CHAPTER 4 ADAPTIVE SHAPE PRIOR MODELING VIA ONLINE DICTIONARY LEARNING

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1 CHAPTER 4 ADAPTIVE SHAPE PRIOR MODELING VIA ONLINE DICTIONARY LEARNING Shaoting Zhang,, Yiqiang Zhan 2, Yan Zhou 3, and Dimitris Metaxas Computer Science Department, University of North Carolina at Charlotte Charlotte, NC, USA 2 CAD R&D, Siemens Healthcare, Malvern, PA, USA 3 Elekta Inc., Maryland Heights, MO, USA szhang6@uncc.edu Shape and appearance, the two pillars of a deformable model, complement each other in object segmentation. In many medical imaging applications, while the low-level appearance information is weak or mis-leading, shape priors play a more important role to guide a correct segmentation. The recently proposed Sparse Shape Composition (SSC) 49,5 opens a new avenue for shape prior modeling. Instead of assuming any parametric model of shape statistics, SSC incorporates shape priors on-the-fly by approximating a shape instance (usually derived from appearance cues) by a sparse combination of shapes in a training repository. Theoretically, one can increase the modeling capability of SSC by including as many training shapes in the repository. However, this strategy confronts two limitations in practice. First, since SSC involves an iterative sparse optimization at run-time, the more shape instances contained in the repository, the less run-time efficiency SSC has. Therefore, a compact and informative shape dictionary is preferred to a large shape repository. Second, in medical imaging applications, training shapes seldom come in one batch. It is very time consuming and sometimes infeasible to re-construct the shape dictionary every time new training shapes appear. In this chapter, we introduce an online learning method to address these two limitations. Our method starts from constructing an initial shape dictionary using the K-SVD algorithm. When new training shapes come, instead of re-constructing the dictionary from the ground up, we update the existing one using a block-coordinates descent approach. Using the dynamically updated dictionary, sparse shape composition can be gracefully scaled up to model shape priors from a large number of training shapes without sacrificing run-time efficiency. Our method is validated on lung localization in X-Ray and cardiac segmentation in MRI time series. Compared to the original SSC, it shows comparable performance while being significantly more efficient.. Introduction Sparse Shape Composition (SSC) 50,49,5 is a recently proposed method for shape prior modeling. Different from previous methods, which often assume a parametric 59

2 60 S. Zhang et al. model for shape statistics, SSC is a non-parametric method that approximates an input shape usually derived from low level appearance features, by a sparse combination of other shapes in a repository. Specifically, SSC models shape priors based on two sparsity observations: ) Given a large shape repository of an organ, a shape instance of the same organ can be approximated by the composition of a sparse set of instances in the shape repository; and 2) a shape instance derived by local appearance cues might have gross errors but these errors are sparse in spatial space. Shape prior modeling is then formulated as a sparse optimization problem. More specifically, an input shape instance, which is usually derived from local appearance cues, is approximated by a sparse combination of other shapes in a repository. In this way, shape priors are incorporated on-the-fly. Thanks to the incorporation of two sparsity priors, SSC is able to correct gross errors of input shape and can preserve shape details even they are not statistically significant in the training repository. Theoretically, the more shape instances contained in the shape repository, the more shape modeling capacity SSC has. However, a repository including a large number of shapes adversely affects the efficiency of SSC, which iteratively performs sparse optimization at run-time. A large shape repository not only increases the computational cost of each round of sparse optimization, but also decreases the convergency speed. Hence, it induces the low run-time efficiency of SSC, which is not acceptable in many applications. In fact, owing to the similar shape characteristics across the population, this large number of shape instances usually contain lots of redundant information. Therefore, a natural solution is to learn a compact and informative dictionary from the large number of training shapes in the repository. Unfortunately, dictionary learning sometimes confronts another limitation. In real world applications, a large number of training shape instances seldom come in one batch. If the dictionary needs to be completely re-learned every time new training shapes come, the learning process will become very time consuming and sometimes infeasible. In this chapter, we introduce an on-line learning method to address these two limitations 48. Our method starts from learning an initial dictionary off-line using available training shapes. K-SVD method is employed to learn the initial dictionary due to its flexibility and accelerated convergency. When new training shapes come, instead of re-constructing the dictionary from the ground up, we use an online dictionary learning method 25 to update the shape dictionary on-the-fly. More specifically, based on stochastic approximation, this online learning algorithm exploits the specific structure of the problem and updates the dictionary efficiently using block-coordinates descent. In this way, our shape dictionary can be gracefully scaled-up to contain a very large number of training shapes. Although the shape dictionary has much less number of atoms compared to the size of training samples, these atoms can still well approximates a shape instance through sparse combination. Hence, rather than using the complete shape repository, our sparse shape composition can be performed on this shape dictionary efficiently without sacrificing the shape modeling capability. This proposed method has been validated on two

3 Adaptive Shape Prior Modeling via Online Dictionary Learning 6 applications: ) 2D lung segmentation in X-Ray images, 2) 2D cardiac segmentation in MRI time series. Compared with traditional SSC, this online dictionary learning based method has similar shape modeling performance with much higher run-time efficiency. 2. Relevant Work We aim to improve the robustness of deformable model-based segmentation methods using shape priors. In various applications of medical image segmentation, deformable models have achieved tremendous success 9,39,6,5,28,32,7, thanks to the combined use of shape and appearance characteristics. While appearance features provide low level clues of organ boundaries, shapes impose high level knowledge to infer and refine deformable models. Therefore, shape prior modeling is crucial to the performance of segmentation methods. Related studies can be traced to two categories, shape modeling and sparse dictionary learning. In the former category, most previous studies 6,5,8,20,35,4,42,60 aim to model shape priors using a parametric model, e.g., multi-variant Gaussian 6 and hierarchical diffusion wavelet 20. Following the seminal work on active shape models, many of these methods have been proposed to alleviate problems in three categories ) complex shape variations 4,29,35,4,6,60,2,53, 2) gross errors of input 9,2,40,3, and 3) loss of shape details 36,7. SSC is the first shape modeling method to handle the above-mentioned three challenges in a unified framework. It is based on the sparse representation theory. Sparsity methods have been widely investigated recently. It has been shown that a sparse signal can be recovered from a small number of its linear measurements with high probability 2,8. These sparsity priors are employed in many computer vision applications, such as, but not limited to, robust face recognition 38, image restoration 26, image bias estimation 56, MR reconstruction 24,6, atlas construction 34, resolution enhancement 54, image bias estimation 57,56 and automatic image annotation 47,3. Sparsity methods have been proved to be very effective at handling gross errors or outliers. To improve the computational efficiency, sparse dictionary learning methods have also been extensively studied in signal processing domain. Popular ones include optimal direction (MOD) and K-SVD. While these methods require the access of all training samples, a recently proposed online dictionary learning 25 allows an efficient dictionary update only based on new samples. Although dictionary learning has been successfully applied on low level image processing tasks, to the best of our knowledge, the proposed method is the first one to employ them for high-level shape prior modeling. 3. Methodology In this section, we will first briefly introduce standard Sparse Shape Composition. Dictionary learning technologies that aims to tackle the two limitations of SSC will be presented afterwards.

4 62 S. Zhang et al. 3.. Sparse Shape Composition SSC is designed based on two observations: ) After being aligned to a common canonical space, any shape can be approximated by a sparse linear combination of other shape instances in the same shape category. Approximation residuals might come from inter-subject variations. 2) If the shape to be approximated is derived by appearance cues, residual errors might include gross errors from detection/segementaion errors. However, such errors are sparse as well. Accordingly, shape priors can be incorporated on-the-fly through shape composition, which is formulated as a sparse optimization problem as follows. Fig. shows an example of the SSC-based shape prior modeling.... Fig.. An illustration of the SSC-based shape prior modeling. Two instances from the training samples are selected to approximate the input shape. In SSC, a shape is represented by a contour (2D) or a triangle mesh (3D) which consists of a set of vertices. Denote the input shape as v, where v R DN is a vector concatenated by coordinates of its N vertices, where D = {2, 3} denotes the dimensionality of the shape modeling problem. (In the remainder of this chapter, any shape instance is defined as a vector in the same way.) Assume D = [d, d 2,..., d K ] R DN K is a large shape repository that includes K accurately annotated and prealigned shape instances d i. The approximation of v by D is then formulated as an optimization problem: arg min x,e,β T (v, β) Dx e λ x + λ 2 e, () where T (v, β) is a global transformation operator with parameter β, which aligns the input shape v to the common canonical space of D. The key idea of SSC lies in the second and third terms of the objective function. In the second term, the L-norm of x ensures that the nonzero elements in x, i.e., the linear combination coefficients, is sparse 2. Hence, only a sparse set of shape instances can be used to approximate the input shape, which prevents the overfitting to errors from missing/misleading appearance cues. In the third term, the same sparse constraint applies on e R DN,

5 Adaptive Shape Prior Modeling via Online Dictionary Learning 63 the large residual errors, which incorporates the observation that gross errors might exist but are occasional. Eq. is optimized using an Expectation-Maximization (EM) style algorithm, which alternatively optimizes β ( E step) and x, e ( M step). M step is a typical sparse optimization, which employs a typical convex solver, e.g., interiorpoint convex solver 27 in this study. By optimizing Eq., shape priors are in fact incorporated on-the-fly through shape composition. Compared to traditional statistical shape models, e.g., active shape model, this method is able to remove gross errors from local appearance cues and preserve shape details even if they are not statistically significant. However, on the other hand, the optimization of Eq. increases the computational cost, hence, limits the run-time efficiency of the SSC Shape Dictionary Learning Theoretically, the more shape instances in D, the larger shape modeling capacity SSC has. However, the run-time efficiency of SSC is also determined by the size of the shape repository matrix D R DN K. More specifically, the computational complexity of the interior-point convex optimization solver is O(N 2 K) per iteration 27, which means the computational cost will increase quickly with the increase of K, the number of the shape instances in the shape repository. Note that O(N 2 K) is the computational complexity for one iteration. Empirically, with larger K, it usually Efficient sparse shape representation with dictionary learning and mesh partitioning: takes more iterations to convergency, which further decreases the algorithm speed.. To efficiently handle many training samples: dictionary learning: Dictionary size = 28 #Coefficients = 800 #Input samples = To efficiently handle thousands of vertices: mesh partitioning: Fig. 2. An illustration of dictionary learning for 2D lung shapes. The input data contains 800 training samples, and a size 28 compact dictionary is generated. Each input shape can be approximately represented as a sparse linear combination of dictionary elements. Note that dictionary elements are virtual shapes, which are not the same as any input shape but can generalize them well. In fact, owing to the similar shape characteristics across the population, these K shape instances usually contain lots of redundant information. Instead of including all of them, D should only contain representative shapes. This is exactly a dictionary learning problem, which has been extensively investigated in signal

6 64 S. Zhang et al. Algorithm The framework of the OMP algorithm. Input: Dictionary D R n k, input data y i R n. Output: coefficients x i R k. Γ =. repeat Select the atom which most reduces the objective arg min j { min x y i D Γ {j} x 2 2 Update the active set: Γ Γ {j}. Update the residual using orthogonal projection } (2) r ( I D Γ (D T Γ D Γ ) D T Γ ) yi (3) where r is the residual and I is the identity matrix. Update the coefficients until Stop criteria x Γ = (D T Γ D Γ ) D T Γ y i (4) processing community. More specifically, a well learned dictionary should have a compact set of atoms that are able to sparsely approximate other signals. In our study, shape dictionary is learned using K-SVD,52, a popular dictionary learning method because of its accelerated converging speed. Mathematically, K-SVD aims to optimize the following objective function with respect to dictionary D and coefficient X: arg min D,X K K i= ( ) 2 y i Dx i x i where y i, i [, K] represents all dataset (all training shapes in our case), D R n k (k K) is the unknown overcomplete dictionary, matrix x i, i [, K] is the sparse coefficients. This equation contains two important properties of the learned dictionary D. First, k K indicates the dictionary has a much more compact size. Second, i, x i guarantees the sparse representation capability of the dictionary. In K-SVD algorithm, Eq. 7 is optimized by two alternative steps, sparse coding and codebook update. Sparse coding is a greedy method which can approximate an input data by finding a sparse set of elements from the codebook. Codebook update is used to generate a better dictionary given sparse coding results. These two steps are alternately performed until convergence. Sparse coding stage: K-SVD algorithm starts from a random D and X and the sparse coding stage uses pursuit algorithms to find the sparse coefficient x i for each signal y i. OMP 3 is employed in this stage. OMP is an iterative greedy algorithm that selects at each step the dictionary element that best correlates with (7)

7 Adaptive Shape Prior Modeling via Online Dictionary Learning 65 Algorithm 2 The framework of the K-SVD algorithm. Input: dictionary D R n k, input data y i R n and coefficients x i R k. Output: D and X. repeat Sparse coding: use OMP to compute coefficient x i for each signal y i, to minimize min x i { y i Dx i 2 2} subject to x i 0 L (5) Codebook update: for i =, 2,..., k, update each column d i in D and also x i T (ith row) Find the group using d i (x i T 0), denoted as ω i Compute error matrix E i Restrict E i by choosing columns corresponding to ω i. The resized error is denoted as Ei R Apply SVD and obtain E R i = UΣV T (6) Update d i as the first column of U. Update nonzero elements in x i T as the first column of V multiplied by Σ(, ) until Stop criterions the residual part of the signal. Then it produces a new approximation by projecting the signal onto those elements already selected 37. The pseudo-codes of OMP is listed in Algorithm. Codebook update stage: In the codebook update stage K-SVD aims to update D and X iteratively. In each iteration D and X are fixed except only one column d i and the coefficients corresponding to d i (ith row in X), denoted as x i T. Eq. 7 can be rewritten as Y k d j x j T j= 2 F = Y 2 d j x j T d i x i T (8) j i F = Ei d i x i 2 T (9) F We need to minimize the difference between E i and d i x i T with fixed E i, by finding alternative d i and x i T. Since SVD finds the closest rank- matrix that approximates E i, it can be used to minimize Eq. 8. Assume E i = UΣV T, d i is updated as the first column of U, which is the eigenvector corresponding to the largest eigenvalue. x i T is updated as the first column of V multiplied by Σ(, ). The updated xi T may not always guarantee sparsity. A simple but effective solution is to discard the zero entries corresponding to the old x i T. The detail algorithms of K-SVD are listed in the Algorithm 2.

8 66 S. Zhang et al. Algorithm 3 Online learn and update dictionary, using mini-batch mode. Input: Initialized dictionary D 0 R n k, input data Y = [y, y 2,..., y K ], y i R n, number of iterations T, regularization parameter λ R. Output: Learned dictionary D T. A 0 = 0, B 0 = 0. for t = T do Randomly draw a set of y t,, y t,2,..., y t,η. for i = η do Sparse coding: x t,i = arg min 2 y t,i D t x λ x. x R k end for A t = βa t + η i= x t,ix T t,i, B t = βb t + η i= y t,ix T t,i, where β = θ+ η θ+, and θ = tη if t < η, θ = η2 + t η otherwise. Dictionary update: Compute D t, so that: t arg min t i= 2 y i Dx i λ x i = D ( arg min 2 T r ( ) D T DA t T r(d T B t ) ). D end for t The learned dictionary D will be used in Eq. at run-time. It is worth noting that an element in D might not be the same as any shape instances in the training set. In other words, the learned shape dictionary consists of virtual shape instances which might not exist in the real world. However, these virtual shapes do have sparse composition capabilities with a significantly more compact size, which can highly improve the run-time efficiency of our sparse shape composition. Fig. 2 shows an illustration of dictionary learning for 2D lung shapes. Ap compact dictionary is generated from input samples, such that each input sample can be approximately represented as a sparse linear combination of dictionary elements. This compact dictionary is used as data matrix D for our model Online Shape Dictionary Update Using the compact dictionary derived by K-SVD, the run-time efficiency of SSC is dramatically improved, as the number of atoms in D is much less than the number of training shapes. However, K-SVD requires all training shapes available in the dictionary update step, which can not be satisfied in a lot of medical applications. For example, owing to the expensive cost, manual annotations of anatomical structures often come gradually from different radiologists/technicions. Re-construction of the dictionary D with every batch of new training shapes is very time consuming and not always feasible. To tackle this problem, we employ a recently proposed online dictionary method 25 to update the shape dictionary. Algorithm 3 shows the framework of online dictionary learning for sparse coding. Starting from an initial dictionary learned by K-SVD, it iteratively employs two

9 Adaptive Shape Prior Modeling via Online Dictionary Learning 67 stages until converge, sparse coding and dictionary update. Sparse coding aims to find the sparse coefficient x i for each signal y i : x i = arg min x R 2 y i Dx λ x (0) k where D is the initialized dictionary or dictionary computed from the previous iteration. LARS-Lasso algorithm 0 is employed to solve this step. The dictionary update stage aims to update D based on all discovered x i, i [, K]: arg min D K K i= 2 y i Dx i λ x i () Based on stochastic approximation, the dictionary is updated efficiently using blockcoordinates descent. It is a parameter-free method and does not require any learning rate tuning. It is important to note that the dictionary update step in Algorithm 3 is significantly different from that of K-SVD. Instead of requiring all training shapes, it only exploits a small batch of newly coming data (i.e., x i, i [, η]). The dictionary update thereby becomes much faster than K-SVD, as η K. In this way, we can efficiently update the shape dictionary online by using new data as selected x i. Using this online updated dictionary, SSC obtains two additional advantages. ) The run-time efficiency of shape composition is not sacrificed with much more training shapes. 2) SSC can be gracefully scaled-up to contain shape priors from, theoretically, infinite number of training shapes. 4. Experiments We validate our algorithm in two applications, lung localization in Chest X-ray, and left ventricle tracking in MRI. 4.. Lung Localization Chest radiography (X-ray) is a widely used medical imaging modality because of the fast imaging speed and low cost. Localization of lungs in chest radiography not only provides lung shapes, which are critical clues for pathology detection, but also paves the way for other medical image analysis tasks, e.g., cardiac measurements. On one hand, owing to the relatively cheap cost of manual/semi-automatic annotations of lungs in X-ray images, it is possible to get a large number of lung shapes for training. On the other hand, however, training lung shapes seldom come in one batch in clinical practices. Instead, clinicians often verify and correct auto-localization results and prefer a system that has self-improvement ability using these corrected shapes as new training shapes. Therefore, lung localization in chest X-ray becomes an ideal use case to test the effectiveness of our online dictionary method. Our lung localization system starts from a set of auto-detected landmarks around the lung (e.g., the bottom-left lung tip) using learning-based methods 55,23,44,45,43,5,

10 68 S. Zhang et al. Fig. 3. Comparisons of the localization results. From left to right: manual label, detection results, PCA, SSC, and the online learning based shape refinement results. Due to the erroneous detection (marked by the box), PCA result moves to the right and is not on the boundary (see the arrow). Zoom in for better view. Table. Quantitative comparisons of the lung localization using shape priors. P, Q, DSC stand for the sensitivity, specificity, and dice similarity coefficient (%), respectively. P Q DSC PCA 87.5 ± ± ± 4.0 SSC 86.7 ± ± ± 3.9 Ours 94.3 ± ± ± 3.6 based on which lung shapes are inferred using shape priors. Note that various factors, e.g., imaging artifacts, lung diseases, etc., might induce missing/wrong landmark detection, which should be corrected by shape prior models. Although the overall system performance depends on multiple components, including initial landmark detection, shape prior modeling and the following deformable segmentation, our comparison focuses on the shape prior modeling part, i.e., other components remain the same in comparsions. Our experimental dataset includes 367 X-ray images from different patients. 32 of them are used as training data to construct the initial data matrix/dictionary D in Eq.. Note that simply stacking more training shapes into D can also improve the capability of shape representation. However, it dramatically reduces the computational efficiency, which highly depends on the scale of D when solving Eq. 27. Three shape prior methods are compared, ) the PCA based prior as used in Active Shape Model 6, 2) SSC 49, and 3) our method. Fig. 3 shows an example of using these methods to infer shapes from auto-detected landmarks. This case is challenging due to the misplaced medical instrument, which causes erroneous detections (marked by a box in Fig. 3). Although all three methods achieve reasonable accuracy, the whole shape of PCA result shifts slightly to the right (where the arrow points in Fig. 3), because PCA is sensitive to outliers. Benefited by the sparse

11 Adaptive Shape Prior Modeling via Online Dictionary Learning 69 Collaborative Trackers Contour Interpolation Edge Detection Sparse Shape Refinement (a) (b) (c) (d) Fig. 4. Tracking pipeline of our method. It includes collaborative tracking, contour interpolation, edge detection and sparse shape refinement. representation and L-norm constraint, SSC and our method can both handle erroneous detections. However, since the initial shape dictionary may not be generative and representative enough, the inferred shape from SSC is not as accurate as the proposed method, which updates the dictionary on-the-fly and improves its capability of shape representations. Table shows the quantitative accuracy (compared to experts annotations) of the three methods, in terms of the sensitivity, specificity, and dice similarity coefficient (DSC). In general, our method achieves significantly better sensitivity, while slightly worse specificity than SSC. The reason is that SSC under-segments some images, which results in low sensitivity but high specificity. Our method achieves much better performance in terms of DSC, which is a more comprehensive measurements (includes both sensitivity and specificity) for localization accuracy. The experiments are performed on a PC with 2.4GHz Intel Quad CPU, 8GB memory, with Python 2.5 and C++ implementations. The whole framework is fully automatic and efficient. The shape inference step takes s, with around 0.06s as an overhead to update the dictionary online, which is negligible. In contrast, re-training the dictionary using K-SVD needs around 5-40s each time Real-time Left Ventricle Tracking Extraction of the boundary contour of a beating heart from cardiac MRI image sequences plays an important role in cardiac disease diagnosis and treatments. MRIguided robotic intervention is potentially important in cardiac procedures such as aortic valve repair. One of the major difficulties is the path planning of the robotic needle, which requires accurate contour segmentation of the left ventricle on a realtime MRI sequence. Thus, the algorithm should be robust, accurate and fast. We use a shape prior based tracking framework to solve this problem. Note that the left ventricle segmentation has been widely investigated 30. Our intent is not to provide a thorough accuracy comparison with state-the-art techniques on standard datasets. Instead, we aim to demonstrate that the proposed method can efficiently deal with this real-time task by incorporating it into a tracking framework. In our method, a collaborative trackers network is employed to provide a deformed mesh and then generate a rough contour as the initialization at each time step 58. Next, this initialized shape model deforms based on low level image appearance. Appearance-based

12 70 S. Zhang et al. Sensitivity Specificity DSC Fig. 5. Box plots for quantitative comparisons. From left to right of each figure: results from the deformation, PCA, SSC, and the proposed method, respectively. deformation may not be accurate since the image information can be ambiguous and noisy. Thus, the shape prior model is employed to refine the deformed contour 59. Fig. 4 shows the cardiac tracking pipeline of our method. Based on this framework, we compare the performance of (a) deformable model based on image appearance, (b) PCA based, (c) SSC based and (d) the online dictionary based shape refinement methods. For computational efficiency consideration, the dictionary size of (b) and (c) is fixed as a small number 8. The SSC method constantly uses this initial dictionary, while the proposed method (c) updates the dictionary on-the-fly by using acquired tracking results as the mini-batch input of Algorithm 3. Fig. 5 shows the quantitative evaluations, in terms of the sensitivity, specificity, and the dice similarity coefficient. Appearance-based deformation results produces inconsistent results when the image information is ambiguous. SSC based shape refinement may not improve the accuracy of the deformed result due to the small size of dictionary. PCA based method achieves good performance. However, it is not able to handle certain new shapes which cannot be generalized from the current PCA results. In general, the proposed method achieves the most accurate result, since it updates the dictionary on-the-fly using newly acquired information. Thus it is more generic and adaptive to new data. Online updating the dictionary takes around 0.03s, which causes very small overhead for the whole system. To track total of 89 frames, our system takes 23.7s. Re-training the dictionary using K-SVD takes around 2s each time, which is not feasible for realtime applications. 5. Conclusions In this chapter, we introduce a robust and efficient shape prior modeling method, Sparse Shape Composition (SSC). Contrary to traditional shape models, SSC incorporates shape priors on-the-fly through sparse shape composition. It is able to handle gross errors, to model complex shape variations and to preserve shape details in a unified framework. Owing to the high computational cost of standard SSC, a straightforward integration might has low runtime efficiency, which impedes

13 Adaptive Shape Prior Modeling via Online Dictionary Learning 7 the segmentation system being clinically accepted. Therefore, we designed two strategies to improve the computational efficiency of the sparse shape composition. First, given a large number of training shape instances, K-SVD is used to learn a much more compact but still informative shape dictionary. Second, when new shapes come, online dictionary learning method is used to update the dictionary on-the-fly. With the dynamic updated dictionary, SSC is adaptive and gracefully scaled-up to contain shape priors from a large number of training shapes without losing the run-time efficiency. The proposed method was validated on two medical applications. Compared to standard SSC, it achieved better shape modeling performance with a much faster speed. It is worth noting our shape prior method can also be coupled with other segmentation algorithms, such as registration-based and patch-based methods 4,62,33,46,22,29. In the future, we would like to apply this shape prior method to more applications such as registration. References. Aharon, M., Elad, M., Bruckstein, A.: K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation. In: IEEE Transaction on Signal Processing. pp (2006) 2. Candes, E., Romberg, J., Tao, T.: Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory 52(2), (2006) 3. Chen, S., Billings, S.A., Luo, W.: Orthogonal least squares methods and their application to non-linear system identification. In: International Journal of Control. pp (989) 4. Cootes, T.F., Taylor, C.J.: A mixture model for representing shape variation. In: Image and Vision Computing. pp. 0 9 (997) 5. Cootes, T., Edwards, G., Taylor, C.: Active appearance models. In: ECCV. pp (998) 6. Cootes, T., Taylor, C., Cooper, D., Graham, J.: Active shape model - their training and application. Computer Vision and Image Understanding 6, (995) 7. Davatzikos, C., Tao, X., Shen, D.: Hierarchical active shape models, using the wavelet transform. IEEE Transactions on Medical Imaging 22(3), (2003) 8. Donoho, D.L.: Compressed sensing. IEEE Transactions on Information Theory 52(4), (2006) 9. Duta, N., Sonka, M.: Segmentation and interpretation of MR brain images. an improved active shape model. IEEE Transactions on Medical Imaging 7(6), (998) 0. Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. The Annals of statistics 32(2), (2004). Engan, K., Aase, S., Hakon Husoy, J.: Method of optimal directions for frame design. In: ICASSP. vol. 5, pp (999) 2. Etyngier, P., Segonne, F., Keriven, R.: Shape priors using manifold learning techniques. In: International Conference on Computer Vision. pp. 8 (2007) 3. Gao, S., Chia, L., Tsang, I.: Multi-layer group sparse codingfor concurrent image classification and annotation. In: IEEE Conference on Computer Vision and Pattern Recognition. pp IEEE (20)

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