Automatic Mitral Leaflet Tracking in Echocardiography by Outlier Detection in the Low-rank Representation

Transcription

1 Automatic Mitral Leaflet Tracking in Echocardiography by Outlier Detection in the Low-rank Representation Xiaowei Zhou, Can Yang, Weichuan Yu Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology, Hong Kong Abstract Tracking the mitral valve leaflet in an ultrasound sequence is a challenging task because of the poor image quality and fast and irregular leaflet motion. Previous algorithms usually applied standard segmentation methods based on edges, object intensity and anatomical information to segment the mitral leaflet in static frames. However, they are limited in practical applications due to the requirement of manual input for initialization or large annotated datasets for training. In this paper we present a completely automatic and unsupervised algorithm for mitral leaflet detection and tracking. We demonstrate that the image sequence of a cardiac cycle can be well approximated with a low-rank matrix, except for the mitral leaflet region with fast motion and tissue deformation. Based on this difference, we propose to track the mitral leaflet by detecting contiguous outliers in the low-rank representation. With this formulation, the leaflet is tracked using the motion cue, but the complicated motion computation is avoided. To the best of our knowledge, the proposed algorithm is the first unsupervised method for mitral leaflet tracking. The algorithm was tested on both 2D and 3D echocardiography, which achieved accurate segmentation with an average distance of 0.87±0.42mm compared to the manual tracing.. Introduction The mitral valve is a thin leaflet structure that lies between the left atrium and the left ventricle to control the direction of blood flow. Mitral valve related disease such as mitral regurgitation is the most common valvular heart disease [4]. Acquiring patient-specific information about the geometry and motion of the mitral valve from medical images is important to better understand the valvular disease and assist the surgical intervention for valve repair. Among various modalities, real-time 3D (RT3D) echocardiography provides an economic and noninvasive way to model the 3D geometry of the mitral valve and capture its fast motion [8]. In order to generate a comprehensive mitral valve model, the mitral leaflet surface needs to be segmented from ultrasound images and tracked throughout the sequence. Manual delineation of the mitral leaflet surface is both laborconsuming and prone to large variance, especially for 3D images when only 2D projection or a slice of the volumetric data can be displayed and processed by the operator at each time. Thus, developing an algorithm that can automatically detect and track the mitral leaflet is critical in clinic applications. While object tracking has been well studied in natural image processing [25], tracking the mitral leaflet from an ultrasound sequence is extremely difficult. Generally, the challenges arise from the following two aspects: Lack of reliable features: Selecting good features that easily distinguish an object of interest from others is critical in designing object tracking algorithms [25]. Common features in images include color (or intensity), edges and texture. However, in ultrasound images the mitral leaflet shares the identical intensity and texture with other tissues such as myocardium. Moreover, the feature quality (e.g. edges) is prone to be influenced by various factors including probe orientation, acoustic speckle, signal dropout, acoustic shadows, etc. [5]. Fast and irregular valve motion: Object tracking usually relies on the computation of optical flow to find feature-point correspondence between frames []. However, motion analysis in ultrasound sequences is extremely difficult and unreliable due to the featuremotion decorrelation problem [2]. Especially, it is more challenging under the conditions of fast motion, large deformation and low frame rate, which is the case in 3D imaging of the mitral valve. The literatures on mitral leaflet tracking are very limited. Most of the previous works tried to segment the mitral leaflet from each frame using active contours [2]. The ac /2/$ IEEE 972

2 (a) Cumulative energy distribution Valve region Reference region Whole image k 0 5 (b) (c) Figure. Matrix spectral analysis. In matrix spectral analysis, we convert the image/sub-image sequence into a matrix with the column index corresponding to the frame index in the sequence. Each column is a vectorized image/sub-image. (a) Manual selected regions for local analysis. The valve region (inside green ellipse) is centered around the valve leaflet, while the reference region covers part of the myocardium and part of the blood pool. (b) The cumulative distribution functions (cdf s) of spectral energy of the matrices corresponding to the valve region, the reference region and the whole image, respectively. The cdf is computed using Eq. (). (c) Results of valve region positioning based on matrix spectral analysis. The first frames of four tested sequences are shown from left to right. The sliding window that gives a matrix with the largest residue of rank-k fitting is displayed. The window is a rectangle, whose size is /7 of the image size. The detail of the procedure is described in Algorithm. tive contour was guided by various features such as edges [3], region statistics [9], optical flow [3] and geometry of the leaflet [4]. These methods required intensive user interaction to either initialize the active contour or correct the biased intermediate results during the segmentation [3]. Graph cuts combined with a thin tissue detector has also been used to find the mitral leaflet surface [7, 8]. While this approach proved to be robust and accurate in practice, it still requires user input to select particular frames and a point that indicates the location of the mitral valve before automatic segmentation. Recently, a learning-based system was proposed for automatic modeling the mitral valve from CT and ultrasound data [9]. In this method, a statistical shape model of the mitral valve was fitted to several anatomical landmarks to find the right position of the mitral valve in the image. Then, the model was refined to match the object boundary, which is defined by the learned boundary detector. Appealing results were obtained with fast speed. However, a large amount of expert-annotated data was needed to learn the shape model and detectors, which is unavailable to public and expensive to collect. In this paper, we aim to develop an unsupervised algorithm to detect and track the mitral leaflet throughout an ultrasound sequence automatically without additional user interaction or datasets for offline training. Instead of focusing on static frames, we would like to make use of the dynamic information throughout the entire sequence, i.e. the difference in motion patterns between the mitral leaflet and myocardium. Existing methods for motion segmentation on natural images tried to solve the problem by either partitioning the motion field [23, 7, 6] or classifying the trajectories of feature points that are extracted from the sequence [22]. Both of them relied on accurate motion estimation and parametric motion models. However, as mentioned previously, the optical flow computed from ultrasound images is very unreliable, especially for the mitral leaflet. Also, most of the cardiac tissues are moving in a complex way, which is difficult to be described by parametric models. Thus, it is hard 973

3 to achieve satisfactory results by simply applying the existing motion segmentation methods on mitral leaflet tracking. To address above challenges, we propose a novel framework for motion-based tracking of the mitral leaflet. Instead of local motion analysis, we explore the global correlation between image frames and try to segment the mitral leaflet by the faithfulness of fitting the image sequence with a lowrank representation [24]. The main contribution of this paper can be summarized as follows: Through matrix spectral analysis, we showed that more bases are needed to describe the mitral leaflet motion compared with the myocardium motion, which provides a promising feature for the mitral leaflet detection. To further illustrate this, we developed a simple algorithm that could locate the mitral valve region in ultrasound images robustly and efficiently. We formulated the mitral leaflet tracking problem as contiguous outlier detection in the low-rank representation. This formulation allows us to segment the object unsupervisedly based on the motion cue while avoiding complicated motion computation. To the best of our knowledge, our algorithm is the first unsupervised algorithm for automatic tracking of the mitral leaflet without any user interaction or training data. The rest of this paper is organized as follows: Section 2 explains the feature used in this paper for mitral leaflet detection. Section 3 introduces the formulation and the algorithm for mitral leaflet tracking. Section 4 reports the experiment results on real sequences. Finally, Section 5 concludes our paper with discussions. 2. Matrix Spectral Analysis Suppose there is an echocardiographic sequence composed of n images. Let D R m n represent a matrix whose columns are vectorized images, i.e. D = [vec(i ) vec(i n )], where m equals to the number of pixels in each image andnis the number of images. If no motion exists in the sequence, all images are identical and rank(d) = by ignoring the noise in images. S- ince cardiac motion exists, the images are changing within a cardiac cycle, which givesrank(d) >. However, due to the correlation among frames, the energy distribution in the matrix spectrum ofd will concentrate on a limited number of principle components. Define the cumulative distribution function (cdf ) of the spectral energy ofd as follows: cdf(k) = k i= σ2 i r i= σ2 i, () Matrix spectrum means the set of eigenvalues of the matrix. Algorithm Locating the valve region. Input: an image sequence{i,,i n }, a set of spatial windows {W l }, constantk. 2. for each l do 3. compute the singular values σ,,σ r for M = [vec(i W l ) vec(i n W l )], where I j W l means the sub-image ofi j insidew l. 4. ǫ 2 l = r i=k+ σ2 i. 5. end for 6. l = argmax l ǫ 2 l 7. Output: W l where r = min(m,n) and σ,,σ r are singular values of D in descending order. As shown in Figure (b), the cdf of the echocardiography sequence has exceeded95% when k > 2. Interestingly, when analyzing different regions locally we find that the corresponding cdf s are different. To demonstrate this, we choose two regions as shown in Figure (a): the valve region including the mitral leaflet; a reference region covering part of myocardium and part of the blood pool. Then, we compute the cdf of each sub-matrix formed by the sub-image sequence of each region. Figure (b) shows that the cdf curve of the valve region lies much lower than that of the reference region, which means that the spectral distribution of the valve region is more spread-out. This is attributed to the fact that the mitral leaflet moves much faster with larger tissue deformation compared with myocardium that moves smoothly across a cardiac cycle. Thus, more principle components are needed to describe the image variation caused by motion of the mitral leaflet. In practice, it is easy to show that 2 : r i=k+ σ 2 i = min D rank(b)=k B 2 F, (2) where B is any rank-k matrix. This means that, given a fixed K, the residue of fitting the valve region with a rank- K matrix will be much larger compared to the residue of fitting other regions. This provides a good feature to detect the mitral leaflet. To illustrate the feasibility of using this feature for mitral leaflet detection, we developed a simple algorithm for valve region positioning as described in Algorithm, with experimental results on real sequences given in Figure (c). In short, a rectangular window is used to define a region, and the sub-image sequence of this region is converted into a matrix with columns being the vectorized subimages. Then, the residue of rank-k fitting r i=k+ σ2 i is computed for the matrix. Next, the window is moved to 2 Three norms of a matrix X are used in this paper. X = ij X ij denotes the l -norm. X F = ij X2 ij is the Frobenius norm. X means the nuclear norm, i.e. sum of singular values. 974

4 cover another region and the residue is computed again, and so on and so forth. Finally, the window that gives a matrix with the largest residue of rank-k fitting is output and displayed. Here, we choosek = 5 for all examples. As shown in Figure (c), all of the output windows roughly cover the mitral leaflet in all examples, which demonstrates the effectiveness of using the proposed feature for mitral leaflet detection. 3. Mitral Leaflet Tracking While Algorithm can roughly locate the valve region, a more sophisticated algorithm is needed to track the leaflet more accurately. In this section, we propose an algorithm to track the mitral leaflet based on leaflet detection over the entire sequence. 3.. Formulation From the discussion in Section 2, we can see that the echocardiographic sequence can be well approximated by a low-rank matrix except for the mitral leaflet. Moreover, the mitral leaflet occupies a relatively small and connected region in the images. Thus, we formulate the problem of mitral leaflet segmentation as contiguous outlier detection in the low-rank representation. Let ij denote the i-th pixel in the j-th frame of the sequence 3 and S {0,} m n denote the outlier support, i.e. S ij = if ij is the mitral leaflet pixel and S ij = 0 otherwise. Thus, the task is to give an optimal estimate to S which labels the outliers of the low-rank model B. Since both B and S are unknown, it is required to estimate them simultaneously to acquire an accurate low-rank model and labeling of outliers. Let P S (X) represent the orthogonal projection of a matrixx onto the linear space of matrices supported bys: { 0, if Sij = 0 P S (X)(i,j) = (3) X ij, if S ij = andp S (X) be its complementary projection, i.e.p S (X)+ P S (X) = X. Then, we propose to minimize the following energy to estimateb ands: min B,S ij {0,} 2 P S (D B) 2 F +β S, s.t. rank(b) K, (4) where K is a predefined constant. The first term penalizes the fitting residue for the non-valvular region, while the second term regularizes the sparsity of the mitral leaflet pixels. Since the leaflet pixels should be contiguous both spatially and temporally, we introduce a smoothness regularizer on S. Consider a graph G = (V,E), where V is the 3 We didn t distinguish between pixel and voxel in this paper because our method is applicable to both 2D and 3D image sequences. set of vertices denoting all m n pixels in the sequence and E is the set of edges connecting spatially and temporally neighboring pixels. Based on the first order Markov Random Fields (MRFs), we adopt the following energy to impose continuity ons: Avec(S) = ω ij,kl S ij S kl, (5) (ij,kl) E where A is the weighted node-edge incidence matrix of G, and the weightω ij,kl is defined as follows: { G ω ij,kl = e D ij DG kl 2 2µ 2, ifij andkl are spatial neighbors, ω t, ifij andkl are temporal neighbors. Here, D G is the Gaussian smoothed image sequence, µ is the standard deviation of Dij G DG kl for spatial neighbors, andω t is the weight for temporal links. We simply fixedω t to be 0. in the implementation. Introducing the spatiallyvaryingω ij,kl makes use of the edge information to improve segmentation. To make the energy minimization tractable, we relax the rank operator onb with the nuclear norm, which has proven to be an effective convex surrogate of the rank operator [6]. Adding the smoothness regularizer and writing (4) in its d- ual form, we obtain the final form of the energy function: min B,S 2 P S (D B) 2 F +α B +β S +γ Avec(S). (6) Here, α, β and γ are all positive parameters, which will be discussed in detail in Section Optimization The objective function defined in (6) is non-convex and it includes both continuous and discrete variables. Joint optimization over B and S is extremely difficult. Hence, we adopt an alternating algorithm that separates the energy minimization over B and S into two steps. B-step is a convex optimization problem and S-step is a combinatorial optimization problem. It turns out that the optimal solutions of B-step and S-step can be computed efficiently. Given an estimate of the support matrixŝ, the minimization in (6) over B turns out to be the matrix completion problem []: min B 2 P Ŝ (D B) 2 F +α B. (7) The original problem is to learn a low-rank matrix from partial observations. The optimal B in (7) can be computed efficiently by the SOFT-IMPUTE algorithm [], which iteratively updates the estimate ˆB using: ˆB Θ α (PŜ (D)+PŜ(ˆB)), (8) 975

5 whereθ α is the singular value thresholding operator: Θ α (X) UΣ α V, Σ α = diag[(d α) +,,(d r α) + ]. (9) HereUΣV is the SVD ofx,d,,d r are singular values of X, and t + = max(t,0). Please refer to [] for more detailed explanations of SOFT-IMPUTE. Next, we minimize the energy in (6) over S given the low-rank matrix ˆB. Noticing that S ij {0,}, the energy can be rewritten as follows: 2 P S (D ˆB) 2 F +β S +γ Avec(S) = (D ij 2 ˆB ij) 2 ( S ij)+β S ij +γ Avec(S) ij ij = (β 2 (Dij ˆB ij) 2 )S ij +γ Avec(S) +C, (0) ij where C = 2 ij (D ij ˆB ij ) 2 is a constant when ˆB is fixed. Above energy is in the standard form of the firstorder MRFs with binary labels, which can be solved exactly in polynomial time using graph cuts [3, 0] Parameter tuning The parameter α in (6) controls rank(ˆb), i.e. K in (4). The low-rank model with an appropriate K should include the myocardium, but exclude the mitral leaflet. According to Figure (b), we choose K = 5 for all experiments. In implementation, we start from a large α. After each run of SOFT-IMPUTE, if rank(ˆb) K, we reduce α by a factor η = 0.9 and repeat SOFT-IMPUTE again until rank(ˆb) > K. Using warm start, this sequential optimization is efficient []. In practice, we find that the result is relatively insensitive to K when K is reasonably large (K 3). This is attributed to the shrinkage effect of the nuclear norm regularization []. The parameter β in (6) controls the sparsity of S. From (0) we can see that Ŝij is more likely to be if 2 (D ij ˆB ij ) 2 > β. Thus the choice of β should depend on the noise level in images and a proper value can be estimated online. Typically we set β = 4.5ˆσ 2, where ˆσ 2 is the variance ofd ij ˆB ij. Since the estimation of ˆB and ˆσ is biased at the beginning iterations, we propose to start our algorithm with a relatively large β, and then reduce β by a factor η 2 = 0.5 after each iteration untilβ reaches4.5ˆσ 2. In other words, we tolerate more error in model fitting at the beginning, since the model itself is not accurate enough. With the model estimation getting better and better, we decrease the threshold and declare more and more outliers. The parameter γ in (6) controls the strength of mutual dependency between neighboring pixels. A larger γ gives more penalty to isolated outliers in order to smooth Ŝ. In all our experiments, we simply chooseγ = β. Algorithm 2 Contiguous Outlier Detection. Input:D = [I,,I n] R m n 2. Initialize: ˆB D,Ŝ 0,α,β. 3. repeat 4. repeat 5. ˆB Θα(PŜ (D)+PŜ(ˆB)); 6. until convergence 7. if rank(ˆb) K then 8. α η α; 9. go to Step 4; 0. end if. estimate ˆσ; 2. β max(η 2β,4.5ˆσ 2 ); 3. Ŝ argmin (β 2 (Dij ˆB ij) 2 )S ij +γ Avec(S) S ij 4. until convergence 5. Output: ˆB,Ŝ Thus, we used the same parameter setting for all experiments. All steps of the algorithm with adaptive parameter tuning are summarized in Algorithm 2. Since we are always minimizing a single function in each step and the parameters (α, β, γ) keep decreasing in adaptive tuning, the energy in (6) decreases monotonically. Furthermore, we can manually set lower bounds for both α and β to stop the iteration. Thus, the convergence of the algorithm is guaranteed. Empirically, Algorithm 2 will terminate in about 0 iterations for a convergence precision of Results To the best of our knowledge, there is no automatic algorithm that can detect the mitral leaflet in previous literatures except for [9], while the method in [9] requires large manually annotated datasets for training feature detectors, which is unavailable to public. Thus, in this paper we mainly focus on the verification of the new algorithm. We evaluated the mitral leaflet tracking method using s- tandard transthoracic echocardiographic (TTE) data 4. All sequences were acquired during a cardiac cycle, using a Philips ie33 Ultrasound System with a S5- probe for 2D imaging and an X3- probe for 3D imaging. We cropped the acquired sequences to remove irrelevant parts and focus on the left chambers. The cropped sequences were fed into Algorithm 2 without other preprocessing. To improve the accuracy of detection, a simple postprocessing was applied to the estimated outlier support of Algorithm 2. We used morphological operations to extract the largest connected component in the mask for each frame, and removed the false detections caused by other tissues with fast motion such as the papillary muscles. 4 While the transesophageal echocardiogram (TEE) is better to image the mitral valve, it is not easy to access. 976

6 (a) (b) (c) (d) Figure 2. Results on 2D echocardiographic sequences. Each sub-figure corresponds to a sequence. Three selected frames during valve opening are displayed from top to bottom. The three columns from left to right show the original images, the segmentation results and the corresponding low-rank components, respectively. 977

7 Firstly, we present the results on four 2D sequences. The sequences were acquired from the apex view. Each sequence has about 30 frames recording a cardiac cycle, with an image size of pixels. The first frames are displayed in Figure (c). The images were cropped to pixels as input to Algorithm 2. To display the results, we selected three frames during valve opening for each sequence. The segmentation results Ŝ and low-rank components ˆB are shown in Figure 2. The mitral leaflet was detected accurately as outliers and excluded from the lowrank component. Moreover, the low-rank model captured the global motion of myocardium. Interestingly, the ultrasound speckles were also depressed in the low-rank components, while the edges of the endocardium border were almost preserved. This also shows the potential of applying the low-rank representation on ultrasound despeckling [2]. Next, we give a quantitative analysis for the segmentation results in Figure 2. The ground truth was provided by manual tracing. Three metrics were used: precision, recall and the mean absolute distance (MAD) between contours, which are defined as follows: precision = Ω Ĉ Ω C, recall = Ω Ĉ Ω C, ΩĈ Ω C MAD(Ĉ,C ) = 2 0 [ d(ĉ(s),c )+d(c (s),ĉ) ] ds. Here,Ĉ means the algorithm-generated contour andc denotes the manual-delineated contour. Ω C means the region enclosed by contourc and Ω is the number of pixels inside regionω. C(s) is a parameterization of contourc with constant velocity C (s). d(p,c) = min s p C(s ) 2 is the minimum distance from point p to contour C. The results are summarized in Table. The MAD has the same order of magnitude with the resolution of the imaging system ( 0.5 mm) and the precision is high. However, the recall is relatively low compared with the resulted precision, which shows that there is a systematic bias of under-segmentation. There are two possible reasons: () The leaflet tissue near the mitral annulus moves similarly with the myocardium. Thus, the method may fail to detect this part. (2) The shrinkage effect of the smoothness constraint (or named small cut behavior) causes under-segmentation. This bias can be corrected by using other information like shape priors. Finally, we demonstrate the applicability of our algorithm to RT3D echocardiography. Due to the lack of highquality TEE data, here we only give some qualitative results on an ECG-gated TTE sequence. The sequence was composed of 7 volumes with voxels per volume. Irrelevant parts were removed and voxels were left for each volume. Since the MRF of the entire sequence ( nodes existed) can t be solved by graph cuts due to the memory limit, the temporal links in the MRF were removed here. Then, graph cuts Table. Quantitative evaluation of the tracking results displayed in Figure 2. The ground truth is provided by manual tracing. Each cell shows the mean value and the standard deviation. precision recall MAD(mm) Sequence (a) 0.93± ± ±0.7 Sequence (b) 0.95± ± ±0.8 Sequence (c) 0.92± ± ±0.63 Sequence (d) 0.9± ± ±0.5 Overall 0.93± ± ±0.42 could be performed volume by volume. The segmentation results on a sample volume are displayed in Figure 3. The algorithm was implemented in MATLAB. All experiments were run on a PC with a 3.4 GHz Intel i7 CPU and 8 GB RAM. It costs around 30 seconds to analyze a 2D sequence and 24.4 minutes to analyze a 3D sequence. The efficiency of the algorithm can be improved by C implementation or GPU computing. 5. Discussion In this paper, we proposed a new feature for mitral leaflet detection and a novel algorithm for leaflet tracking. The algorithm is completely automatic without manual initialization and annotated training data. According to our knowledge, this is the first unsupervised algorithm for mitral leaflet tracking. Our formulation is similar to the sparse and low-rank decomposition or Principle Component Pursuit (PCP) [5], which becomes popular recently. There are several key differences between our algorithm and PCP: PCP uses a convex relaxation by replacing the penalty on the cardinality of outliers with the l -norm. This relaxation requires outliers to be randomly distributed [5]. In our formulation, we directly optimize the nonconvex penalty (notice that S in (6) equals to the cardinality of outliers) to make use of the robustness of the nonconvex penalty in outlier detection [20]. PCP does not use the contiguous property of outliers. In our formulation, the MRF is integrated to model the contiguity prior to further improve the outlier detection accuracy. PCP makes the sparse and low-rank decomposition while ignoring the noise part. In our method, the noise is also considered, which is more general in practice. In future, we plan to conduct a more comprehensive e- valuation of the algorithm on more RT3D data, including high-quality TEE sequences. Also, we would like to integrate more features such as the geometry of the mitral valve into our framework. 978

8 X-Z (two-chamber) Y-Z (four-chamber) X-Y (short-axis) 3-D Figure 3. Segmentation results on a RT3D echocardiographic sequence. Only the result on a sample frame is displayed. From left to right are three orthogonal 2D slices across the mitral leaflet and a 3D model of the mitral leaflet generated by surface rendering. References [] S. Beauchemin and J. Barron. The computation of optical flow. ACM Computing Surveys, 27(3): , 995. [2] A. Blake and M. Isard. Active contours. Springer, 998. [3] Y. Boykov, O. Veksler, and R. Zabih. Fast approximate energy minimization via graph cuts. PAMI, 23(): , [4] P. Burlina, C. Sprouse, D. DeMenthon, A. Jorstad, R. Juang, F. Contijoch, T. Abraham, D. Yuh, and E. McVeigh. Patientspecific modeling and analysis of the mitral valve using 3D- TEE. Information Processing in Computer-Assisted Interventions, pages 35 46, [5] E. Candes, X. Li, Y. Ma, and J. Wright. Robust Principal Component Analysis? Journal of ACM, 58(): 37, [6] A. Chan and N. Vasconcelos. Layered dynamic textures. PA- MI, 3(0): , [7] D. Cremers and S. Soatto. Motion competition: A variational approach to piecewise parametric motion segmentation. IJCV, 62(3): , [8] J. Hung, R. Lang, F. Flachskampf, S. Shernan, M. McCulloch, D. Adams, J. Thomas, M. Vannan, and T. Ryan. 3d echocardiography: a review of the current status and future directions. Journal of the American Society of Echocardiography, 20(3):23 233, [9] R. Ionasec, I. Voigt, B. Georgescu, Y. Wang, H. Houle, F. Vega-Higuera, N. Navab, and D. Comaniciu. Patientspecific modeling and quantification of the aortic and mitral valves from 4-D cardiac CT and TEE. IEEE Trans. Medical Imaging, 29(9):636 65, , 5 [0] V. Kolmogorov and R. Zabih. What Energy Functions Can Be Minimizedvia Graph Cuts? PAMI, 26(2):47 59, [] R. Mazumder, T. Hastie, and R. Tibshirani. Spectral Regularization Algorithms for Learning Large Incomplete Matrices. JMLR, : , , 5 [2] J. Meunier and M. Bertrand. Ultrasonic texture motion analysis: theory and simulation. IEEE Trans. Medical Imaging, 4(2): , 995. [3] I. Mikic, S. Krucinski, and J. Thomas. Segmentation and tracking in echocardiographic sequences: Active contours guided by optical flow estimates. IEEE Trans. Medical Imaging, 7(2): , [4] V. Nkomo, J. Gardin, T. Skelton, J. Gottdiener, C. Scott, and M. Enriquez-Sarano. Burden of valvular heart diseases: a population-based study. The Lancet, 368(9540):005 0, [5] J. Noble and D. Boukerroui. Ultrasound image segmentation: A survey. IEEE Trans. Medical Imaging, 25(8):987 00, [6] B. Recht, M. Fazel, and P. Parrilo. Guaranteed Minimum- Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization. SIAM Review, 52(3):47 50, [7] R. Schneider, D. Perrin, N. Vasilyev, G. Marx, P. del Nido, and R. Howe. Mitral annulus segmentation from 3D ultrasound using graph cuts. IEEE Trans. Medical Imaging, 29(9): , [8] R. Schneider, N. Tenenholtz, D. Perrin, G. Marx, P. del Nido, and R. Howe. Patient-specific mitral leaflet segmentation from 4D ultrasound. In MICCAI, [9] Y. Shang, X. Yang, L. Zhu, R. Deklerck, and E. Nyssen. Region competition based active contour for medical object extraction. Computerized Medical Imaging and Graphics, 32(2):09 7, [20] Y. She and A. Owen. Outlier detection using nonconvex penalized regression. Journal of the American Statistical Association, 06(494): , [2] P. Tay, C. Garson, S. Acton, and J. Hossack. Ultrasound despeckling for contrast enhancement. IEEE Transactions on Image Processing, 9(7): , [22] R. Vidal and Y. Ma. A unified algebraic approach to 2-d and 3-d motion segmentation. In ECCV, [23] Y. Weiss. Smoothness in layers: Motion segmentation using nonparametric mixture estimation. In CVPR, [24] J. Wright, Y. Ma, J. Mairal, G. Sapiro, T. Huang, and S. Yan. Sparse representation for computer vision and pattern recognition. Proceedings of the IEEE, 98(6):03 044, [25] A. Yilmaz, O. Javed, and M. Shah. Object tracking: A survey. ACM computing surveys, 38(4): 45,