Intelligent nonconvex compressive sensing using prior information for image reconstruction by sparse representation

Author’s Accepted Manuscript Intelligent Nonconvex Compressive Sensing Using Prior Information for Image Reconstruction by Sparse Representation Qiang Wang, Dan Li, Yi Shen www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(16)31292-9 http://dx.doi.org/10.1016/j.neucom.2016.10.051 NEUCOM17687

To appear in: Neurocomputing Received date: 17 May 2016 Revised date: 16 August 2016 Accepted date: 31 October 2016 Cite this article as: Qiang Wang, Dan Li and Yi Shen, Intelligent Nonconvex Compressive Sensing Using Prior Information for Image Reconstruction by Sparse Representation, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.10.051 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Intelligent Noncovex Compressive Sensing Using Prior Information for Image Reconstruction by Sparse Representation Qiang Wang Member IEEE, Dan Li, Yi Shen Member IEEE 1 Control Science and Engineering, Harbin Institute of Technology, No.92, West Da-Zhi Street, Nangang District, Harbin, China, 150001 [email protected], [email protected], [email protected]

Abstract Image reconstruction by sparse representation, which is based on the fact that natural images are intrinsically sparse under some over-completed dictionaries, has shown promising results in many applications. However, due to the down-sampled measurements, the results of image reconstruction by sparse representation are sometimes not accurate enough. In this paper, we propose a novel intelligent nonconvex compressive sensing (INCS) algorithm using prior information for image reconstruction by sparse representation. First of all, the over-completed dictionary of Ridgelet is used to introduce the sparse level for each image block. Then we use the nonlocal self-similarity property and joint sparsity to obtain the basic prior information to guide the reconstruction, which contributes a lot to improving the reconstruction accuracy and reducing the computational complexity. To enhance the guidance accuracy of prior information, the property that natural image blocks spatially nearby share the similar structures is exploited to extract more information to enrich the basic prior information. Under the guidance of prior information, the intelligent optimization algorithm, which performs superiorly in solving combinatorial optimization problems and global searching, is utilized to solve the nonconvex l0 minimization problem essentially. By means of the prior information and the intelligent searching strategy, the proposed INCS can not only improve the reconstruction accuracy significantly but also reduce the computational complexity to accel-

Preprint submitted to Journal of LATEX Templates

August 16, 2016

erate the reconstruction speed. Extensive experiments on five natural images are conducted to verify the performance of our proposed method INCS. The experimental results demonstrate that INCS outperforms the state-of-the-art algorithms in terms of PSNR, SSIM and visual quality. Keywords: Image reconstruction, Sparse representation, Nonconvex l0 minimization, Intelligent optimization, Prior information.

1. Introduction Sparse representation, which is based on the fact that natural images are intrinsically sparse in some domains, has shown promising performances in various image applications, such as image reconstruction [1, 2, 3], image denoising [4, 5, 6] and super-resolution [7, 8, 9]. Image reconstruction by sparse representation aims to reconstruct each image block by finding the sparse representation from the over-completed dictionary based on the down-sampled measurements. The over-completed dictionary [10, 11], which has more columns than rows, can represent the natural image blocks more sparsely and better characterize the image structures. Although the natural images can be sparsely represented by some orthogonal dictionaries with the advantage of fast implementation, the over-completed dictionary can offer more flexible and adaptive sparse representations for natural image structures based on the redundant property. Suppose the original image block is represented by x ∈ Rn , we can obtain the down-sampled measurements y ∈ Rm by the measurement matrix φ ∈ Rm∗n (m << n), where y = φ ∗ x. As the original image block x can be sparsely represented by the over-completed dictionary ψ, then y = φ ∗ ψ ∗ s, where s is the sparse representation of x. In compressive sensing theory [12, 13], the goal of reconstruction is formulated as Eq.(1). s∗ = argmins  s 0

subject to

y =φ∗ψ∗s

(1)

where  s 0 is the l0 norm of s, which counts the nonzero components of the sparse vector. 2

If the sparsity level K is known as a prior, the CS reconstruction problem can be given by Eq.(2). s∗ = argmins  y − φ ∗ ψ ∗ s 22

subject to

 s 0 ≤ K

(2)

The sparse representation of the natural image block can be calculated by Eq.(3). x=ψ∗s=



si ∗ di = D ∗ S

(3)

i∈Δ

where Δ presents the indexes collection of the nonzero components of the sparse vector. di indicates the column of ψ corresponding to the nonzero index i in Δ. D ∈ Rn∗K consists of all the corresponding columns of ψ and S consists of all the nonzero coefficients. Based on Eq.(3), the problem of image reconstruction by sparse representation can be modeled by Eq.(4).

(D∗ , s∗ ) = argminD,s  y − φ ∗ ψ ∗ s 22

subject to

 D p,0 ≤ K (4)

where  D p,0 denotes the indexes of nonzero columns of D. After the estimated matrix D∗ is obtained, the estimated spare representation s∗ can be calculated by Eq.(5). s∗Idx(D) = (φ ∗ D∗ )† ∗ y

and

s∗Idx(ψ)−Idx(D∗) = 0

(5)

where Idx(.) is the operation which aims to extract the indexes of the atoms (.) in the over-completed dictionary ψ and Idx(ψ) represents all the indexes {1, 2, ..., n}. (.)† is the pseudo-inverse operation, which can be calculated by (.)† = ((.) ∗ (.))−1 ∗ (.) . Then we obtain the reconstructed image block by x∗ = D∗ ∗ s∗ . However, Eq.(4) is a nonconvex l0 minimization problem, which has a high computational complexity and is difficult to be achieved by traditional algo-

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rithms. The common methods for solving nonconvex l0 minimization is greedy algorithms, such as orthogonal matching pursuit (OMP) algorithm [14] and compressive sampling matching pursuit (CoSaMP) algorithm [15]. Nevertheless, greedy algorithms always need a relatively large measurement rate to obtain a better reconstruction accuracy and it is more likely to fall into a sub-optimal solution. Intelligent optimization algorithm can be utilized to solve the nonconvex l0 minimization essentially since it has a superior performance in solving combinatorial optimization problems and global searching. In our previous works [16] and [17], we proposed two reconstruction methods based on genetic algorithm (GA) and artificial immune algorithm (AIA) to solve the l0 minimization directly, which achieved significant reconstruction accuracy but slow reconstruction speed. Simulated annealing (SA) algorithm [18, 19] was also widely used to solve the nonconvex l0 minimization, such as simulated annealing algorithm for sparse reconstruction (SASR) [20], hybrid simulated annealing thresholding (HSAT) algorithm [21] and heuristic search algorithm for multiple measurement vectors problem (HSAMMV)[22]. To improve the reconstruction performance when the measurement rate is relatively small, we proposed an intelligent greedy pursuit model in [23] to solve the nonconvex l0 minimization essentially by combining the superiorities of intelligent optimization algorithm and greedy algorithms. However, based on the fact that intelligent searching needs more iterations to find the global optimal solution, the computational complexity is also relatively high. Although intelligent optimization algorithm can be used to solve the nonconvex l0 minimization, it is challenging to find the global optimal solution D∗ from the over-completed dictionary, which has a large amount of columns. To solve this problem, it is advisable to extract some prior information based on the particular structures of natural images to guide the reconstruction. Prior information [24, 25], which is beneficial to improving the reconstruction accuracy and reducing the computational complexity, plays an important role in image reconstruction. Some particular structures in images, such as edge, texture and tree structures, can be exploited to obtain the prior information. In 4

deed, many recent works were based on prior information. In our previous work [26, 27], two novel methods for image sequences reconstruction and image reconstruction were proposed respectively, which both utilize the rough edge as the prior information to guide the reconstruction and improve the reconstruction quality significantly. However, the methods in [26, 27] just perform well for images with obvious edge and high sparsity in wavelet domain, such as computed tomography (CT) images and magnetic resonance (MR) images. For natural images, the nonlocal self-similarity [28, 29, 30] property has found successful applications in many image restoration methods, such as reconstruction, denoising, super-resolution, etc. [28] proposed a centralized sparse representation (NCSR) model for image reconstruction, which exploits the image nonlocal self-similarity to obtain good estimations of the sparse coding coefficients of the original image and then centralize the sparse coding coefficients of the observed image to those estimates nonlocally. In [31], a novel image denoising strategy, which is based on an enhanced sparse representation in transform domain using nonlocal self-similarity property, was proposed for image denoising and achieved a significant improvement. [32] proposed a novel method for image deblurring and super-resolution, where the image nonlocal self-similarity was introduced as a regularization term of the optimization function and the regularization parameters were adaptively estimated for better reconstruction performance. In this paper, we propose a novel intelligent nonconvex compressive sensing (INCS) algorithm using prior information for natural image reconstruction by sparse representation. First of all, the sparsity prior of each image block can be obtained by the over-completed dictionary of Ridgelet [33], which generates the atoms by scaling, shifting and rotating a prototype function as in [34]. Then, we exploit the nonlocal self-similarity property and joint sparsity [35] to obtaining the prior information to guide the reconstruction, which contributes a lot to reducing the computational complexity. In addition, based on the property that image blocks spatially nearby share the similar structures, more information can be extracted to enrich the prior information, which can enhance the guidance of the prior information and improve the reconstruction accuracy significant5

ly. The proposed method INCS aims to find the sparse representation of image blocks by solving the nonconvex l0 minimization essentially and searching for the global optimal solution intelligently under the guidance of the prior information. To this end, we use intelligent optimization algorithm, which performs superiorly in combinatorial optimization problems and global searching, to design the intelligent searching strategies for INCS. Moreover, the matching strategies of greedy algorithm are exploited to optimize the searching strategies to accelerate the reconstruction speed. By means of the guidance of prior information and the superior performance of intelligent optimization algorithm in solving the nonconvex l0 minimization, the proposed method INCS achieves a significant reconstruction accuracy and a reasonable reconstruction speed. Experiments on five natural images are implemented to verify the superior performance of our proposed method INCS. Compared with orthogonal matching pursuit (OMP) algorithm, simultaneous OMP (SOMP) algorithm based on joint sparsity [36] and collaborate OMP (COMP) algorithm using prior information, the experimental results demonstrate that our proposed method INCS is superior to the state-of-the-art algorithms in terms of PSNR, SSIM and visual quality. The major contribution of this paper is threefold: 1. We exploit the nonlocal self-similarity property of natural images and joint sparsity to extracting the prior information to guide the reconstruction, which contributes a lot to reducing the computational complexity. 2. Based on the fact that image blocks spatially nearby in natural images share the similar structures, more information is extracted to enrich the prior information, which can enhance the guidance of the prior information and improve the reconstruction accuracy significantly. 3. We utilize the superior performance of intelligent optimization algorithm in solving combinatorial optimization problems and global searching to solve the nonconvex l0 minimization essentially and accelerate the reconstruction speed by introducing the matching strategies of greedy algorith-

6

m, which can reconstruct the image blocks accurately when the measurement rate is relatively small. The layout of this paper is organized as follows: In section 2, the framework of the proposed INCS and the computational complexity analysis are provide. Experimental results and analysis are given in section 3 to verify the superior performance of INCS. Section 4 concludes this paper.

2. The proposed INCS In this section, we first introduce the over-completed dictionary and how to obtain the prior information. Then the proposed method INCS is described in detail. In the end, we analyse the computational complexity. 2.1. Over-completed Dictionary In our proposed method INCS, we use the over-completed dictionary of Ridgelet, which is constructed by scaling, shifting and rotating a prototype Ridgelet function, to introduce the sparsity level for each image block. There are three parameters in the Ridgelet function, e.g. direction θ, scale a and shift b, where the direction θ is the most important parameter for representing the structures of natural images. In this paper, we fix the sampling intervals as pi/36, 0.2 and 1 for θ, a and b respectively. As a result, the obtained overcompleted dictionary is enriched by 11521 atoms with different orientations. To verify the performance of the over-completed dictionary, we conduct experiments on two natural images, where OMP algorithm is used for the image reconstruction by sparse representation. The test images are Lena and Barbara with size of 512 ∗ 512 and the size of image blocks is 16 ∗ 16. The experimental results are shown in Fig.(1). From Fig.(1), we can see that the performance of image reconstruction by sparse representation achieves a significant visual quality improvement while without obvious block artifacts, which can further verify the effectiveness of the over-completed dictionary for image reconstruction.

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Figure 1: The results of image reconstruction by sparse representation when the parameters of the over-completed dictionary of Ridgelet are π/36, 0.2 and 1 respectively. Each image block with size of 16∗16 can be sparsely represented by the combination of 32 atoms. (a) Reconstructed Lena, PSNR=36.3829, SSIM=0.9835. (b) Reconstructed Barbara, PSNR=33.2797, SSIM=0.9798.

2.2. Prior information We use the two particular properties of natural images: (a) the nonlocal self-similarity property and (b) the property that natural images are changing locally slowly, to obtain the prior information to guide the reconstruction. The nonlocal self-similarity property and joint sparsity are exploited to extract the basic prior information, and the property that image blocks spatially nearby have the similar structures is used to enrich the basic prior information to enhance the guidance accuracy. First of all, we introduce the steps of obtaining the basic prior information based on the nonlocal self-similarity property and joint sparsity. (1) Parti√

√

tion. We first partition the original image Xori ∈ R n∗ n to L non-overlapped √ √ image blocks x1 , x2 , ..., xL ∈ RB∗L with size of B ∗ B. Then the measurement vectors y1 , y2 , ..., yL ∈ RmB ∗L can be obtained by the Gaussian random measurement matrix φ. (2) Clustering. In this paper, the Affinity Propagation (AP) clustering [37] method can be directly used to yi , i = 1, 2, ..., L to obtain the classes Yi , i = 1, 2, , C. The main reason is that the distance be8

tween the down-sampled measurement vectors can represent the distance between image blocks based on the fact that the Gaussian projection is approximately distance preserving. (3) Common atoms. After AP clustering, the blocks in the same class share the similar structures, so all the blocks in the same class can be sparsely represented by the common atoms chosen from the over-completed dictionary but different coefficients. Based on the fact, we can find a group of common atoms from the over-completed dictionary to sparsely represent all the blocks in the same class. The problem turns to solve Eq.(6), where Yi = {yi /i ∈ Ri }, i = 1, 2, ..., C, Si represents the coefficients matrix calculated by Si = (φ ∗ Di )† ∗ Yi , and the obtain Di∗ is the matrix including the common atoms. In this stage, we use the simultaneous orthogonal matching pursuit (SOMP) algorithm to obtain the common atoms δj , j = 1, 2, ..., C. for each class, which is shown in Algorithm 1. In this case, we can obtain the common atoms of each image block not only based on its own measurements but also based on a group measurements of other blocks which share the similar structures with it, which indicates that the optimal solution can be more accurate and stable even when the measurement rate is relatively small. For each image block, the common atoms collection can be represented by Pi1 = δj , yi ∈ Yj , j = 1, 2, ..., C, i = 1, 2, ..., L. The basic prior information is the indexes of Pi1 , i = 1, 2, ..., L. Fig.(2) shows the results of AP clustering and SOMP reconstruction for Barbara and Lena. (Di∗ , Si∗ ) = argminDi ,Si  Yi − φ ∗ Di ∗ Si 22 subject to

 Di p,0 ≤ K, i = 1, 2, ..., C.

(6)

Secondly, we introduce how to enrich the basic prior information to enhance the guidance accuracy based on the property that natural images are changing locally slowly and the blocks spatially nearby have the similar structures. (1) Selection. We select the 8 local nearby blocks as the local neighbors. (2) Enhancing atoms. Extract the common atoms of these classes, which the local neighbors belong to, to construct the enhancing atoms Pi2 , i = 1, 2, ..., L. (3) Combination. Combine Pi2 , i = 1, 2, ..., L with Pi1 , i = 1, 2, ..., L to construct 9

Algorithm 1 SOMP Require: Over-completed dictionary ψ ; Measurement matrix φ ; Classes Yj , j = 1, 2, ..., C.; The sparsity level K; Ensure: 1: for each class Yj , j = 1, 2, ..., C. do 2: Initialize the collection of common atoms as Δj = ∅ and the residual as R = Yj 3: while t ≤ K do 4: find the atomjj which satisfy jj = argminjj  (φ ∗ djj ) ∗ R 22 5: Set Δj = Δj djj 6: update the residual by R = Yj − φ ∗ Δj ∗ ((φ ∗ Δj )† ∗ Yj ) 7: t=t+1 8: end while 9: obtain the collection of common atoms Δj for class Yj 10: end for Output: Δj , j = 1, 2, ..., C. the collection Pi1



Pi2 , i = 1, 2, ..., L = {dz /θz , az , bz }, i = 1, 2, ..., L, which is

beneficial to enriching the common atoms to enhance the guidance accuracy. Furtherly, we extract more atoms from the over-completed dictionary, which  are very similar to the atoms of Pi1 Pi2 , i = 1, 2, ..., L, to enrich the prior information to enhance the guidance accuracy. The idea is to extract the atoms which have the same directions and scales but different shifts as the atoms of  the collection Pi1 Pi2 , i = 1, 2, ..., L, which is formulated by Pi , i = 1, 2, ..., L in Eq.(7). Pi = {dp /θp = θz , ap = az , dp ∈ ψ}, i = 1, 2, ..., L.

(7)

Finally, the indexes of the collection Pi , i = 1, 2, ..., L can be recognized as the final prior information to guide the reconstruction, which is represented in Eq.(8). SSi = Idx(Pi ) = {ii/dii ∈ Pi }, i = 1, 2, ..., L.

10

(8)

Figure 2: The results of AP clustering and SOMP reconstruction for Barbana and Lena. The first column represents the original images, and the second column donates a class of images blocks based on AP clustering and the third column donates the obtained common atoms for each class based on reconstruction.

2.3. Intelligent nonconvex compressive sensing As intelligent optimization algorithm has superior performance in solving combinatorial optimization problems and global searching, we use the operations of genetic algorithm (GA) and simulated annealing (SA) algorithm to design the intelligent searching strategies of INCS to solve the nonconvex l0 minimization essentially and search for the global optimal solution intelligently. GA and SA are both iterative random searching algorithms, where GA performs well in global searching and SA performs well in local searching. Also, SA permits

11

accepting a less optimal solution and even some random solutions with a positive probability by implementing Metropolis rule, which can avoid falling into a suboptimal solution and is more likely to find the global optimal solution. However, the intelligent optimization algorithm needs more iterations to find the global optimal solution, which leads to the high computational complexity and slow reconstruction speed. To improve it, we introduce the matching strategies of greedy algorithm, which performs quite well in reconstruction speed, to design the update mechanism to reduce the computational complexity. Also, the prior information described in Section 2.2 is used to guide the reconstruction, which can reduce the computational complexity significantly. The flowchart of the proposed method INCS is shown in Algorithm 2 and the details are described as follows. (1) Prior information. The prior information obtained in Section 2.2 is set as the feasible searching area for the intelligent searching. The objective of INCS is to search for the estimated K indexes from the feasible area Pi , i = 1, 2, ..., L instead of the indexes collection {1, 2, ..., N } to sparsely represent each image block. The prior information contributes a lot to reducing the searching area, which can reduce the computational complexity and improve the reconstruction accuracy significantly. (2) Fitness function. In this paper, the sparsity level K can be obtained as a prior based on the over-completed dictionary of Ridgelet. The proposed method INCS aims to find the estimated support collection I with K indexes from Pi , i = 1, 2, ..., L, where the corresponding atoms can represent the image block sparsely. If the estimated collection I is the unique solution of Eq.(4), it must satisfy the condition that φ∗DI ∗((φ∗DI )† ∗y) = y. As φ∗DI ∗((φ∗DI )† ∗y) = y is equivalent to  φ∗DI ∗((φ∗DI )† ∗y)−y 22 = 0 and  φ∗DI ∗((φ∗DI )† ∗y)−y 22 satisfies  φ∗DI ∗((φ∗DI )† ∗y)−y 22 ≥ 0, the fitness function can be set as Eq.(9). Then we can obtain the estimated collection I by optimizing Eq.(9). After the collection I is obtained, the reconstructed image block can be calculated by Eq.(10).

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Algorithm 2 INCS for Image Reconstruction Require: Over-completed dictionary ψ, Measurement matrix φ; Minimum error Error, Maximum iterations Mmax and Nmax ; The parameters q, T1 and popsize; Ensure: 1: Partition the original image into nonoverlapped 16 ∗ 16 image blocks and obtain the measurements yi , i = 1, 2, ..., L using φ 2: Implement AP clustering method to yi , i = 1, 2, ..., L and obtain the classes Yj , j = 1, 2, ..., C 3: for each class Yj , j = 1, 2, ..., C do 4: Implement Algorithm 1 to Yj to obtain the collection of the common atoms δj 5: For each image block, the collection of common atoms is calculated by Pi1 = {δj , yi ∈ Yj } 6: end for 7: for each image block yi do 2 8: select the common atoms  2 Pi of the 8 local neighbours and combine it 1 1 with Pi to obtain Pi  Pi = {dz /θz , az , bz } 9: enrich the collection Pi1 Pi2 by Pi = {dp /θp = θz , ap = az , dp ∈ ψ} 10: Obtain the prior information by SSi = Idx(Pi ) 11: Initialize the group pop with popsize individuals 12: while jj ≤ Mmax or e ≤ E do 13: Implement the mutation operation on all the individuals and select the better individuals 14: for each individual pop0 do 15: while ii ≤ Nmax or e ≤ E do 16: select q elements from SSi to combine with pop0 as Q 17: calculate recxi = (φ ∗ ψQ )† ∗ yi and select the indexes with the K bigger values as pop1 18: choose the updating individual based on Metropolis rule 19: calculate the error e = y − φ ∗ ψpop0 ∗ ((φ ∗ ψpop0 )† ∗ yi ) 22 20: ii = ii + 1 21: end while 22: end for 23: select the bestindividual based on the fitness function 24: Tjj = T1 /(jj + 1) and jj = jj + 1 25: calculate the error e = y−φ∗ψbestindividual ∗((φ∗ψbestindividual )† ∗yi ) 22 26: end while 27: obtain the reconstructed image block recs by Eq.(10). 28: end for 29: rearrange the reconstructed image blocks to obtain the reconstructed image recX Output: recX

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minI f (I) = φ ∗ DI ∗ ((φ ∗ DI )† ∗ y) − y 22

recsI = (φ ∗ ψI )† ∗ y

and

recsS−I = 0

(9)

(10)

(3) Initialization. One of the initial individuals is set as the indexes of the common atoms of this class, which the image block belongs to, and the other popsize − 1 initial individuals are generated by randomly choosing K indexes from the feasible searching area SS. (4) Iteration. Firstly, implement mutation operation on all the individuals and choose the better individuals based on the fitness function. For each individual Im , calculate xm = (φ ∗ ψIm )† ∗ y and replace the index, which has the smallest numerical value of xm , with the randomly selected index from the collection SS − Im . Then, for each individual, we implement the following operations: (a) select q indexes from the feasible searching area SS and combine it with D1 as the test collection Q. (b) calculate the coefficients recx = (φ∗ψQ )† ∗y, where ψQ = {di /i ∈ Q, di ∈ ψ}. (c) select the indexes with the K largest coefficients to generate a new collection Di . (d) calculate the fitness of the new individual and the old individual and select one individual based on Metropolis rule. Metropolis rule is that: if p ≤ η, where p is calculated by Eq.(11) and η is a random number chosen in [0, 1], select the new individual pop1 , otherwise, select the old individual pop0 . Thirdly, calculate the fitness value for all the individuals and set the individual with the smallest fitness value as the bestindividual. Finally, calculate the error e and update the parameter Tii = T1 /log(ii + 1). p = min(1, exp(−(f (pop0 ) − f (pop1 ))/Tii )).

(11)

(5) The terminal condition is that the number of iterations reach the maximum number or the error is smaller than the minimum value E. After the estimated support collection bestindividual is obtained, the sparse representation of the reconstructed image block can be calculated by Eq.(10).

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Then we can rearrange the reconstructed blocks to obtain the reconstructed image. The whole flowchart of the proposed method INCS using prior information for image reconstruction by sparse representation is shown in Fig.(3).

Figure 3: The flowchart of INCS using prior information for image reconstruction by sparse representation

2.4. Computational Complexity Analysis From Algorithm 2, we can find that the main computational complexity concentrates on the iterations because of the matrix multiplication. In each iteration, matrix multiplication lies in step 13, step 17 and step 23. The total

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number of matrix multiplication in step 13, step 17 and step 23 can be calculated by 2 ∗ popsize, Mmax ∗ Nmax and popsize respectively. As the computational complexity of the matrix multiplication is upper bounded by O(B 3 ), where B is the length of the reconstructed signal, the computational complexity of the proposed method INCS can be upper bounded by Eq.(12). popsize ∗ (Mmax ∗ Nmax + 3) ∗ O(B 3 )

(12)

Eq.(12) indicates that the computational complexity of INCS is influenced by the popsize and the iteration number significantly. A relatively large popsize can make the individuals search parallelly, which is more likely to find the global optimal solution. So the computational complexity can be reduced by reducing the number of iterations. As mentioned above, the prior information contributes a lot to reducing the computational complexity. To solve the nonconvex l0 minimization, which is NP-hard problem with a very high computational complexity, we have to find the only global optimal solution from the feasible area (intelligent searching area) K possible solutions. It is very difficult to find the global optimal including CN

solution when the length of the original signal N is relatively large. However, in our proposed method INCS, under the guidance of the prior information, the K size of the feasible area can be reduced to CN N , where N N = length(SS) is

the length of the prior information SS and it is smaller than N significantly. In this case, our proposed method INCS can find the global optimal solution by using less iterations as the intelligent searching area is shrunk significantly. In conclusion, the prior information can not only reduce the computational complexity to accelerate the reconstruction speed but also be more likely to find the global optimal solution to improve the reconstruction accuracy.

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3. Experimental results 3.1. Experimental Configuration In this section, the experiments are conducted on five natural images to verify the the superior performance of our proposed method INCS. The test natural images are Barbara, Lena, Boat, House and Peppers, which are shown in Fig.(4). The sizes of Barbara, Lena and Boat are 512 ∗ 512 and that of House and Peppers are 256∗256 respectively. In all the experiments, the test images are partitioned into nononverlapped blocks with size of 16 ∗ 16. The measurement matrix φ is obtained by extracting m rows of the N ∗ N orthogonal random matrix. The parameters of the over-completed dictionary are set as π/36, 0.2 and 1 for θ, a and b respectively. In all the experiments, we suppose that each block is sparsely represented by 32 atoms chosen from the over-completed dictionary, namely the sparsity level K = 32 is set as a prior. We also set the main parameters Mmax = 20, Nmax = 100, E = 10−5 of our proposed method INCS for all the experiments. All the experiments are implemented in Matlab 2011a on the PC with 3.2Hz Intel Core i5 processor and 8.0GB memory running the Windows 7 system. Also, the compared method OMP is implemented using SolveOMP in SparseLab toolbox [38]. 3.2. Experimental results and analysis In this experiment, our proposed method INCS is compared with three stateof-the-art algorithms, including orthogonal matching pursuit (OMP) algorithm, simultaneous orthogonal matching pursuit (SOMP) algorithm and collaborative orthogonal matching pursuit (COMP) algorithm, to verify the significant improvement of the reconstruction performance. SOMP exploits the nonlocal self-similarity property and reconstructs the original image blocks based on joint sparsity. The idea of COMP is also to utilize the nonlocal self-similarity property and the property that image blocks spatially nearby share the similar structures to extract the prior information, and then use the OMP algorithm to

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Figure 4: The test natural images

reconstruct the original image blocks under the guidance of prior information. The measurement rate, which is calculated by mB /B, changes from 0.3 to 0.6. The reconstruction performance is evaluated in terms of PSNR, SSIM [39], Time and visual quality. The results of PSNR, SSIM and Time in all the experiments are calculated by averaging the results of 20 tests and shown in Table 1, Table 2 and Table 3 respectively. The visual performance of these four algorithms for Barbara and Lena are shown in Fig.(5) and Fig.(6). From Table 1 and Table 2, we can find that our proposed method INCS achieves the highest PSNR and the highest SSIM, namely the best reconstruction accuracy, for all the test natural images when the measurement rate changes from 0.3 to 0.6. Table 1 shows that our proposed method INCS has several dB improvement than OMP algorithm. The first main reason is that INCS exploits

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Table 1: The PSNR(dB) results of the reconstructed images based on the different methods when the measurement rate changing from 0.3 to 0.6 Image barbara

Algorithm OMP SOMP COMP INCS

0.3 22.4936 25.6038 25.8525 26.4533

0.4 24.0040 26.1738 27.0191 28.5662

0.5 25.1280 27.8262 28.5662 30.3909

0.6 26.7455 28.7083 29.9278 30.4176

lena

OMP SOMP COMP INCS

25.6905 29.2807 29.8918 30.4197

26.5311 29.5493 30.4197 32.1369

29.2347 30.8390 31.5354 32.6781

30.4304 32.1369 32.3263 34.3829

boats

OMP SOMP COMP INCS

22.2720 27.4471 27.8701 28.0455

25.5689 27.9516 28.1780 29.7879

26.7337 29.4009 30.1818 30.5414

27.8895 30.3617 31.9985 33.6792

house

OMP SOMP COMP INCS

25.2661 28.7768 29.3147 30.0653

27.5867 29.3432 29.8999 31.0685

28.7580 30.6714 31.0685 32.3137

29.6054 31.1232 31.3123 32.6740

peppers

OMP SOMP COMP INCS

24.0234 26.3196 26.7885 27.4727

25.5082 28.0947 28.5390 29.3864

26.6011 28.9704 29.5913 30.0029

27.0532 30.1625 30.9638 31.7359

the nonlocal self-similarity property and the property that image blocks spatially nearby share the similar structures to obtain the prior information to guide the reconstruction, which can improve the reconstruction accuracy significantly, especially when the measurement rate is relatively small. The second main reason is that we take advantage of intelligent optimization algorithm to solve the nonconvex l0 minimization essentially, which also contributes a lot to improving the reconstruction accuracy. Under the guidance of the prior information, INCS is more likely to find the global optimal solution by intelligent searching. From Table 1, we can find that the PSNR of INCS is almost 1dB to 1.5dB higher than that of SOMP. As we all know, SOMP also utilizes the nonlocal self-similarity property and joint sparsity to reconstruct the original image blocks as INCS does. However, INCS increases a step to refine the reconstruction accuracy of SOMP by solving the nonconvex l0 minimization essentially and intelligently searching the global optimal solution. Table 1 also shows that the PSNR of our

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Table 2: The SSIM results of the reconstructed images based on the different methods when the measurement rate changing from 0.3 to 0.6 Image barbara

Algorithm OMP SOMP COMP INCS

0.3 0.8088 0.8730 0.8791 0.9084

0.4 0.8732 0.8992 0.9186 0.9422

0.5 0.8891 0.9306 0.9422 0.9612

0.6 0.9188 0.9448 0.9586 0.9614

lena

OMP SOMP COMP INCS

0.8763 0.9164 0.9272 0.9473

0.8918 0.9421 0.9473 0.9717

0.9387 0.9452 0.9527 0.9621

0.9533 0.9623 0.9717 0.9884

boats

OMP SOMP COMP INCS

0.6921 0.8651 0.8709 0.8751

0.8611 0.8746 0.8927 0.9480

0.8882 0.9475 0.9489 0.9576

0.9062 0.9228 0.9397 0.9746

house

OMP SOMP COMP INCS

0.5986 0.7862 0.7996 0.8072

0.7909 0.7981 0.8037 0.8305

0.7970 0.8069 0.8211 0.8315

0.8183 0.8240 0.8333 0.9039

peppers

OMP SOMP COMP INCS

0.6410 0.7154 0.7450 0.7643

0.7436 0.8166 0.8217 0.8679

0.7686 0.8185 0.8538 0.8773

0.8276 0.8302 0.8797 0.9032

proposed method INCS is almost 0.5dB to 1dB overpass that of COMP algorithm. Although COMP exploits the same method with INCS to obtain the prior information to guide the reconstruction, INCS can solve the nonconvex l0 minimization essentially and is more likely to find the global optimal solution by combining the superiorities of intelligent optimization algorithm with matching strategies of greedy algorithms. In other words, the intelligent searching strategies of INCS is more effective than that of COMP. By comparing with those three state-of-the-art algorithms, we can illustrate the superior performance of our proposed method INCS through utilizing the nonlocal self-similarity property to extract the basic prior information, utilizing the property that image blocks spatially nearby share the similar structures to enrich the basic prior information, and taking advantage of intelligent optimization algorithm to solve the nonconvex l0 minimization essentially. As we can find from Table 3, the reconstruction speed of the compared algo-

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Table 3: The Time (Sec.) results of the reconstructed images based on the different methods when the measurement rate changing from 0.3 to 0.6 Image barbara

Algorithm OMP SOMP COMP INCS

0.3 1531.8400 688.6987 773.2297 1136.2056

0.4 1864.7011 736.4892 841.9035 1067.2266

0.5 2164.3239 788.1072 872.3301 1031.1454

0.6 2559.4813 798.2015 903.6129 940.3546

lena

OMP SOMP COMP INCS

1243.6085 744.6069 863.8033 1276.3443

1469.6051 804.6341 920.2160 1237.2094

1697.3329 899.0113 996.6749 1173.7113

1865.0149 970.7218 1094.9994 1121.0085

boats

OMP SOMP COMP INCS

1753.6598 1693.8609 1862.5901 2577.1654

1969.0335 1773.0098 1991.5729 2502.9085

2060.8502 1872.9397 2036.7921 2395.0129

2231.5532 1938.9420 2145.7921 2228.0349

house

OMP SOMP COMP INCS

490.3236 366.3050 401.4267 619.0445

506.8125 390.3670 425.0688 556.0129

524.6164 430.9162 439.4077 503.4985

561.4017 456.5071 453.2982 483.6655

peppers

OMP SOMP COMP INCS

395.8517 394.6204 451.9651 752.4074

426.0140 412.0563 476.0208 663.0617

463.2201 438.7224 490.4564 623.1907

495.3496 465.8974 528.2455 583.0366

rithms decrease with the increasing measurement rate, but that of our proposed method INCS increases with the increasing measurement rate. Also, SOMP algorithm achieves the fastest reconstruction speed and INCS is slower than the SOMP and COMP algorithms. The main reason of the reconstruction speed of COMP algorithm and INCS slowing down is that they increase an extra process to refine the reconstruction performance of SOMP algorithm. Although COMP and INCS algorithms utilize the same method to obtain the prior information to guide the reconstruction, our proposed method INCS has a slower reconstruction speed due to that the intelligent searching strategies of INCS needs more iterations to find the global optimal solution in comparison with COMP algorithm. However, with the increasing measurement rate, the intelligent searching strategies of INCS becomes more effective, which can reduce the necessary iterations of intelligent searching to accelerate the reconstruction speed. Moreover, as the prior information has a strong guidance for the reconstruction

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and reduces the intelligent searching area significantly, the proposed method INCS is more likely to find the global optimal solution, which contributes a lot to reducing the necessary iterations and make the reconstruction speed of INCS more reasonable. To verify the superior performance of our proposed INCS, Fig.(5) and Fig.(6) present the visual quality of the reconstructed images based on those four algorithms when the measurement rate is fixed as 0.5. In Fig.(5) and Fig.(6), the images in the first column represent the whole reconstructed images based on these four algorithms and that of the second column and the third column represent the enlarged parts of the corresponding reconstructed images. As we can see, our proposed method INCS achieves a significantly better visual quality than OMP algorithm. Although the improvement of the PSNR by our proposed method INCS does not achieve even 1dB than that of SOMP and COMP, INCS achieves the faithful visual quality without much block effects and obvious artifacts. In other words, INCS achieves the best visual quality and outperforms the compared algorithms for all the test images.

4. Conclusion This paper proposes a novel intelligent nonconvex compressive sensing method using prior information for image reconstruction by sparse representation. First of all, INCS extracts the prior information by exploiting the nonlocal selfsimilarity property, which contributes a lot to reducing the computational complexity. Also, more information are extracted based on the property that image blocks spatially nearby share the similar structures to enrich the prior information to enhance the guidance accuracy. Then, INCS solves the nonconvex l0 minimization essentially by combining the superiorities of intelligent optimization algorithm with the matching strategies of greedy algorithms, which improves the reconstruction accuracy significantly. Under the guidance of prior information, INCS is more likely to find the global optimal solution by intelligent searching, which reduces the computational complexity significantly and

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Figure 5: The visual quality of the reconstruction based on OMP, SOMP, COMP and INCS for Barbara when the measurement is 0.5.

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Figure 6: The visual quality of the reconstruction based on OMP, SOMP, COMP and INCS for Lena when the measurement is 0.5.

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makes the reconstruction speed of INCS more reasonable. Experimental results on five natural images demonstrate that our proposed method INCS outperforms the state-of-the-art algorithms, including OMP, SOMP and COMP, on PSNR, SSIM and visual quality. Although our proposed method INCS improves the reconstruction quality significantly, it still needs future improvements. The possible extensions include: extracting more prior information based on other particular structures of images to enhance the reconstruction accuracy, establishing more efficient model to obtain the more accurate prior information and designing more efficient updating mechanism of intelligent searching to accelerate the reconstruction speed.

ACKNOWLEDGMENT This work is financially supported by National Science Foundations of China (No.61174016) and (No.61171197).

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