CN102789633B - Based on the image noise reduction system and method for K-SVD and locally linear embedding - Google Patents

Abstract

The invention discloses a kind of image noise reduction system and method based on K-SVD and locally linear embedding, be specifically related to the Image Denoising of sparse signal representation based on dictionary learning and reconstruct and manifold learning.Its employing K-SVD method is the sparse signal representation based on dictionary learning and the reconfiguration technique of framework, when solving sparse signal representation, locally linear embedding is added objective function as constraint condition, the contact between the sparse coefficient that decomposites is strengthened with this, overcome the impact of random noise on sparse coefficient, thus obtain the image noise reduction effect more excellent than former K-SVD method.

Description

Based on the image noise reduction system and method for K-SVD and locally linear embedding

Technical field

The invention belongs to technical field of image processing, relate to a kind of image noise reduction system and method, particularly a kind of image noise reduction system and method based on K-SVD and locally linear embedding.

Background technology

In actual applications, image is inevitably subject to the interference of various noise signal in acquisition, transmitting procedure.Therefore, must process noisy image at receiving end, improve the signal to noise ratio (S/N ratio) of image, improve picture quality, from noisy image, extract true, effective original image information as far as possible.Image noise reduction is a hot issue in image processing field always, and scholars also promotes the signal to noise ratio (S/N ratio) of image by various signal processing means.

In recent years, deepen continuously along with based on the signal transacting of dictionary learning and rarefaction representation and the research of reconstructing method, these class methods are applied to image noise reduction field and also achieve certain achievement.At Michael Elad; Michal Aharon.Image Denoising Via Sparse andRedundant Representations Over Learned Dictionaries [J] .IEEETransactions on image processing, 2006, show in 15 (12): the 3736-3745. documents delivered, by the K-SVD algorithm application of classics in image noise reduction, can obtain and be better than traditional image reconstructed based on Global Dictionary and the complete discrete cosine dictionary of mistake, promote noise reduction.But, criterion due to the image noise reduction based on K-SVD method approaches noisy image by be multiplied with the sparse signal reconstructed image of gained of dictionary, and noise has stronger randomness, therefore such noise reduction criterion must reduce the stronger correlativity of image local, thus is unfavorable for the raising of reconstructed image quality.

On the other hand, manifold learning in signal transacting is a kind of effective unsupervised learning method, and it can find, the implicit variable (low-dimensional data) of minority in high dimensional data, these implicit variablees are then nested in higher-dimension theorem in Euclid space, exist with combined non-linearity manifold.Although may higher than original signal dimension based on the dimension of sparse signal in the signal transacting of dictionary learning and rarefaction representation and reconstructing method, but opennessly but this sparse signal can be considered as low dimensional signal, thus ensure the application of manifold learning in the signal transacting field of dictionary learning and rarefaction representation.At Miao Zheng, Jiajun Bu, , Chun Chen, et.al.GraphRegularized Sparse Coding for Image Representation [J] .IEEETransaction on image processing, 2011, show in 20 (5): the 1327-1335. documents delivered, during by locally linear embedding (LLE) methods combining sparse signal process in manifold learning, effectively can realize the Classification and clustering of image, from certain angle illustrate locally linear embedding (LLE) method manifold learning can embody when Image Reconstruction in image local correlations.

Summary of the invention

The object of the invention is the K-SVD method based on dictionary learning and sparse signal representation to combine with locally linear embedding, be applied to image processing field, realize image noise reduction.

Technical scheme of the present invention is from the viewpoint of following two: 1. dictionary learning and sparse signal representation aspect, because picture signal itself has certain structural information, as the profile texture etc. of image, therefore the method for dictionary learning can obtain this peculiar structure of signal by study, makes the sparse coefficient on this dictionary must be the maximization performance of signal structure feature.Picture noise does not then generally possess this structure, has stronger randomness.Therefore image the approaching noisy image by being reconstructed by sparse coefficient on dictionary, can remove noise effectively.And K-SVD algorithm is a kind of efficient, practical algorithm wherein, therefore the present invention with K-SVD algorithm for framework carries out framework.2. in manifold learning, consider image structural information can pollute by noise, and due to original image unknown, reconstructed image can only approach noisy image, thus must cause there is a part of false structure information in the image reconstructed.Therefore utilize manifold learning arithmetic, force to set up contact between the sparse coefficient of image block, thus be conducive to the real structure information of outstanding image, and effectively suppress false structure information.The present invention then sets up contact by locally linear embedding algorithm realization between the sparse coefficient of image block.Finally, realize a kind of image noise reduction system and method for base locally linear embedding under K-SVD framework, and obtain the effect more excellent than traditional K-SVD image denoising method.

Main technical content of the present invention is as follows:

A kind of image noise reduction system based on K-SVD and locally linear embedding, comprise with lower module: sampling module, calculate Laplacian Matrix L module, objective function and dictionary, sparse coefficient optimization module, estimated image block acquisition module, overall estimation image block acquisition module;

Noisy image → sampling module → calculating Laplacian Matrix L module → objective function and dictionary, sparse coefficient optimization module → estimated image block acquisition module → overall estimation image block acquisition module → denoising image;

Described objective function and dictionary, sparse coefficient are optimized module and are comprised overall goals function construction module, each sparse coefficient are optimized to module, optimize module to dictionary D;

Describedly module is optimized to each sparse coefficient comprise two parts: the structure of sparse coefficient objective function and the optimization of sparse coefficient;

Described to dictionary D optimize module comprise two parts: the structure of the objective function of dictionary D and the optimization of dictionary D.

Based on an image denoising method for K-SVD and locally linear embedding, comprise following steps:

(1), in sampling module, input noisy image, N number of M is gone out to this image sampling ₁× M ₂the image block of pixel size, and record the position of each image block in former figure, to a kth image block picture element matrix B _kcarry out order by row to pile up, k=1,2 ..., N, forms (a M ₁m ₂the column vector Y of) × 1 _k;

(2), in calculating Laplacian Matrix L module to the column vector corresponding to all image blocks according to locally linear embedding method, according to the structure of each vector point in space geometry, calculate the weight matrix W of distance, and and then structure Laplacian Matrix L;

(3), optimize in module in objective function and dictionary, sparse coefficient and utilize K-SVD algorithm construction objective function to be: constraint condition is || Y _k-DX _k|| ₂≤ ε carries out the Optimization Solution of parameter D and X,

Wherein, Y=[Y ₁y ₂... Y _n], D is dictionary, X _kfor Y _kthe sparse coefficient that D projects, X=[X ₁x ₂... X _n], || .|| _ffor Frobenius norm, || .|| ₂be 2 norms, Tr (.) is the computing of Matrix Calculating mark, (.) ^tfor matrix turns order computing, β is weighting coefficient, β=0.1, and ε is resolution error thresholding;

(4) in estimated image block acquisition module, utilize dictionary D and the sparse coefficient X of optimization _k, to column vector Y _kestimate, its solution formula is k=1,2 ..., N, and by this estimator carry out sequential breakdown by row, constructing corresponding size is M ₁× M ₂estimated image block picture element matrix

(5), in overall estimation image collection module by image block that step (4) estimates, according to the positional information of image block in former figure in step (1), be covered to the correspondence position of former figure, and record in former figure the estimated image block number w each pixel needing cover _i,j, i, j are location of pixels coordinate, and the pixel value of the estimated image block of correspondence k=1,2 ..., w _i,j;

(6), weight w is utilized _i,jright carry out weights addition, the pixel value of image correspondence position after final acquisition noise reduction ${\tilde{P}}_{i,j}=(\underset{w_{i,j}}{\Σ}{\hat{P}}_{i,j}+{\mathrm{\αP}}_{i,j})/(w_{i,j}+\mathrm{\α}),$ Wherein P _i,jthe pixel value of original image and corresponding weighting coefficient is respectively, α=1 with α.

Utilization described in step (3) based on K-SVD algorithm to formula: constraint condition is || Y _k-DX _k|| ₂≤ ε carries out the Optimization Solution of D and X, and it carries out as follows:

(3a), construct at the sparse coefficient objective function optimized in module each sparse coefficient: k=1,2 ..., N, this optimization formula is equivalent to following objective function:

$min\left|\right|Y_k-{\mathrm{DX}}_k|{|}_2^2+{\mathrm{\βL}}_{k,k}||X_k+\frac{1}{L_{k,k}}\underset{i\≠k}{\Σ}{L}_{k,i}{X}_i|{|}_2^2,$ Constraint condition is || Y _k-DX _k|| ₂≤ ε,

Wherein, L _i,jfor the i-th row jth column element of the Laplacian Matrix L described in step (2);

(3b), in step (3a) $min\left|\right|Y_k-{\mathrm{DX}}_k|{|}_2^2+{\mathrm{\βL}}_{k,k}||X_k+\frac{1}{L_{k,k}}\underset{i\≠k}{\Σ}{L}_{k,i}{X}_i|{|}_2^2$ Following formula is adopted to carry out equivalence:

$min\left|\right|\left(\begin{array}{c}{Y}_k\\ -\sqrt{{\mathrm{\βL}}_{k,k}}{\mathrm{XP}}_k\end{array}\right)-\left(\begin{array}{c}D\\ \sqrt{{\mathrm{\βL}}_{k,k}}I\end{array}\right)X_k|{|}_2^2$

Wherein, I is (M ₁m ₂) × (M ₁m ₂) unit matrix of size, P _kfor corresponding to X _ka vector, be defined as:

$P_{k,i}=\left\{\begin{array}{cc}\frac{L_{k,i}}{L_{i,i}}& i\≠k\\ 0& i=k\end{array}\right.$

Wherein, P _k,ifor P _kin the value of the i-th row;

(3c), be optimized in the sparse coefficient optimized in module each sparse coefficient: right $\left|\right|\left(\begin{array}{c}{Y}_k\\ -\sqrt{{\mathrm{\βL}}_{k,k}}{\mathrm{XP}}_k\end{array}\right)-\left(\begin{array}{c}D\\ \sqrt{{\mathrm{\βL}}_{k,k}}I\end{array}\right)X_k|{|}_2^2$ In all X _k, k=1,2 ..., N, calculates according to sparse coefficient derivation algorithm in K-SVD, when || Y _k-DX _k|| ₂during≤ε, stop sparse coefficient X _ksparsely to solve;

(3d), construct at the objective function of the dictionary D optimized in module dictionary D: k=1,2 ..., N, this optimization formula is equivalent to following objective function:

constraint condition is || Y _k-DX _k|| ₂≤ ε,

(3e), at the objective function of the dictionary D optimized in module dictionary D be optimized: according to K-SVD, dictionary optimized algorithm calculates, when for all k=1,2 ..., N, || Y _k-DX _k|| ₂during≤ε, stop the Optimization Solution to dictionary D, export sparse coefficient X _kwith corresponding dictionary, enter step (4), otherwise repeat to step (3a).

The beneficial effect that the present invention reaches:

Popular approach and dictionary learning and sparse signal representation and reconstructing method combine by the present invention, specifically, K-SVD method and locally linear embedding are organically combined, realize the object of image noise reduction.The method, than the image denoising method based on single K-SVD method, more can give prominence to the internal information structure of true picture, thus promotes the Y-PSNR (PSNR) of reconstructed image further, obtains more excellent noise reduction.

Accompanying drawing explanation

Fig. 1 is the overall framework figure of the image denoising method that the present invention is based on K-SVD and locally linear embedding;

The concrete enforcement block diagram that Fig. 2 is objective function and dictionary, sparse coefficient optimizes module;

Fig. 3 is the noise reduction figure of the noisy image Pepper of the present invention;

Fig. 4 is the noise reduction figure of the noisy image Cameraman of the present invention;

Fig. 5 is the noise reduction figure of the noisy image Lena of the present invention.

Embodiment

Below in conjunction with accompanying drawing, the image denoising method based on K-SVD and locally linear embedding of the present invention is further elaborated.

As shown in Figure 1 and Figure 2, a kind of image noise reduction system based on K-SVD and locally linear embedding, comprise with lower module: sampling module, calculate Laplacian Matrix L module, objective function and dictionary, sparse coefficient optimization module, estimated image block acquisition module, overall estimation image block acquisition module;

Noisy image → sampling module → calculating Laplacian Matrix L module → objective function and dictionary, sparse coefficient optimization module → estimated image block acquisition module → overall estimation image block acquisition module → denoising image;

Described objective function and dictionary, sparse coefficient are optimized module and are comprised overall goals function construction module, each sparse coefficient are optimized to module, optimize module to dictionary D;

Describedly module is optimized to each sparse coefficient comprise two parts: the structure of sparse coefficient objective function and the optimization of sparse coefficient;

Described to dictionary D optimize module comprise two parts: the structure of the objective function of dictionary D and the optimization of dictionary D.

Based on an image denoising method for K-SVD and locally linear embedding, comprise following steps:

(1), in sampling module, input noisy image, N number of M is gone out to this image sampling ₁× M ₂the image block of pixel size, and record the position of each image block in former figure, to a kth image block picture element matrix B _kcarry out order by row to pile up, k=1,2 ..., N, forms (a M ₁m ₂the column vector Y of) × 1 _k;

(2), in calculating Laplacian Matrix L module to the column vector corresponding to all image blocks according to locally linear embedding method, according to the structure of each vector point in space geometry, calculate the weight matrix W of distance, and and then structure Laplacian Matrix L;

(3), optimize in module in objective function and dictionary, sparse coefficient and utilize K-SVD algorithm construction objective function to be: constraint condition is || Y _k-DX _k|| ₂≤ ε carries out the Optimization Solution of parameter D and X,

Wherein, Y=[Y ₁y ₂... Y _n], D is dictionary, X _kfor Y _kthe sparse coefficient that D projects, X=[X ₁x ₂... X _n], || .|| _ffor Frobenius norm, || .|| ₂be 2 norms, Tr (.) is the computing of Matrix Calculating mark, (.) ^tfor matrix turns order computing, β is weighting coefficient, β=0.1, and ε is resolution error thresholding;

(4) in estimated image block acquisition module, utilize dictionary D and the sparse coefficient X of optimization _k, to column vector Y _kestimate, its solution formula is k=1,2 ..., N, and by this estimator carry out sequential breakdown by row, constructing corresponding size is M ₁× M ₂estimated image block picture element matrix

(5), in overall estimation image collection module by image block that step (4) estimates, according to the positional information of image block in former figure in step (1), be covered to the correspondence position of former figure, and record in former figure the estimated image block number w each pixel needing cover _i,j, i, j are location of pixels coordinate, and the pixel value of the estimated image block of correspondence k=1,2 ..., w _i,j;

(6), weight w is utilized _i,jright carry out weights addition, the pixel value of image correspondence position after final acquisition noise reduction ${\tilde{P}}_{i,j}=(\underset{w_{i,j}}{\Σ}{\hat{P}}_{i,j}+{\mathrm{\αP}}_{i,j})/(w_{i,j}+\mathrm{\α}),$ Wherein P _i,jthe pixel value of original image and corresponding weighting coefficient is respectively, α=1 with α.

Utilization described in step (3) based on K-SVD algorithm to formula: constraint condition is || Y _k-DX _k|| ₂≤ ε carries out the Optimization Solution of D and X, and it carries out as follows:

(3a), construct at the sparse coefficient objective function optimized in module each sparse coefficient: k=1,2 ..., N, this optimization formula is equivalent to following objective function:

$min\left|\right|Y_k-{\mathrm{DX}}_k|{|}_2^2+{\mathrm{\βL}}_{k,k}||X_k+\frac{1}{L_{k,k}}\underset{i\≠k}{\Σ}{L}_{k,i}{X}_i|{|}_2^2,$ Constraint condition is || Y _k-DX _k|| ₂≤ ε,

Wherein, L _i,jfor the i-th row jth column element of the Laplacian Matrix L described in step (2);

(3b), in step (3a) $min\left|\right|Y_k-{\mathrm{DX}}_k|{|}_2^2+{\mathrm{\βL}}_{k,k}||X_k+\frac{1}{L_{k,k}}\underset{i\≠k}{\Σ}{L}_{k,i}{X}_i|{|}_2^2$ Following formula is adopted to carry out equivalence:

$min\left|\right|\left(\begin{array}{c}{Y}_k\\ -\sqrt{{\mathrm{\βL}}_{k,k}}{\mathrm{XP}}_k\end{array}\right)-\left(\begin{array}{c}D\\ \sqrt{{\mathrm{\βL}}_{k,k}}I\end{array}\right)X_k|{|}_2^2$

Wherein, I is (M ₁m ₂) × (M ₁m ₂) unit matrix of size, P _kfor corresponding to X _ka vector, be defined as:

$P_{k,i}=\left\{\begin{array}{cc}\frac{L_{k,i}}{L_{i,i}}& i\≠k\\ 0& i=k\end{array}\right.$

Wherein, P _k,ifor P _kin the value of the i-th row;

(3c), be optimized in the sparse coefficient optimized in module each sparse coefficient: right $\left|\right|\left(\begin{array}{c}{Y}_k\\ -\sqrt{{\mathrm{\βL}}_{k,k}}{\mathrm{XP}}_k\end{array}\right)-\left(\begin{array}{c}D\\ \sqrt{{\mathrm{\βL}}_{k,k}}I\end{array}\right)X_k|{|}_2^2$ In all X _k, k=1,2 ..., N, calculates according to sparse coefficient derivation algorithm in K-SVD, when || Y _k-DX _k|| ₂during≤ε, stop sparse coefficient X _ksparsely to solve;

(3d), construct at the objective function of the dictionary D optimized in module dictionary D: k=1,2 ..., N, this optimization formula is equivalent to following objective function:

constraint condition is || Y _k-DX _k|| ₂≤ ε,

(3e), at the objective function of the dictionary D optimized in module dictionary D be optimized: according to K-SVD, dictionary optimized algorithm calculates, when for all k=1,2 ..., N, || Y _k-DX _k|| ₂during≤ε, stop the Optimization Solution to dictionary D, export sparse coefficient X _kwith corresponding dictionary, enter step (4), otherwise repeat to step (3a).

Effect of the present invention can further illustrate with crossing experiment:

1) experiment condition

This experiment adopts standard testing image Lena, Pepper, Camerman as experimental data, and adopt Matlab7.0 as emulation tool, allocation of computer is Intel Duo i2410/2G.

2) experiment content

Utilize former K-SVD method and the image denoising method based on K-SVD and locally linear embedding of the present invention respectively, under different noise variances, noise reduction is carried out to all kinds of input test image.Setting dictionary Atom number in experiment is 128, and the level sampling of image and Vertical Sampling interval are all 4 pixels, and sampled images block size is the block of pixels of 8 × 8:

First, carry out noise reduction to the Pepper image that noise variance is 20, as shown in Figure 3, wherein Fig. 3 (a) is the noisy image of input to result, and Fig. 3 (b) is former K-SVD methods and results, and Fig. 3 (c) is the inventive method result;

Secondly, carry out noise reduction to the Cameraman image that noise variance is 30, as shown in Figure 4, wherein Fig. 4 (a) is the noisy image of input to result, and Fig. 4 (b) is former K-SVD methods and results, and Fig. 4 (c) is the inventive method result;

Finally, carry out noise reduction to the Lena image that noise variance is 40, as shown in Figure 5, wherein Fig. 5 (a) is the noisy image of input to result, and Fig. 5 (b) is former K-SVD methods and results, and Fig. 5 (c) is the inventive method result;

3) interpretation

As can be seen from Fig. 3, Fig. 4, Fig. 5, the present invention is better than former K-SVD algorithm and is about about 0.2dB in the PSNR evaluation of image noise reduction, it is relatively good that the grain details information of image all keeps, at polytype input picture as Pepper, Cameraman, Lena, can obtain good quality reconstruction.

The above is only the preferred embodiment of the present invention; it should be pointed out that for those skilled in the art, under the prerequisite not departing from the technology of the present invention principle; can also make some improvement and distortion, these improve and distortion also should be considered as protection scope of the present invention.

Claims (2)

1. the image denoising method based on K-SVD and locally linear embedding, it is characterized in that utilizing with lower module: sampling module, calculate Laplacian Matrix L module, objective function and dictionary, sparse coefficient optimization module, estimated image block acquisition module, overall estimation image block acquisition module;

Described method comprises following steps:

(1), in sampling module, input noisy image, N number of M is gone out to this image sampling ₁× M ₂the image block of pixel size, and record the position of each image block in noisy image, to a kth image block picture element matrix B _kcarry out order by row to pile up, k=1,2 ..., N, forms (a M ₁m ₂the column vector Y of) × 1 _k;

(2), in calculating Laplacian Matrix L module to the column vector corresponding to all image blocks according to locally linear embedding method, according to the structure of each vector point in space geometry, calculate the weight matrix W of distance, and and then structure Laplacian Matrix L;

(3), optimize in module in objective function and dictionary, sparse coefficient and utilize K-SVD algorithm construction objective function to be: constraint condition is || Y _k-DX _k|| ₂≤ ε carries out the Optimization Solution of parameter D and X,

Wherein, Y=[Y ₁y ₂... Y _n], D is dictionary, X _kfor Y _kthe sparse coefficient that D projects, k=1,2 ..., N, X=[X ₁x ₂... X _n], || .|| _ffor Frobenius norm, || .|| ₂be 2 norms, Tr (.) is the computing of Matrix Calculating mark, (.) ^tfor matrix turns order computing, β is weighting coefficient, and ε is resolution error thresholding; Y _kbeing a column vector, is row of matrix Y;

(4), in estimated image block acquisition module, dictionary D and sparse coefficient X is utilized _k, to column vector Y _kestimate, its solution formula is k=1,2 ..., N, and by estimator carry out sequential breakdown by row, constructing corresponding size is M ₁× M ₂estimated image block picture element matrix

(5), in overall estimation image collection module by image block that step (4) estimates, according to the positional information of image block in noisy image in step (1), be covered to the correspondence position of noisy image, and record in noisy image the estimated image block number w each pixel needing cover _i,j, i, j are location of pixels coordinate, and the pixel value of the estimated image block of correspondence k=1,2 ..., w _i,j;

(6) the estimated image block number w each pixel needing cover, is utilized in noisy image _i,j, right carry out weights addition, the pixel value of image correspondence position after final acquisition noise reduction the pixel value of noisy image and corresponding weighting coefficient is respectively with α.

2. the image denoising method based on K-SVD and locally linear embedding according to claim 1, is characterized in that the K-SVD algorithm construction objective function that utilizes described in step (3) is: constraint condition is || Y _k-DX _k|| ₂≤ ε carries out the Optimization Solution of parameter D and X, and it carries out as follows:

(a), constructing at the sparse coefficient objective function optimized in module each sparse coefficient: k=1,2 ..., N, this formula is equivalent to following objective function:

$min\left|\right|Y_k-{\mathrm{DX}}_k|{|}_2^2+{\mathrm{\βL}}_{k,k}||X_k+\frac{1}{L_{k,k}}\underset{i\≠k}{\Σ}{L}_{k,i}{X}_i|{|}_2^2,$ Constraint condition is || Y _k-DX _k|| ₂≤ ε,

Wherein, L _i,jfor the i-th row jth column element of the Laplacian Matrix L described in step (2);

(b), in step (a) $min\left|\right|Y_k-{\mathrm{DX}}_k|{|}_2^2+{\mathrm{\βL}}_{k,k}||X_k+\frac{1}{L_{k,k}}\underset{i\≠k}{\Σ}{L}_{k,i}{X}_i|{|}_2^2$ Following formula is adopted to carry out equivalence:

$min\left|\right|\left(\begin{array}{c}{Y}_k\\ -\sqrt{{\mathrm{\βL}}_{k,k}}{\mathrm{XP}}_k\end{array}\right)-\left(\begin{array}{c}D\\ \sqrt{{\mathrm{\βL}}_{k,k}}I\end{array}\right)X_k|{|}_2^2$

Wherein, I is (M ₁m ₂) × (M ₁m ₂) unit matrix of size, P _kfor corresponding to X _ka vector, be defined as:

$P_{k,i}=\left\{\begin{array}{cc}\frac{L_{k,i}}{L_{i,i}}& i\≠k\\ 0& i=k\end{array}\right.$

Wherein, P _k,ifor P _kin the value of the i-th row;

(c), be optimized in the sparse coefficient optimized in module each sparse coefficient: right in all X _k, k=1,2 ..., N, calculates according to sparse coefficient derivation algorithm in K-SVD, when || Y _k-DX _k|| ₂during≤ε, stop sparse coefficient X _ksparsely to solve;

(d), constructing at the objective function of the dictionary D optimized in module dictionary D: k=1,2 ..., N, optimizes formula and is equivalent to following objective function:

constraint condition is || Y _k-DX _k|| ₂≤ ε,

(e), be optimized at the objective function of the dictionary D optimized in module dictionary D: according to K-SVD, dictionary optimized algorithm calculates, when for all k=1,2 ..., N, || Y _k-DX _k|| ₂during≤ε, stop the Optimization Solution to dictionary D, export sparse coefficient X _kwith corresponding dictionary, enter step (4), otherwise repeat to step (a).