cvpr cvpr2013 cvpr2013-33 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Xiaowei Zhou, Xiaojie Huang, James S. Duncan, Weichuan Yu
Abstract: Active contours are widely used in image segmentation. To cope with missing or misleading features in images, researchers have introduced various ways to model the prior of shapes and use the prior to constrain active contours. However, the shape prior is usually learnt from a large set of annotated data, which is not always accessible in practice. Moreover, it is often doubted that the existing shapes in the training set will be sufficient to model the new instance in the testing image. In this paper, we propose to use the group similarity of object shapes in multiple images as a prior to aid segmentation, which can be interpreted as an unsupervised approach of shape prior modeling. We show that the rank of the matrix consisting of multiple shapes is a good measure of the group similarity of the shapes, and the nuclear norm minimization is a simple and effective way to impose the proposed constraint on existing active contour models. Moreover, we develop a fast algorithm to solve the proposed model by using the accelerated proximal method. Experiments using echocardiographic image sequences acquired from acute canine experiments demonstrate that the proposed method can consistently improve the performance of active contour models and increase the robustness against image defects such as missing boundaries.
Reference: text
sentIndex sentText sentNum sentScore
1 Duncan‡, Weichuan Yu† † Department of ECE, The Hong Kong University of Science and Technology ‡ Image Analysis and Processing Group, Yale University Abstract Active contours are widely used in image segmentation. [sent-2, score-0.245]
2 To cope with missing or misleading features in images, researchers have introduced various ways to model the prior of shapes and use the prior to constrain active contours. [sent-3, score-0.866]
3 However, the shape prior is usually learnt from a large set of annotated data, which is not always accessible in practice. [sent-4, score-0.235]
4 Moreover, it is often doubted that the existing shapes in the training set will be sufficient to model the new instance in the testing image. [sent-5, score-0.264]
5 In this paper, we propose to use the group similarity of object shapes in multiple images as a prior to aid segmentation, which can be interpreted as an unsupervised approach of shape prior modeling. [sent-6, score-0.633]
6 We show that the rank of the matrix consisting of multiple shapes is a good measure of the group similarity of the shapes, and the nuclear norm minimization is a simple and effective way to impose the proposed constraint on existing active contour models. [sent-7, score-1.631]
7 Experiments using echocardiographic image sequences acquired from acute canine experiments demonstrate that the proposed method can consistently improve the performance of active contour models and increase the robustness against image defects such as missing boundaries. [sent-9, score-1.072]
8 Among various techniques, the active contour model is widely used. [sent-12, score-0.719]
9 A contour is evolved by minimizing certain energies to match the object boundary while preserving the smoothness of the contour [2]. [sent-13, score-0.736]
10 The active contour is usually represented by landmarks [18] or level sets [20, 8]. [sent-14, score-0.76]
11 A variety of image features have been used to guide the active contour, typically including image gradient [7, 3 1], region statistics [34, 8], color and texture [14]. [sent-15, score-0.391]
12 In real applications, the performance of the active contour model is prone to be corrupted by missing or misleading features. [sent-16, score-0.976]
13 For example, segmentation of the left ventricle in ultrasound images is still an unresolved problem due Figure1. [sent-17, score-0.442]
14 Theimagesinthe top row show the segmentation results of the region-based active contour applied on each image separately. [sent-19, score-0.809]
15 The images in the bottom row show the corresponding results with a group similarity constraint on the shapes, which makes the shapes consistent with each other and less affected by local misleading features. [sent-20, score-0.65]
16 This property is desired in many applications, such as segmentation of the left ventricle from a cardiac image sequence. [sent-21, score-0.364]
17 to the characteristic artefacts in ultrasound such as attenuation, speckle and signal dropout [23]. [sent-22, score-0.254]
18 To improve the robustness of active contours, the shape prior is often used. [sent-23, score-0.549]
19 The prior knowledge of the shape to be segmented is modeled based on a set of manually-annotated shapes to guide the segmentation. [sent-24, score-0.4]
20 In more recent works, the shape prior was applied by regularizing the distance from the active contour to the template in a level-set framework [10, 24, 9]. [sent-26, score-0.917]
21 Another category of methods popularly used for shape prior modeling is the active shape model or point distribution model [11]. [sent-27, score-0.699]
22 Briefly speaking, each shape is denoted by a vector and regarded as a point in the shape space. [sent-28, score-0.334]
23 During the segmentation of a new image, the candidate shape is constrained in the shape space [19, 29]. [sent-30, score-0.39]
24 Other extensions of the active shape 222999666977 model include manifold learning [15] and sparse representation [33], to name a few. [sent-32, score-0.501]
25 Previous methods for shape prior modeling require a large set of annotated data, which is not always accessible in practice. [sent-34, score-0.235]
26 It is often doubted that the existing shapes in the training set will be sufficient to model the object shape in a new image. [sent-36, score-0.414]
27 In this paper, we propose to use the similarity among object shapes as a prior for segmentation. [sent-37, score-0.346]
28 A practical application is the segmentation of cardiac images, where the shape of the left ventricle shall keep consistent throughout a cardiac cycle although it deforms globally due to the heart beating. [sent-38, score-0.704]
29 If the active contour is applied on each of them separately, the segmentation result will be affected by the misleading features. [sent-41, score-0.968]
30 Our results are presented in the bottom row, where the similarity among the object shapes is used as a constraint to make the active contours more robust. [sent-42, score-0.998]
31 We showed that the vectors representing a group of similar shapes would form a low-rank matrix, even if they are different with each other due to certain global coordinate transformation. [sent-44, score-0.291]
32 Based on the low-rank property of similar shapes, we proposed to use the nuclear norm (convex surrogate of rank) to regularize the group similarity of shapes in segmentation. [sent-46, score-0.678]
33 The regularizer could be conveniently integrated into existing active contour models. [sent-47, score-0.761]
34 The experiments showed that the proposed constraint made the active contour model better regularized and require fewer iterations to converge. [sent-50, score-0.898]
35 We applied the proposed method to ultrasound image segmentation and demonstrated that the groupsimilarity regularization could significantly improve the robustness of the active contour model. [sent-52, score-1.055]
36 Group similarity measure To apply a group similarity constraint to active contours, a proper similarity measure is desired. [sent-60, score-0.832]
37 Typically, the similarity between two contours is measured by calculating the distances between the corresponding points on the contours, and the group similarity can be measured by the sum of pairwise distances between contours. [sent-61, score-0.526]
38 The main drawback of this method is that the contour distance is not invariant under similarity transformation. [sent-62, score-0.464]
39 For example, two contours with identical shapes but different sizes may be regarded as dissimilar based on the distance between them, which is not desired in our task of segmenting the left ventricle. [sent-63, score-0.509]
40 Here, we propose to use the matrix rank to measure the group similarity of shapes. [sent-64, score-0.379]
41 For example, the rank equals to 1 if the shapes are identical, and the rank may increase if some shapes change. [sent-68, score-0.728]
42 Moreover, we can show that the shape matrix is still lowrank if the shape change is due to the similarity transforma- tion such as translation, scaling and rotation. [sent-69, score-0.428]
43 of the shape matrix describes the degree of freedom of the shape change. [sent-86, score-0.332]
44 The low-rank constraint will allow the global change of contours such as translation, scaling, rotation and principal deformation to fit the image data while truncating the local variation caused by image defects. [sent-87, score-0.38]
45 Energy function Given a sequence of images I1, · · · , In, we try to find a set of contours C1, · · · , Cn to segment Ithe object in these images. [sent-90, score-0.317]
46 wfih h(Ceri)e i Xs th =e energy o·f , an active contour model to evolve the co(nCtour in each frame, such as snake [18], geodesic active contour [7], and region-based models [34, 8]. [sent-93, score-1.531]
47 Since rank is a discrete operator which is both difficult to optimize and too rigid as a regularization method, we propose to use the following relaxed form as the objective function: n mXini? [sent-97, score-0.232]
48 (4) Here, rank(X) in (2) is replaced by the nuclear norm ? [sent-101, score-0.227]
49 As a tight convex surrogate to the rank operator [16], the nuclear norm has several good properties: Firstly, the convexity of the nuclear norm makes it possible to develop fast and convergent algorithms in optimization. [sent-109, score-0.688]
50 Secondly, the nuclear norm is a continuous function, which is important for a good regularizer in many applications. [sent-110, score-0.269]
51 For instance, in our problem, the small perturbation in the shapes may result in a large increase of rank(X), while ? [sent-111, score-0.202]
52 gives the curve evolution steps in typical active contour models. [sent-121, score-0.822]
53 g The intuition of our algorithm is that, at each iteration, we first evolve the active contours according to the imagebased forces and then impose the group similarity regularization via singular value thresholding. [sent-179, score-0.96]
54 Results In this section, we evaluate the proposed method on both synthesized data and ultrasound images. [sent-196, score-0.215]
55 To demonstrate the advantages of the group similarity constraint, we compare the results of the same active contour model before and after applying the proposed constraint. [sent-197, score-0.904]
56 We select the regionbased active contour in (3) as the basic model, which is less sensitive to initialization and has fewer parameters to tune compared with edge-based methods. [sent-198, score-0.748]
57 In our implementation, we initialize the active contours as X0 = [C0, · · · , C0], where C0 is a rough outline of the object placed manually ]i,n w an image. [sent-199, score-0.596]
58 β in (3) controls the smoothness of each contour, λ in (4) controls the group similarity of contours, and μ in (7) controls the step-length of curve evolution in each iteration. [sent-201, score-0.378]
59 The dark region in a heart shape is the object to be segmented, which has different sizes and positions in different images. [sent-208, score-0.203]
60 Our goal is to find the heart shapes in all images based on the prior that the shapes are similar to each other in these images. [sent-211, score-0.505]
61 As shown in Figure 2, our results with the shape similarity constraint are more robust against the local defects in images compared to the results without such constraint. [sent-212, score-0.405]
62 Ultrasound image segmentation We apply our method to five 2-D short-axis sequences of canine echocardiography. [sent-216, score-0.215]
63 The echocardiography segmentation is a very challenging problem due to various misleading features in ultrasound images. [sent-217, score-0.464]
64 For each panel, the top row and the bottom row present the results of region-based active contours without and with the proposed constraint, respectively. [sent-222, score-0.596]
65 Sequence information including the data size, the rank of output X, the number of iterations for the algorithm to stop, and the computational time. [sent-237, score-0.208]
66 The results of the region-based active contour without the proposed constraint are given in the top rows. [sent-242, score-0.823]
67 For example, in frames 7, 9 and 13 of sequence 1, the active contour fail- s to stop near the true endocardial border on the right due to the low contrast over there. [sent-244, score-0.891]
68 Moreover, the active contour is prone to be trapped by the misleading features, e. [sent-246, score-0.878]
69 the upper part of the contour in frame 1 of sequence 2 is attracted to the bright pattern in the blood pool. [sent-248, score-0.561]
70 Firstly, the contour shapes are globally consistent with each other throughout the sequence, which is attributed to the group similarity constraint. [sent-251, score-0.755]
71 Hence, the contours are more resistent to local misleading features. [sent-252, score-0.404]
72 For example, the segmented shapes in frame 1 and frame 13 of sequence 2 are largely different and the shape change is far away from the affine transformation. [sent-254, score-0.584]
73 Therefore, the region-based active contours cannot attach closely to the true boundary. [sent-257, score-0.596]
74 Quantitative evaluation of the segmentation results w/ and w/o the proposed constraint on the region-based active contours. [sent-289, score-0.545]
75 Let C1 and C2 be two contours to aben compared, cthoeenff MAD(C1,C2) =? [sent-295, score-0.245]
76 s (13) Here, d(p, C) is the minimum distance from point p to contour C, |C| represents the contour length, ΩC denotes the region ,in |sCi|de re pC,r e|ΩseCn |t means othnteo area onfg tthhe, region, and sup riendgiiocant eisn stihdee supremum. [sent-300, score-0.79]
77 This is due to the fact that part of the segmentation result is corrupted by the missing boundary while this error can be corrected by adding the shape constraint. [sent-310, score-0.338]
78 As shown in Table 2, the standard deviation with the proposed constraint is distinctly lower than that without the constraint, which shows the significance of the proposed constraint to improve the robustness of the active contour model. [sent-312, score-0.957]
79 Effect of λ The most important parameter in our method is the weight λ of the nuclear norm regularization in (4). [sent-315, score-0.258]
80 The larger λ is, the lower rank(X) will be, which makes the output contours more similar to each other. [sent-316, score-0.245]
81 Another alternative way is to choose a constant K specifying the degree of freedom allowed for shape variation and then solve the model with a decreasing sequence of λ until rank(X) reaches K. [sent-322, score-0.25]
82 The experiments showed that the algorithm with the shape constraint converged faster than that without shape constraint. [sent-328, score-0.404]
83 This can be explained by the fact that the added constraint will make the active contour model better regularized, which results in faster convergency and fewer iterations. [sent-329, score-0.852]
84 For instance, in the case of missing boundary, the active contour without the shape constraint may evolve further away from the ground truth and converge slowly, where we have to stop it manually after a number of iterations, e. [sent-330, score-1.132]
85 Discussion In this paper, we proposed a simple and effective way to regularize the group similarity of shapes in the active contour model based on low-rank modeling and rank minimization. [sent-338, score-1.299]
86 In the active shape model [11], a candidate shape is constrained in a shape space described by C(w) = C + Φw, where C is the mean shape, sΦc i sb a m byatr Cix(w w)it =h c Co l+um Φnws representing tdhieffe mreeantn smhoadpees, of shape variation, and w is a vector of coefficients. [sent-341, score-0.951]
87 Hence, the active shape model also admits a low-dimensional assumption, which is similar to our low-rank assumption. [sent-343, score-0.501]
88 The difference is that the shape space in the active shape model is constructed from offline training, while the low-rank model in our method is constructed unsupervisely along with the segmentation. [sent-344, score-0.651]
89 Another topic closely related to our work is shape analysis such as shape clustering [26, 25]. [sent-347, score-0.3]
90 The difference is that sample shapes are provided as inputs in shape analysis while they are the outputs of segmentation. [sent-348, score-0.352]
91 We use the landmarks to represent the contour instead of level sets. [sent-350, score-0.409]
92 For instance, if there are n contours represented by the zero-level sets of n signed distance functions (SDFs), and the contours are identical in shape but different in location, the matrix consisting of the vectorized SDFs has a rank of n, which is full-rank. [sent-352, score-0.862]
93 A limitation of using the shape similarity constraint is the possibility of removing frame-specific details of the shapes. [sent-355, score-0.35]
94 A possible solution in our problem is to refine the segmentation by running an active contour model that is more sensitive to local features with our results being both initialization and templates to constrain the curve evolution. [sent-357, score-0.873]
95 In this paper, we demonstrate the advantages of our method based on 2-D active contours. [sent-359, score-0.351]
96 Appendix All ultrasound images were acquired using an open chest canine preparation at the Yale Translational Research Imaging Center and the studies were performed with appropriate institutional approvals. [sent-369, score-0.361]
97 2D short-axis images were acquired at a frame rate of 122-149 fps using a Philips iE33 ultrasound system with X7-2 phased array transducer at a nominal frequency of 4. [sent-370, score-0.348]
98 Using prior shapes in geometric active contours in a variational framework. [sent-434, score-0.846]
99 Towards robust and effective shape modeling: Sparse shape composition. [sent-574, score-0.3]
100 Segmentation of the left ventricle from cardiac mr images using a subject-specific dynamical model. [sent-586, score-0.304]
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