COMPRESSED SENSING MRI WITH BAYESIAN DICTIONARY LEARNING.

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1 COMPRESSED SENSING MRI WITH BAYESIAN DICTIONARY LEARNING Xinghao Ding 1, John Paisley 2, Yue Huang 1, Xianbo Chen 1, Feng Huang 3 and Xiao-ping Zhang 1,4 1 Department of Communications Engineering, Xiamen University, Fujian, China 2 Department of EECS, University of California, Berkeley, USA 3 Philips Research China, Shanghai, China 4 Department of Electrical and Computer Engineering, Ryerson University, Toronto, Canada dxh@xmu.edu.cn ABSTRACT We present an inversion algorithm for magnetic resonance images (MRI) that are highly undersampled in k-space. The proposed method incorporates spatial finite differences (total variation) and patch-wise sparsity through in situ dictionary learning. We use the beta-bernoulli process as a Bayesian prior for dictionary learning, which adaptively infers the dictionary size, the sparsity of each patch and the noise parameters. In addition, we employ an efficient numerical algorithm based on the alternating direction method of multipliers (ADMM). We present empirical results on two MR images. Index Terms compressed sensing, MRI reconstruction, dictionary learning, Bayesian models 1. INTRODUCTION Magnetic resonance (MR) imaging is a widely used technique for visualizing the anatomical structure and physiological functioning of the body. A limitation of MR imaging is its slow scan speed during data acquisition, which is a drawback especially in dynamic imaging applications. Recent advances in signal reconstruction from measurements sampled below the Nyquist rate, called compressed sensing [1, 2], have had a major impact on MRI imaging [3]. CS-MRI allows for significant undersampling in k-space, where each k-space measurement is a Fourier transform coefficient of the MR image. MRI reconstruction using undersampled k-space data is a case of an ill-posed inverse problem. However, compressed sensing (CS) theory has shown that it is possible to reconstruct a signal from significantly fewer measurements than mandated by traditional Nyquist sampling if the signal is sparse in a particular transform domain [1, 2]. CS-MRI reconstruction algorithms tend to fall into two categories: Those which enforce sparsity withing an imagelevel transform domain (e.g., [3] [7]), and those which en- This project is supported by NSFC (No.9003, ), the Natural Science Foundation of Fujian Province of China (NO: 12J05160), The National Key Technology R&D Program (12BAI07B06), the Fundamental Research Funds for the Central Universities (No.: ). force sparsity on a patch-level (e.g., [8] [11]). Most CS-MRI reconstruction algorithms belong to the first category. For example Sparse MRI [3], the leading study in CS-MRI, performs MR image reconstruction by enforcing sparsity in both the wavelet domain and the total variation (TV) of the reconstructed image. Algorithms with image-level sparsity constraints such as Sparse MRI typically employ off-the-shelf dictionaries, which can usually capture only one feature of the image. The second group of algorithms learn a sparse basis on image subregions called patches that is adapted to the image class of interest. One approach to adapted basis learning is dictionary learning. Recent studies in the image processing literature have shown that dictionary learning can find sparse representations of images on the patch-level [12] [13], [18]. These algorithms learn a patch-level basis (i.e., dictionary) by exploiting structural similarities between patches extracted from images within a class of interest (for example K-SVD [13]). Among these approaches, adaptive dictionary learning usually outperforms analytical dictionary approaches in denoising, super-resolution reconstruction, interpolation, inpainting, classification and other applications, since the adaptively learned dictionary suits the signals of interest [13] [15]. Dictionary learning has been applied to CS-MRI as a sparse basis for reconstruction (e.g., LOST [9] and DLMRI [10]). Results using this approach demonstrate a significant improvement when compared with previous CS-MRI methods. However, these methods still have restrictions in that the dictionary size, patch sparsity and noise levels must be preset. In addition, algorithms such as dictionary learning that are based on only local image sparsity do not take into account additional image-level constraints, such as total variation, which can improve reconstruction. In this paper, we address the issues discussed above by proposing a new inversion framework for CS-MRI. Our work makes two contributions: 1) We propose a combination of in situ dictionary learning and total variation as a sparsity constraint for the inverse CS-MRI problem. We use the alternating direction method of multipliers (ADMM) to derive an /13/$ IEEE 2319 ICIP 13

2 efficient optimization procedure [21]; 2) We use a Bayesian approach to dictionary learning based on the beta-bernoulli process [16] [18]. This approach has three advantages: (i) it can learn the size of the dictionary from the data, (ii) it can learn the sparsity pattern on a patch-by-patch level, (iii) it can adaptively learn regularization weights, which correspond to noise variance in the Bayesian framework. 2. COMPRESSED SENSING MRI WITH BPFA In this section, we briefly review the problem of CS-MRI. We then present our CS inversion approach that uses a Bayesian method for dictionary learning called beta process factor analysis (BPFA). Let x R N be a N N image in vectorized form. Let F u C u N, u < N, be the under-sampled Fourier encoding matrix and y = F u x be the sub-sampled set of k-space measurements. The goal is to estimate x from the small fraction of k-space measurements y. For dictionary learning, let R i be the ith patch extraction operator. The operator R i is a P N matrix of all zeros except for a one in each row that extracts a vectorized P P patch from the image, x i = R i x R P for i = 1,..., N. We represent the CS-MRI inversion problem as optimizing an unconstrained function of the form arg min x h(x) + λ 2 F ux y 2 2, (1) where F u x y 2 2 is a data fidelity term, λ > 0 is a regularization parameter and h(x) is a regularization function that controls desired properties of the image specifically, that the reconstruction is sparse in the selected domain. As our regularization scheme for CS-MRI inversion, we propose a combination of total variation and Bayesian dictionary learning. We write this optimization problem as arg min x,ϕ λ gh g (x) + h l (x) + λ 2 F ux y 2 2, (2) h g (x) := T V (x), h l (x) := i γ ε 2 R i x Dα i f(ϕ). For the patch-wise regularization function h l (x) we use BPFA as given in Algorithm 1 [16]-[18]. This is a Bayesian model for generating images from dictionaries that are sparsely used; the optimization problem is equivalently an inference problem in Bayesian terms. The parameters to be optimized for BPFA are contained in the set ϕ = {D, s, z, γ ε, π}, and are defined in Algorithm 1. The regularization term γ ε is a model variable that corresponds to an inverse variance parameter of the multivariate Gaussian likelihood. In (2), the function f can be expanded to include the negative log likelihood from Algorithm 1, which we omit here. This term acts as the sparse basis for the image and also aids in producing a denoised reconstruction [16]. Algorithm 1 Generating an image with BPFA 1. Construct a dictionary D = [d 1,..., d K ]: d k N(0, P 1 I P ), k = 1,..., K. 2. Draw a probability π k [0, 1] for each d k : π k Beta(a 0, b 0 ), k = 1,..., K. 3. Draw precision values γ ε Gamma(g 0, h 0 ) and γ s Gamma(e 0, f 0 ). 4. For the ith patch in x: (a) Draw the vector s i N(0, γs 1 I K ). (b) Draw the binary vector z ik Bernoulli(π k ). (c) Define α i = s i z i by an element-wise product. (d) Construct the patch R i x = Dα i + ε i with noise ε i N(0, γε 1 I P ). 5. Construct the image x as the average of all R i x that overlap on a given pixel. The regularization function h g (x) is the total variation of the image. This term encourages homogeneity within contiguous regions of the image, while still allowing for sharp jumps in pixel value at edges due to the underlying l 1 penalty. For the total variation penalty T V (x) we use the isotropic TV model []. To review, let ψ i be the 2 N difference operator for pixel i. Each row of ψ i contains a 1 centered on pixel i, and 1 on the pixel directly above (for the first row of ψ i ) or to the right (for the second row of ψ i ) of pixel i, and zeros elsewhere. Let Ψ = [ψ1 T,..., ψn T ]T be the resulting 2N N difference matrix for the entire image. The TV coefficients are β = Ψx R 2N, and the isotropic TV penalty is β2i β2 2i, where i ranges T V (x) = i ψ ix 2 = i over the pixels in the MR image. Several algorithms have been proposed for TV minimization, for example using Newton s method [22] or graph cuts [23]. Recently, a simple and efficient method based on the alternating direction method of multipliers (ADMM) [21], also called the split Bregman method, has been proposed [19] for TV denoising models. We adopt this approach in our optimization algorithm. 3. ALGORITHM We sketch an algorithm for finding a local optimal solution to the non-convex objective function given in (2). This involves a stochastic optimization (inference) portion for updating the BPFA parameters, and an ADMM algorithm for the total variation penalty. Our use of ADMM for total variation minimization follows []. Also, a general introduction to the ADMM algorithm is given in [21]. Recall from the discussion in Section 2 that the total variation part of the objective is T V (x) = λ g i ψ ix 2. We use the ADMM algorithm to split x from this TV penalty. We begin by defining the TV coefficients for the ith pixel as 23

3 β i := ψ i x and we insert β for ψ i x in the objective under the constraint that they are equal. For each β i, we introduce a vector of Lagrange multipliers η i. We then split β i from ψ i x by relaxing their equality via an augmented Lagrangian. This results in the objective function L(x, β, η, ϕ) = λ g i β i 2 + ηi T (ψ ix β i ) + ρ 2 ψ ix β i γ ε i 2 R i x Dα i f(ϕ) + λ 2 F ux y 2 2. (3) The first line is the ADMM expansion of the TV penalty, which contains a Lagrangian term plus a squared error term. From the ADMM theory, holding everything else fixed, this objective will have optimal values β i and x with β i = ψ ix, and so the equality constraints will be satisfied. As written in (3), optimizing this function can be split into three separate sub-problems: one for the TV penalty β i, one for BPFA ϕ = {D, s, z, γ ε, π} and one for the image reconstruction x. Following the discussion in [21], we simplify the ADMM part by defining u i = (1/ρ)η i and completing the square in the first line of (3). We then cycle through the following three sub-problems using the current values of all parameters: (P 1) β i = arg min β λ g β i 2 + ρ 2 ψ ix β i + u i 2 2, (P 2) ϕ = arg min ϕ i γ ε 2 R i x Dα i f(α i, D), (P 3) x ρ = arg min x i 2 ψ ix β i + u i γ ε i 2 R i x D α i λ 2 F ux y 2 2, u i = u i + ψ i x β i, i = 1,..., N. P 1 has a simple, globally optimal and closed form solution that can be found in, e.g., []. The update for u i follows from the general ADMM algorithm [, 21]. Since P 2 is non-convex, we cannot perform the desired minimization, and so an approximation is required. Furthermore, this problem requires iterating through several parameters. Our approach is to use stochastic optimization for problem P 2 by Gibbs sampling each variable in BPFA conditioned on current values for all other variables. A Gibbs sampling algorithm for this model can be found in [18]. P 3 is a least squares problem with a closed form solution. However, we observe that P 3 requires the inversion of a prohibitively large matrix as written. Instead, we work in the Fourier domain as done in [], which results in the inverted matrix being diagonal. We update each Fourier coefficient one-by-one, and then take to inverse transform to obtain x. 4. EXPERIMENTS We perform experiments using our CS-MRI inversion approach on two 6 6 MR images as shown in Figure 2b Fig. 1. Example masks for sub-sampling % of k-space. (left) Cartesian mask, (middle) random mask, (right) radial mask. The Cartesian mask is the most practical, followed by the radial mask. and 3b. We consider sub-sampling 10%, %, %, % and 35% of k-space. We also consider three sub-sampling masks: Cartesian, (pseudo) random and radial, which determine the sampling pattern in k-space. We show examples of these masks in Figure 1. We compare our algorithm (BPFA+TV) with three other algorithms: (i) BPFA without TV, (ii) FCSA [], (iii) SparseMRI [3]. For FCSA and SparseMRI we use the codes available from the authors websites, as well as their default parameterizations. To measure performance, we use the peak-signal-to-noise ratio (PSNR). In Figures 2 and 3 we show qualitative results from the algorithms. In Figure 2 we consider a shoulder scan with 35% sampling using a Cartesian mask. The inset shows a close-up of a region of the original image and reconstructions. For this example, BPFA+TV is smoother than FCSA, but has a better resolution than SparseMRI. We show similar results for the coronal image in Figure 3 using a random mask that samples % in k space. The results appear to again slightly favor BPFA+TV for this example. In Figure 4 we show PSNR results for all sampling percentages and masks. The performance for the BPFA models is the best for these problems. We also see that in general, adding a total variation term to the penalty improves the reconstruction while not significantly increasing the running time. We report that the model learned a dictionary from the images that captured the salient features in a sparse way. We also note that performance significantly improved as a result of adaptively learning BPFA-related parameters such as regularization weights, rather than fixing them in advance. 5. CONCLUSION We have presented an algorithm for CS-MRI inversion that uses total variation and Bayesian dictionary learning as sparsity constraints. We used the ADMM algorithm for efficient optimization of the total variation penalty and Bayesian model that learns the dictionary from the data during image reconstruction. Results on two MR images show that the proposed algorithm performs favorably compared with other available software for CS-MRI. 2321

4 (a) (b) (c) (d) (e) Fig. 2. Reconstruction of the shoulder image with a 35% sampling in k-space using Cartesian sampling: (a) sampling mask, (b) original image, (c) BPFA+TV, (d) FCSA, (e) SparseMRI. The inset shows a region in more detail. (a) (b) (c) (d) (e) Fig. 3. Reconstruction of the coronal image with a % sampling in k-space using random sampling: (a) sampling mask, (b) original image, (c) BPFA+TV, (d) FCSA, (e) SparseMRI. The inset shows a region in more detail (a) Shoulder/Cartesian (b) Shoulder/Random (c) Shoulder/Radial (d) Coronal/Cartesian 15 (e) Coronal/Random 18 (f) Coronal/Radial Fig. 4. Reconstruction PSNR as a function of percentage sampled in k-space. 2322

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