nips nips2012 nips2012-78 knowledge-graph by maker-knowledge-mining

78 nips-2012-Compressive Sensing MRI with Wavelet Tree Sparsity

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Author: Chen Chen, Junzhou Huang

Abstract: In Compressive Sensing Magnetic Resonance Imaging (CS-MRI), one can reconstruct a MR image with good quality from only a small number of measurements. This can signiﬁcantly reduce MR scanning time. According to structured sparsity theory, the measurements can be further reduced to O(K + log n) for tree-sparse data instead of O(K + K log n) for standard K-sparse data with length n. However, few of existing algorithms have utilized this for CS-MRI, while most of them model the problem with total variation and wavelet sparse regularization. On the other side, some algorithms have been proposed for tree sparse regularization, but few of them have validated the beneﬁt of wavelet tree structure in CS-MRI. In this paper, we propose a fast convex optimization algorithm to improve CS-MRI. Wavelet sparsity, gradient sparsity and tree sparsity are all considered in our model for real MR images. The original complex problem is decomposed into three simpler subproblems then each of the subproblems can be efﬁciently solved with an iterative scheme. Numerous experiments have been conducted and show that the proposed algorithm outperforms the state-of-the-art CS-MRI algorithms, and gain better reconstructions results on real MR images than general tree based solvers or algorithms. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract In Compressive Sensing Magnetic Resonance Imaging (CS-MRI), one can reconstruct a MR image with good quality from only a small number of measurements. [sent-4, score-0.105]

2 According to structured sparsity theory, the measurements can be further reduced to O(K + log n) for tree-sparse data instead of O(K + K log n) for standard K-sparse data with length n. [sent-6, score-0.232]

3 However, few of existing algorithms have utilized this for CS-MRI, while most of them model the problem with total variation and wavelet sparse regularization. [sent-7, score-0.558]

4 On the other side, some algorithms have been proposed for tree sparse regularization, but few of them have validated the beneﬁt of wavelet tree structure in CS-MRI. [sent-8, score-0.955]

5 Wavelet sparsity, gradient sparsity and tree sparsity are all considered in our model for real MR images. [sent-10, score-0.482]

6 The original complex problem is decomposed into three simpler subproblems then each of the subproblems can be efﬁciently solved with an iterative scheme. [sent-11, score-0.166]

7 Numerous experiments have been conducted and show that the proposed algorithm outperforms the state-of-the-art CS-MRI algorithms, and gain better reconstructions results on real MR images than general tree based solvers or algorithms. [sent-12, score-0.401]

8 One limitation of MRI is its imaging speed, including both scanning speed and reconstruction speed. [sent-14, score-0.307]

9 Long waiting time and slow scanning may result in patients’ annoyance and blur on images due to local motion such as breathing, heart beating etc. [sent-15, score-0.194]

10 According to compressive sensing (CS) [1,2] theory, only a small number of measurements is enough to recover an image with good quality. [sent-16, score-0.44]

11 This is an extension of Nyquist-Shannon sampling theorem when data is sparse or can be sparsely represented. [sent-17, score-0.099]

12 Compressive Sensing Magnetic Resonance Imaging (CSMRI) becomes one of the most successful applications of compressive sensing, since MR scanning time is directly related to the number of sampling measurements [3]. [sent-18, score-0.339]

13 As most images can be transferred to some sparse domain (wavelet etc. [sent-19, score-0.125]

14 ), only O(K + K log n) samples are enough to obtain robust MR image reconstruction. [sent-20, score-0.105]

15 Recent works on structured sparsity show that the required number of sampling measurements could be further reduced to O(K + log n) by exploring the tree structure [4-6]. [sent-22, score-0.517]

16 A typical relationship in tree sparsity is that, if a parent coefﬁcient has a large/small value, its children also tend to be large/small. [sent-23, score-0.362]

17 Some methods have been proposed to improve standard CS reconstruction by utilizing this prior. [sent-24, score-0.141]

18 Specially, two convex models are proposed to handle the treebased reconstruction problem [7]. [sent-25, score-0.175]

19 In Bayesian compressive sensing, Markov Chain Monte Carlo (MCMC) and variational Bayesian (VB) are used to solve the tree-based hierarchical models [8][9]. [sent-27, score-0.2]

20 Turbo AMP [10] also well exploits tree sparsity for compressive sensing with an iterative approximate message 1 passing approach. [sent-28, score-0.594]

21 However, none of them has conducted numerous experiments on MR images to validate their superiority. [sent-29, score-0.198]

22 In existing CS-MRI models, the linear combination of total variation and wavelet sparse regularization is very popular [3,12-15]. [sent-30, score-0.528]

23 They are the state-of-the-art algorithms for CS-MRI, but none of them utilizes tree sparsity prior to enhance their performance. [sent-34, score-0.398]

24 In this paper, we propose a new model for CS-MRI, which combines wavelet sparsity, gradient sparsity and tree sparsity seamlessly. [sent-35, score-0.856]

25 In tree structure modeling, we assign each pair of parentchild wavelet coefﬁcients to one group, which forces them to be zeros or non-zeros simultaneously. [sent-36, score-0.662]

26 This is an overlapping group problem and hard to be solved directly. [sent-37, score-0.11]

27 Then each of subproblems has closed form solution or can be solved efﬁciently by existing techniques. [sent-39, score-0.084]

28 We conduct extensive experiments to compare the proposed algorithm with the state-of-the-art CS-MRI algorithms and several tree sparsity algorithms. [sent-40, score-0.365]

29 (3) Numerous experiments have been conducted to compare the proposed algorithm with the state-of-the-art CS-MRI algorithms and several general tree-based algorithms or solvers. [sent-44, score-0.098]

30 1 Related work Tree based compressive sensing If a signal is sparse or can be sparsely represented, the necessary samples to reconstruct it can be signiﬁcantly smaller than that needed in Nyquist-Shannon sampling theorem. [sent-47, score-0.39]

31 Moreover, if we know some prior about the structure of the original signal, such as group or graph structure, the measurements can be further reduced [4,5]. [sent-48, score-0.184]

32 Some previous algorithms have utilized the tree structure of wavelet coefﬁcients to improve CS reconstruction [7-10]. [sent-49, score-0.833]

33 OGL [7] is a convex approach to model the tree structure: 1 ˆ θ = arg min{F (θ) = ||b − AΦT θ||2 + λg 2 θ 2 n g∈G 1 ||θg ||2 + τ 2 2 i=1 j (θi − θi )2 } (1) j∈Ji where θ is a set of the wavelet coefﬁcients. [sent-50, score-0.69]

34 A represents a partial Fourier transform for MR reconstruction problem and b is the measurements data. [sent-51, score-0.239]

35 When wavelet coefﬁcients are recovered, they can be transformed to the recovered image by an inverse wavelet transform. [sent-55, score-0.917]

36 This method well explores the tree structure assumption, but may be slow in general as following reasons: a) the parent-child relationship in their model is hard to maintain; b) it applies SpaRSA [11] to solve (1). [sent-56, score-0.334]

37 In (2), x is the original image to be reconstructed and w is Gaussian white noise. [sent-59, score-0.17]

38 In these approaches, graphical models are used to represent the wavelet tree structure and the distribution of each coefﬁcient is decided by its parent’s value. [sent-60, score-0.662]

39 2 Efﬁcient MR image reconstruction algorithms In existing CS-MRI algorithms, the linear combination of total variation and wavelet sparsity constrains has shown good property for MR images. [sent-62, score-0.876]

40 α and β are two positive parameters, and Φ denotes the wavelet transform. [sent-64, score-0.406]

41 These approaches are very effective on real MR image reconstruction, but none of them utilizes the wavelet tree structure to get further enhancement. [sent-69, score-0.832]

42 3 1 Ax − b 2 2 2 +α x TV + β Φx 1 } (3) Convex overlapped group sparsity solvers SLEP [18] (Sparse Learning with Efﬁcient Projections) has the package for tree structured group lasso (4). [sent-71, score-0.503]

43 Its main function is to iteratively solve the tree structured denoising problem. [sent-72, score-0.33]

44 When it comes to reconstruction problem, it applies FISTA to transfer the problem to denoising. [sent-73, score-0.141]

45 x = arg min{ ˆ x 1 Ax − b 2 2 2 + β Φx tree } (4) YALL1 [19] (Your Algorithms for L1) can solve the general overlapping group sparse problem efﬁciently. [sent-74, score-0.423]

46 It ﬁrst relaxes the constrained overlapping group minimization to unconstrained problem by Lagrangian method. [sent-76, score-0.085]

47 Then the minimization of the x and z subproblems can be written as: x = arg min{ ˆ x,z β2 Ax − b 2 s 2 2 + λT GΦx + 1 β1 wi ||zi ||2 } ||z − GΦx||2 − λT Ax + 2 2 2 i=1 (5) where G indicates the grouping index with all its elements to be 1 or 0. [sent-77, score-0.089]

48 3 Algorithm Observations tell us that the wavelet coefﬁcients of real MR images tend to be quadtree structured [20], although not strictly. [sent-80, score-0.561]

49 Moreover they are generally sparse in wavelet and gradient domain. [sent-81, score-0.449]

50 Tree-based MRI problem can be formulated as follows: min{F (x) = x 1 Ax − b 2 2 2 +α x TV + β( Φx 1 ||Φxg ||2 )} + (6) g∈G The total variation and L1 term in fact have complemented the tree structure assumption, which make our model more robust on real MR images. [sent-84, score-0.367]

51 This is a main difference with previous tree structured algorithms or solvers. [sent-85, score-0.291]

52 1 Solution As mentioned above, z-subproblem in (7) can be written as: zgi = arg min{β||zgi ||2 + zgi λ ||zgi − (GΦx)gi ||2 }, i = 1, 2, . [sent-93, score-0.276]

53 , s 2 2 (8) where gi is the i-th group and s is number of total groups. [sent-96, score-0.112]

54 It has a closed form solution by soft thresholding: zgi = max(||ri ||2 − β ri , 0) , i = 1, 2, . [sent-97, score-0.149]

55 2 Algorithm analysis Suppose x represents an image with n pixels and z contains n elements. [sent-106, score-0.105]

56 Step 4 takes O(n log n) when the fast wavelet transform is applied. [sent-111, score-0.45]

57 Note that n ≤ 2n since we assign every parent-child coefﬁcients to one group and leave every wavelet scaling in one group. [sent-113, score-0.455]

58 We introduce the wavelet tree structure constrain in our model, without increasing the total computation complexity. [sent-115, score-0.694]

59 In the MR imaging problem, A is partial Fourier transform with m rows and n columns. [sent-119, score-0.109]

60 The fewer measurements we samples, the less MR scanning time is need. [sent-121, score-0.155]

61 So MR imaging is always interested in low sampling ratio cases. [sent-122, score-0.116]

62 We conduct experiments on four MR images : ”Cardiac”, ”Brain”, ”Chest” and ”Shoulder” ( Figure 1). [sent-131, score-0.082]

63 We ﬁrst compare our algorithm with the classical and fastest MR image reconstruction algorithms: CG[3], TVCMRI[12], RecPF[13], FCSA[14,15], and then with general tree based algorithms or solvers: AMP[10], VB[9], YALL1[19], SLEP[18]. [sent-132, score-0.496]

64 2 Comparisons with MR image reconstruction algorithms We ﬁrst compare our method with the state-of-the-art MR image reconstruction algorithms. [sent-145, score-0.522]

65 For convenience, all test images are resized to 256×256. [sent-146, score-0.082]

66 We decompose the wavelet coefﬁcients to 4 levels here since more levels would increase the computation cost and less levels will weaken tree structure beneﬁt. [sent-149, score-0.687]

67 Although tree structure deﬁnitely cost a little more time to solve, it always achieves the best performance in terms of SNR and CPU time. [sent-151, score-0.256]

68 We have conducted experiments on other images and obtain similar results that the proposed algorithm always has the best performance in terms of SNR and CPU time. [sent-152, score-0.12]

69 This result is reasonable because 5 we exploit wavelet tree structure in our model, which can reduce requirement for the number of measurements or increase the accuracy of the solution with the same measurements. [sent-153, score-0.738]

70 Figure 2: Brain image reconstruction with 20% sampling. [sent-154, score-0.246]

71 Visual results from left to right, top to bottom are original image, images reconstructed by CG [3], TVCMRI [12], RecPF [13], FCSA [14,15], and the proposed algorithm. [sent-155, score-0.171]

72 3 Comparisons with general algorithms of tree structure We also compare our algorithm with existing algorithms for tree sparsity with statistical inference and convex optimization. [sent-164, score-0.685]

73 Due to the higher space requirement and time complexity of VB, we resize all images to 128 × 128. [sent-169, score-0.082]

74 Figure 3 shows the reconstruction results on ”Brain” image with only 20% measurements. [sent-171, score-0.246]

75 These results are reasonable because none of the other algorithms uses the sparse prior of MR images in wavelet and gradient domain simultaneously. [sent-176, score-0.594]

76 Table 1: Comparisons of SNR (db) on four MR images Algorithms Iterations Cardiac Brain Chest Shoulder AMP [10] VB [9] SLEP [18] YALL1 [19] Proposed 10 10 50 50 50 11. [sent-177, score-0.082]

77 Although statistical algorithms are slow in general, they have the convenience without tuning parameters, as all parameters are learned from data. [sent-218, score-0.086]

78 Fortunately, good parameters for MR image reconstruction are easy to tune in our model. [sent-219, score-0.246]

79 Except the proposed algorithm, all other algorithms have a strong assumption of the tree structure. [sent-220, score-0.25]

80 However for real MR data, many images do not strictly follow this assumption. [sent-221, score-0.114]

81 14 Figure 3: Brain image reconstruction with 20% sampling. [sent-262, score-0.246]

82 Visual results from left to right, top to bottom are original image, images reconstructed by AMP [10], VB [9], SLEP [18], YALL1 [19], and the proposed algorithm. [sent-263, score-0.171]

83 To show the beneﬁt of proposed model, we design another experiment on a toy MR image, which more strictly follow the tree structure assumption. [sent-273, score-0.284]

84 First we set the wavelet coefﬁcients who have the smallest 0. [sent-274, score-0.406]

85 Figure 4 shows the original toy brain image and corresponding results of different algorithms. [sent-278, score-0.227]

86 From Figure 4 and 3, we could ﬁnd that the proposed algorithm has great superiority on real MR image, because we combined TV and wavelet sparsity, which ”soften” and complement the tree structure assumption for real MR data. [sent-280, score-0.726]

87 Other tree based algorithms depend on ”hard” tree structure only, which makes them hard to perform well on CS-MRI. [sent-281, score-0.506]

88 7 Figure 4: Toy image reconstruction with 20% sampling. [sent-287, score-0.246]

89 Visual results from left to right, top to bottom are original image, images reconstructed by AMP [10], VB [9], SLEP [18], YALL1 [19], and the proposed algorithm. [sent-288, score-0.171]

90 All algorithms terminate after 50 iterations except AMP [10] and VB [9] terminate after 10 iterations. [sent-297, score-0.128]

91 5 Conclusions Real MR images not only tend to be tree structured sparse, but also are sparse in wavelet and gradient domain . [sent-299, score-0.792]

92 To solve this model, we decompose the original problem into three simpler ones and solve each of them very efﬁciently. [sent-301, score-0.138]

93 Compared with the state-of-the-art algorithms in CS-MRI, the tree structure in our model can help reduce the required measurements, and leads to better performance. [sent-304, score-0.286]

94 Compared with general tree sparsity algorithms, our algorithm can obtain more robust results on real MR data. [sent-305, score-0.367]

95 Future work will be combining the proposed algorithm with the nonlocal total variation [22] for multi-contrast MRI [21]. [sent-306, score-0.112]

96 (2006) Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. [sent-314, score-0.173]

97 (2007) Sparse MRI: The application of compressed sensing for rapid MR imaging. [sent-320, score-0.158]

98 (2008) An efﬁcient algorithm for compressed MR imaging using total variation and wavelets. [sent-376, score-0.22]

99 (2010) A fast alternating direction method for tvl1-l2 signal reconstruction from partial fourier data. [sent-382, score-0.244]

100 (2009) Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. [sent-401, score-0.238]

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