F3-A2/F3-A3: Tensor-based formulation for Spectral Computed Tomography (CT) with novel regularization techniques

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1 F3-A2/F3-A3: Tensor-based formulation for Spectral Computed Tomography (CT) with novel regularization techniques Abstract Spectral computed tomography (CT) has become possible with the development of photon counting X-ray detector technology. Energy selective measurement capabilities of these devices open the doors to many exciting directions in CT research. In this work we assume perfect energy resolution at detectors, which results in a family of monochromatic CT problems. We propose a tensor based iterative algorithm that simultaneously reconstructs the X-ray attenuation distribution for each energy level. Specifically, we model the multi-spectral unknown as a 3rd order tensor where first two dimensions are in space and the 3rd dimension is in energy. This approach allows the design of a regularizer based on low rank assumptions on the multi-spectral unknown where we apply tensor spectral norm penalties. In addition, when accompanied to total variation (TV) it enhances the regularization capability and provides superior reconstructions. Additionally, we have developed an adaptively weighted L2 norm regularizer with excellent edge preserving capabilities. The problem is cast as a convex optimization problem that is solved using the alternating direction method of multipliers (ADMM). Simulation results show that the proposed regularizer is applicable to the spectral CT problem and reliable in recovering multi-linear structures in an inverse problem set up. I. PARTICIPANTS Faculty/Staff Name Title Institution Phone Eric Miller Professor Tufts University eric.miller@tufts edu Brian Tracey Research Assistant Tufts University btracey@eecs.tufts Professor edu Students Name Degree Pursued Institution Intended Year of Graduation Oguz Semerici PhD Tufts University Oguz.Semerci@ tufts.edu 2012 II. PROJECT OVERVIEW AND SIGNIFICANCE Preventing the access of explosive materials to aviation facilities is the first step of airport security. Therefore, automatic detection of explosives is crucial. This aim of this project is to design X-ray CT image formation methods with potential application for luggage screening in airports. Our work introduces a novel iterative algorithm for spectral CT problem. Accurate identification of the spectral characteristics of the medium under investigation is required as the first step to material characterization. Our approach is specifically designed to increase the detection probability of explosives as well as to decrease the false alarm rate.

2 III. RESEARCH AND EDUCATION ACTIVITY A. State-of-the-Art and Technical Approach A conventional computed tomography (CT) imaging system utilizes energy integrating detector technology [1] and provides a monochromatic reconstruction of the linear attenuation coefficient distribution of an object under investigation. The polychromatic nature of the X-ray spectra is either neglected [2], [3] or incorporated into the model in an iterative reconstruction method to achieve more accurate results [4], [5]. Energy resolving (photon counting) detector technology [6], on the other hand, provides the possibility of energy selective measurement and opens the door to spectral CT technology. Spectral CT promises improved diagnostic imaging [7], [8] and applicability to the security domain [9] due to the contrast enhancement and material characterization capabilities. In this work, we propose an iterative reconstruction method for the spectral CT problem where we model the multi spectral unknown as a low rank 3rd order tensor. Recently, there has been considerable work on recovering corrupted matrices or tensors based on low-rank and sparse decomposition [10] or solely on low-rank assumptions [11], [12], [13], [14]. Our goal here is to adopt the generalized tensor rank formulation [15] to regularize the spectral CT problem. Similar studies where the multi spectral unknown is modeled as a superposition of low rank and sparse matrices had been conducted for 4D cone beam CT [16] and spectral tomography [17]. In these approaches, the multi spectral unknown is represented as a matrix with row dimension in space and column dimension in energy. Applying the low rank prior to the multi-spectral matrix is a special case of our tensor model where only the unfolding in the energy dimension is considered. However, more powerful regularization can be achieved, especially when the number of energy bins is limited, if the redundancy in the spatial dimensions is exploited with the incorporation of unfoldings in spatial dimensions [14]. One of the purposes of this work is to investigate the effect of low-rank assumptions on the unfoldings in the spatial dimensions. At the photon counting detectors the photons in the polychromatic spectrum are classified into energy bins. If we assume perfect energy resolution (i.e., infinitesimal bin width) the polychromatic X-ray CT problem reduces to N E linear monochromatic problems: Ax i = y i + n i, i = 1,..., N E Here A is the CT system matrix discretizing the Radon transform, x i is the vectorized 2D linear attenuation coefficient image, n i is the noise 1 vector and y i is the measurement vector for the i th energy bin. Let us define the 3rd order tensor where N1 and N2 are the number of pixels in spatial dimensions. Note that can be obtained by lexicographical ordering of the N 1 x N 2 attenuation distribution (frontal slice) at the i th energy. A depiction of the multi-spectral phantom used in this study along with the corresponding attenuation curves are given in Figure 1. With this notation we can define the forward operator K in a block diagonal form as follows: Figure 1: (Left) Polyenergetic phantom to be recovered from processing. (Right) Attenuation curves of matericals present in phantom. For the tensor unfolding the mode-k is a matrix whose

3 columns are mode-k fibers, where mode-k fibers are vectors in R Nk that are obtained by varying the index in k th dimension and fixing the others [18]. It is easy to visualize unfolding operations in terms of frontal, horizontal and lateral slices. Figure 2 depicts the 1st and 2nd unfoldings for a 3 rd order tensor. Matrix 2: The polyenergetic phantom is viewed as a rank three tensor, whose unfoldings (arramgent of entries into matix form) provides the basis for our novel regularization methods. The nuclear norm of a matrix X is defined as where σ i s are the singular values. It is shown that the nuclear norm provides the tightest convex relaxation for the rank operation in matrices [12]. Consequently, convex optimization methods for low-rank matrix completion and rank estimation via minimization of the nuclear norm [12] were proposed. This idea has been generalized to tensors for an N-way tensor [15] as: Except for K-edge materials X-ray attenuation at neighboring energies are highly correlated. Therefore, for spectral CT, one expects the third unfolding x 3 to be low-rank [17]. However, structural redundancies can also be exploited by enforcing low-rank structure on other two unfoldings. To this end, we use a more general form of the tensor nuclear norm as a regularizer: In addition to the nuclear norm regularizer, this year we have also developed a new, weighted L2 type regularizer designed to perform edge enhancement within the context of standard linear least squares processing algorithms. The regularizer is defined as: where is discrete gradient operator in x and y directions and is the diagonal weighting matrix with (W) ii s close to one if the i th pixel is an edge and zero otherwise. As the location of edges is not known a prior, W is constructed in an iterative manner. We now use a slight abuse of notation and rename WD as D k where k refers to the iteration count. The reconstruction algorithm solves series of minimization problems and at each step it updates according to the results from previous step as follows: Here f: [01] -> [01] is monotonic decreasing function that adaptively generates weights inversely propor-

4 tional to the edge magnitude. We suggest two functions, which are demonstrated in Figure 3, for these purposes. We seek a solution for the spectral CT problem via the convex optimization problem: where R(x) is replaced by R L2 (x) or total variation regularizer (TV) regularizer for comparison purposes. We compared following methods Figure 3. in our simulations: 1. Filtered back projection (FBP) algorithm applied to each energy bin separately. 2. Only TV regularization at each energy bin separately (i.e., λ 1 = 0 and R (x) = TV (x)). 3. Only nuclear norm regularization (i.e., λ 2 = 0). 4. TV and nuclear norm regularization (i.e., R(x) = R L2 (x)) Figure 4 - Ground Truth 80,20 kev 5. Weighted L2 and nuclear norm regularization (i.e., R(x) = TV (x)) We simulated multi-spectral data for 12 energies between 20 and 80 kev with 60dB SNR for 16 uniformly distributed angles between 0 and 180 degrees. We have added some texture on the objects and a small linear variation to the background of the phantom (128x128 pixels) given in Figure 1. The ground truth images for 20 and 80 kev bins are given in Fig. 4. Fig. 5 shows reconstruction results for 20 and 80 kev images. Figure 5 - Reconstructions with methods given above. IV. FUTURE PLANS The current approach can be extended to the detection of explosives by inducing the prior knowledge of their different chemical properties. Future research will be dedicated to the development of such priors.

5 V. LEVERAGING OF RESOURCES We have been in contact with Analogic Corporation concerning collaboration on this project. Dr. Miller and Mr. Semerci visited Analogic in November 2012 to discuss acquiring data from their scanners and the associated models for use in our processing schemes that could be used to further validate the methods we are developing. To date, intellectual property issues have prevented this collaboration from moving forward. VI. DOCUMENTATION A. Publications and Conference Presentations 1. Oguz Semerci and Eric Miller, An iterative reconstruction method for spectral CT with tensor based formulation and nuclear norm regularization, International Conference on Image Formation in X ray Computed Tomography, 2012, Salt Lake City, USA. 2. O. Semerci, N. Hao, M. E. Kilmer, and E. Miller, Tensor-based Formulation for Spectral Computed Tomography, 2012 SIAM Conference on Imaging Science, May 20-22, 2012, Philadelphia PA. 3. O. Semerici and E. L. Miller, Parametric Level Set Approach to Simultaneous Object Identification and Background Reconstruction for Dual Energy Computed Tomography, IEEE Transactions on Image Processing, Vol. 21, No. 5, pp , May Oguz Semerci and Eric Miller, Tensor-based formulation for Spectral Computed Tomography (CT) with novel regularization techniques. To be submitted to Inverse Problems. B. Technology Transfer We are in the process of filing a provisional patent for the weighted L2 regularizer. C. Seminars, Workshops and Short Courses Prof. Miller and Mr. Semerci has presented this work as part of the following seminars over the past 12 months: 1. E. L. Miller, Model-Based Ideas for Sensor Fusion, Algorithm Development for Security Applications (ADSA) Workshop 06: Development of Fused Explosive Detection Equipment with Specific Application to Advanced Imaging Technology, November 8-9, 2011, Northeastern University, Boston, MA 2. E. Miller, Model-Based, Variational Methods for Segmentation with Applications to Inverse Problems, Lawrence Livermore National Laboratory Seminar, January 26, O. Semerci and E. L. Miller, Iterative Reconstruction Methods for Dual and Multi Energy Computed Tomography, Algorithm Development for Security Applications (ADSA) Workshop Development of Fused Explosive Detection Equipment with Specific Application to Advanced Imaging Technology, May 15-16, 2012, Northeastern University, Boston, MA VII. REFERENCES [1] B. Whiting, P. Massoumzadeh, O. Earl, J. O Sullivan, D. Snyder, and J. Williamson, Properties of preprocessed sinogram data in X-ray computed tomography, Medical physics, vol. 33, p. 3290, [2] C. Bouman and K. Sauer, A unified approach to statistical tomography using coordinate descent optimization, Image Processing, IEEE Transactions on, vol. 5, no. 3, pp , mar [3] X. Pan, E. Sidky, and M. Vannier, Why do commercial ct scanners still employ traditional, filtered back-

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