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6th International Conference on Applied Human Factors and Ergonomics (AHFE 2015) and the

Affiliated Conferences, AHFE 2015

Graph-based sparse representation for image denoising

Qi Gea *, Xiaogang Chenga, Wenze Shaoa,Yue Dongb,Wenqin Zhuanga, Haibo Lia,t

aCollege of Telecommunications & Information Engineering, Nanjing University of Posts and Telecommunications,Nanjing, Jiangsu, China bInstitute of Advanced Technology, Nanjing University of Posts & Telecommunication, Nanjing, Jiangsu, China

Abstract

Sparse representation has shown the effectiveness in solving image restoration and classification problems. To improve the performance of sparse representation, thepatch-based and graph-based regularization term with respect to the sparse coding areproposedto solve image restoration and classification problems, respectively. In this paper, the local manifold structure of intensity on patches is exploitedby a graph Laplacian operator forperforming more precise estimation of sparse coding. Additionally, an improved nonlocal regularization term with the local manifold structure information is proposed to preserve the texture more effectively compared with the traditional nonlocal regularization term.The experimental results on image denoising show the promising performance in terms of both peak signal noise ratio and visual perception.

© 2015 Published by Elsevier B.V.Thisis an open access article under the CC BY-NC-ND license

(http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of AHFE Conference

Keywords: Sparse representation;Graph-based regularization term; Manifold structure; Image denoising

1. Introduction

Image denoisingis an important application in several fields, including astronomy[1], remote sensing[2], biomedical imaging[3], etc. Image denoising problem isoften formulated as an optimization problem, where the criterionis a linear combination of dataconsistency error term and a regularization term. The classical regularization term, e.g., regularization term based on total variation (TV) [4-6], have been widely used in image restoration for its

* Corresponding author. Tel.: +086-1351-2510-326.

E-mail address: geqi@njupt. edu. cn ^Corresponding author. Tel.: 086-1395-2090-959. E-mail address: lihb@njupt.edu. cn

2351-9789 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of AHFE Conference

doi: 10.1016/j.promfg.2015.07.253

effectiveness in removing the noise while preserving the edges. However, the TV-regularization term results in reconstructed images with staircase artifacts and over-smooth. To figure out these problems, the sparsity-based regularization term is introduced to solve the image restoration problems. By the sparse representation, the data is represented as a linear combination of a few numbers of basis such that it can be interpreted in a elegant way. Moreover, the over-complete sparse representation gives a redundant representation over trained dictionaries. Benefit from these advantages, the sparse representation is widely utilized in image restoration [13-22], face recognition and classification[23-25].

The research in [24-25] shows that the sparse learning performance can be significantly improved by the geometrical structure of the learning data. The graph-based regularization term of sparse coding proposed in [9] exploits the local geometrical structure of the sample data to learn the sparse representation of the sample data. The results on image classification in [9]showthat the graph-based regularization term of sparse coding promotes the discrimination of the classifier.

Motivated by the recent works in sparse coding based on the manifold structure, we propose a novel image denoising algorithm with a graph-based regularization term by encoding the geometrical structural information into the sparse representation model. By the technique of graph Laplacian operator, the similarity in a k-nearest neighbor of each patch is extracted. Through this process, the obtained sparse representation is more discriminating compared with the traditional sparse coding algorithms based on the patch information. The experimental results of the proposed method will show the outperformance in comparison with other image denoising algorithms based on sparse representation.

The rest of the paper is organized as follows: in Section 2, we review the related works on the image restoration based on sparse representation. In Section 3, we introduce the modeling of the proposed algorithm and the numerical implements. In Section 4, the experimentalresults are represented and in Section 5 concludes the paper.

2. Background

2.1. Sparse representation of image problem

In the scenario of image denoising, the observed image is modeled as y = x + v, where v is assumed to be a Gaussian distributed white noise process of a specified standard deviation <r . The sparse representation model assumes that aimage x » Da , where D e RKyM (K < M ) is an over-complete dictionary, and most entries of the coefficient vector a are zero. The sparse decomposition of x can be obtained by solving an l0 -minimization problem, formulated as,

ax = argmin||a|L , s.t. I|x-Dall <£ (1)

a 110 II II2

where 11*|| 0 is a pseudo norm that counts the number of non-zero entries in a , and^ is a small constant controlling the approximation error. Since l0 -minimization problem is an NP-hard combinational optimization problem, it can be relaxed to the convex l1 -minimization. The objective function of sparse representation model for image restoration can be formulated as follows,

a = argmin j||x - Da|| 2 + ,«|| a| (2)

where /u is the regularization parameter. It is a trade-off between the minimization the sparse error of x and the sparse coefficients a, the first term is the fidelity term and the second term is the regularization term with respect to the sparse coefficients a . The basis pursuit (BP) has been adopted in solving the l1 -minimization problem Eq. (2) in

2.2. Non-locally centralized sparse representation model

The non-local patch similarity prior is extracted for regularizing the sparse coefficients a [13,14], since the nonlocal prior has led to promising results in image restoration, especially in image denoising. By adding the non-local regularization term of the sparse coefficients a , the co-called non-locally centralized sparse representation (NCSR) model is formulated as follows:

a = argnun j||y - ® °D«||2 + p^ll«i I1 + ¿11«i " Pi||p} (3)

where ® is a degradation matrix, ® is an identity matrix for image denoising problem, ai is the sparse coefficients of each local patch, pi is the nonlocal-prior-estimation of ai , X is a regularization parameter of the non-local regularization term, and p is 1 or 2. Compared with the classical sparse representation model for image denoising [22], the nonlocal regularization term with respect to the sparse coefficients is added in the Eq.(3). This regularization term utilize the nonlocal redundancy of the images such that the local structure of the images can be preserved better than the traditional patch-learned dictionary method.

3. Graph-based sparse representation for image denoising

Traditional dictionaries of DCT, wavelet and curvelet dictionaries are lack of structure information of images. The dictionaries learning from example patch [22] enlarge the redundant such that the dictionaries can represent more local structures. However, there is little image structure information in the dictionaries. Additionally, the traditional nonlocal regularization term measures the similarity between two patches depending on the intensity in each patch while not the geometrical structure of the intensity. In real applications of image process, the intensity distribution is more likely to reside on a low-dimensional sub-manifold embedded in a high-dimensional manifold. Consequently, it is necessary to employ the local manifold structure to model the intensity distribution.

3.1. Objective function

In the patch-based sparse representation model [22], it tries to find a product of dictionary D and a sparse coefficient ai to approximate each patch zi of the image X. Each row of the dictionary D corresponds to the pixels in the patch. Analogously, the similarity between two adjacent rows of the dictionary D equals to the similarity between two corresponding pixels. Assuming that the similarity of the intensity of the two pixels are close based on the intrinsic geometry of data distribution, then the two corresponding rows of D are also similar to each other.

Given the set of image patches

MM. (4)

the size of the given patch zh is 4n xyfn , h is the index of each patch , then the pixels on each patch can be represented as x1 —, xn. Suppose the size of the image is VN xVN , N >> n . We construct a graph G = (V,E)with n vertices and edges eij e E c V x V, i, j = 1—n, where each vertex represents a pixel and an edge etJ spans two vertices xi, xj. Let:

W = (w )j (5)

be the weight matrix of G. The typical Gaussian weighting function is utilized in this work,

Wjj = exp {-fi{lt -1J )2) (6)

where Ii indicates the image intensity at xi . The degree of one vertex is

4 = ^ wij for all edges eij incident on xi. (7)

Fig.1. Graph-based nonlocal estimation w.r.t. the sparse coding. The red arrows represent the local similarity extracted by the combinatorial Laplacian matrix; while the blue arrows represent the similarity between the two patches.

With the degrees of the vertex, the degree matrix of every vertex is defined asD = diag(cll,d2,---dn j. Then the combinatorial Laplacian matrix is defined as Lh = D - W [10]. According to the graph-based segmentation model [11, 12], the intrinsic Riemann structure of dictionary D can be formulated by a combinatorial formulation S' = DTLhD with the graph Laplacian matrix Lh. The matrix S* is a symmetrical matrix whose elements equal to a Dirichlet integral of each atom of the dictionary D. The Dirichlet integral of each atom is equivalent to the regularization term which keeps the similarity between two adjacent elements of the atom, if the corresponding pixels of the two elements are similar. To the contrary, it boosts the dissimilarity if the corresponding pixels are unlike. Consequently, the sparsity of the dictionary is improved. To keep the orthogonality of the dictionary D, we normalize the matrix S* by S = DTLhD/||D||2. Hence, we propose the following sparse coding model as follows:

[ah, x\ = argmin

Xb - x\\2 IKIli + XI\Rhx - DSah112 h h

hh - NL_G(«h)||,

where Lh is the Laplacian matrix of the h-th patch, matrix Rh represents an matrix that extracts the h-th block from the image, ah is the sparse coding of the h-th patch, and NL _ G (ah) is the nonlocal graph estimation of patch sparse coding a h.

3.2. The nonlocal graph estimation of sparse coding

The traditional nonlocal regularization term of the sparse coding ah,h = 1—M on a given patch is estimated as the weighted averages of the other patches in a search window, i.e., the nonlocal similar patches. In [13],the authors cluster the set of the patches {z based on the pixel intensity to form several subsets |nj} , where C is the number of the subsets, j is the index of the clusters. Then the nonlocal estimation of the sparse coding ah is defined as [13]:

Pi = ZtexP HIDah " Da,

where ah k is the sparse coding corresponding to the patch zk , d is the scale parameter of the Gaussian function, and £ is the normalization factor. Based on this traditional nonlocal estimation, we develop a novel nonlocal graph estimation NL _ G (ah) of sparse coefficient ah on each patch as follows,

NL _ G (ah) = X Texp (-11 h (D«h) - Lh „ (Da 117f) S h

ten. C \ '

where Lhk is the Laplacian matrix of the patch zhk. Each element of the row vectors of Lhk corresponds to the similarity between the adjacent pixels in a patch. The matrix Lht generalizes to a Laplace-Beltrami operator [30] to calculate the estimated value of the observed pixel according to the adjacent pixel. Under the framework of the iteratively optimization method [27], minimizing the object function Eq.(8) consists two steps: learning graph regularized sparse coefficients {ah}h^ while fixing the dictionary D, and learn dictionary D while fixing the coefficients {ah\M=i. We show the function of the nonlocal graph estimation Eq.(10) in Fig. 1.

3.3. Numerical solution

3.3.1. Learning sparse coefficients with graph regularization

We update the sparse coefficients {ah}"j by assuming the underlying dictionary D is known. The minimization problem Eq.(10) has two unknowns, the sparse coefficient and output image x. Based on the numerical algorithm in [22], the problem Eq.(10) becomes

a h = argmin -j |a„| [ + XH^* - DSaJ2 + |a„ " NL _ G (« h )|| p [ . (11)

"h ,x l h h h J

This minimization problem with l1 -minimization is non-differentiable and convex. Similar to the numerical solution of [13], we use the surrogate algorithm in [14] to solve Eq.(11). On the (k + 1)-th iteration, the proposed shrinkage operator for the j-th element of ah is

aj+1 = ST{vk - NL _ G (a))) + NL _ G (a)) (12)

whereis the soft-thresholding operator and vk = Dr(Rhx- Dak)/c + ak, where r = // c , and c is the parameter guaranteeing the convexity of the surrogate function.

3.3.2. Learning the dictionary D

When the sparse coefficients {ah}M^1 are fixed, the dictionary D is learnt by solving the minimizing problem as follows:

Vh e{1---M},min||1,s.t.|Rhx-DS«h| <&2 (13)

For each column of D = {dj,---,dN}, find the set of patches that use this atom, Q.l = {h|ah (l) * 0,l = 1—N}. Foreach index h efll, solving the representation error

eh = RhXh -XdnSn«h (n) (14)

Sn eS = (S1 •■■ Sn),S = ^^. (15)

Set the El as the matrix whose columns are the

Then the SVD algorithm is applied in decomposition El = UAVT . Choose the first column of U as the updateddictionary dl.

We summarize the algorithm of minimizing the proposed energy functional as follows,

Algorithm 1

Step1. Initialization:

Set the initial estimate as x — y for image denoising and deblurring, and the parameters of regularization term Я,(Л,у . Step2. Do the iteration:

Substep2.1. Update the dictionaries jdn j by solving Eq.(8) with SVD algorithm. Substep2.2. while \\Rhx - DSaJ <£2,£- 0.2 Do the loop as follows:

(a) xJ+1/2 = xJ + S(^y — xJ ^ where Ö is step size parameter.

(b) Compute VJ by vJ = DT (.Rhx - DaJ )/c + aj .

(c) Compute the sparse coefficients tt J according to Eq.(6).

(d) Update the denoising image by computing

x = + ZRrR] (ЛУ+RTS (DaA))

4. Experimental results

In this section, we demonstrate the denoising results that are achieved by the proposed method on the typical test images. And we compare the results of the several related methods and the proposed method. For the convenient of comparison, the patch size is set as 7 x 7 , the number of clusters is k = 70 , S = 0.02 ,which are the same as the nonlocal centralized sparse representation model(NCSR)[13]. We add the noise variation <j = 10,20,50,80,100 for comparing the results of the methods of SAPCA-BM3D[26], the nonlocal sparse representation for image restoration (NSR) [19], the nonlocal centralized sparse representation (NCSR)[13], and the proposed method. For evaluating the quality of the denoised images, the peak signal noise ratio (PSNR) is presented in the Table 1.

Table 1. Results of peak signal noise ratio (PSNR), top left- SAPCA-BM3D[26], top right- NSR[19], down left-NSCR, down right-the proposed method.

О 10 20 50 80 100

Lena 36.07 35.83 33.20 32.85 29.08 28.96 2l.03 27.26 25.3l 25.96

35.l8 35.81 32.92 32.95 28.88 29.02 2l.11 2l.12 25.58 25.86

Monarch 34.76 34.48 30.91 30.58 26.28 25.60 24.03 23.36 22.52 21.82

34.5l 34.60 30.l1 30.l3 25.69 25.83 23.42 24.01 22.05 22.22

Barbara 35.07 34.95 31.97 31.53 27.51 2l.13 24.93 25.28 23.08 23.56

34.99 34.98 31.l2 31.l4 2l.10 2l.43 25.08 25.32 23.22 23.56

Boat 34.10 33.99 31.02 30.8l 26.89 26.l6 24.87 25.08 23.l0 23.94

33.8l 33.44 30.l4 30.80 26.59 26.l8 24.42 24.ll 23.62 23.85

Cameraman 34.52 34.14 30.86 30.54 26.89 26.l6 24.62 24.l3 22.8l 23.14

34.10 34.13 30.44 30.50 26.5l 26.64 24.43 24.69 22.82 23.05

Couple 34.13 33.9l 30.83 30.l2 26.48 26.31 24.51 24.l0 23.19 23.34

33.94 33.98 30.5l 30.64 26.22 26.45 24.65 24.73 23.35 23.57

Fig.2.Denoising performance comparison on the Barbara image. (a) Original image. (b) noisy image

( ct - 1°° ), (C) denoised image by SAPCA-BM3D[33], (d) denoised image by [13] with the traditional nonlocal regularization; (e) denoised image by the proposed method with the traditional nonlocal-regularization; (f) denoised result of the proposed method.

In Figs.2, we show the comparison of denoising results on four typical test images. We add the noise level on both strong noise corruption (noise level is a = 100). To show the effectiveness of the graph-based regularization term, we show the comparison on the NCSR method, the proposed method with the traditional nonlocal regularization term and the proposed method. It can be seen from Figs.2(c) that some blocky effect is generated by the SAPCA-BM3D method. As shown in Fig.2(d), although the NCSR method avoids the blocky effect, it smoothes the textures and the edges are not well preserved. By comparing Fig.2(d) to Fig.2(e), it can be seen that with the local geometrical information, the proposed method is more effective than the image denoising methods without the local geometrical information. By comparing Fig.2(e) to Fig.2-3(f), the proposed method is more effective in preserving the texture part than the proposed model with the traditional nonlocal regularization term. The textures region is better preserved by the proposed method than the NCSR method, and the smooth part of the image is also much clearer than the NCSR method and the proposed method with the traditional nonlocal regularization term.

5. Conclusion

In this paper, we propose a graph-based sparse representation model for image denoising. The traditional sparsity-based methods ignores the local manifold structure of the image intensity. To this end, we apply the graph-based method to exploit the manifold feature on each patch and apply it to construct the nonlocal regularization term. Experimental results demonstrate that the proposed method has highly competitive performance and robustness on various kinds of images comparing with state-of-art denoising methods based on sparsity.

Acknowledgements

The authors would like to thank the anonymous reviewers for their many valuable comments and suggestions that helped to improve both the technical content and the presentation quality of this paper. This research is supported by the Talent Introduction Foundation of NJUPT (NY213007),the Foundation Institute of Advanced Technology (XJKY14012).

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