and from raw data are presented in Figures 7.1 and 7.2 Figure 7.1: FA image of coronal slice of raw rat brain data. Figure 7.2: FA results for smoothed data. 42

53 Fiber Tracking LIC Figures 7.3 and 7.4 depict the computed fiber tracts for the reconstructed rat brain data. The intensity of the LIC texture has be modulated with the FA image to emphasize the most anisotropic region of each image. Figure 7.3: LIC fiber tracts in coronal slice of smoothed rat brain data. Figure 7.4: LIC fiber tracts in axial slice of smoothed rat brain data.

54 Streamtubes We will now compare our computed streamtubes with a fluoroscopic image. Fibers are evident in in fluoroscopic images as high intensity treelike structures. The fluoro image shown on the left of 7.5 shows a known anatomical feature, the fiber crossings in the corticospinal tract at the base of the brain. The streamtube map from the same region is shown on the right of 7.5. The streamtube map was Figure 7.5: Comparison of fluoroscopic image (left) with extracted streamtubes (right). generated by starting streamline integration at each point of a sparse grid within the data if F A > 0.3. Tracking stopped at F A < The streamtube radius is a function of FA Particles Figure 7.8 shows a fiber tracing sequence obtained by seeding fiber in the corpus callosum region of the rat brain data. The brighter particles signify high FA value. Dimmer particles can be seen tracing into grey matter.

55 Figure 7.6: Axial view of streamtubes in corpus callosum. 45

56 Figure 7.7: Details of corpus callosum. 46

57 Figure 7.8: Sequence from Fiber visualization (2 seconds between images). 47