CN109101978B - Saliency target detection method and system based on weighted low-rank matrix recovery model - Google Patents

Saliency target detection method and system based on weighted low-rank matrix recovery model Download PDF

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CN109101978B
CN109101978B CN201810739591.2A CN201810739591A CN109101978B CN 109101978 B CN109101978 B CN 109101978B CN 201810739591 A CN201810739591 A CN 201810739591A CN 109101978 B CN109101978 B CN 109101978B
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唐厂
万诚
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China University of Geosciences
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Abstract

The invention discloses a salient object detection method and a system based on a weighted low-rank matrix recovery model, which are used for improving the accuracy of foreground and background separation in an image. Through the weighted low-rank matrix recovery, the background and the salient objects can be effectively separated, and the method can work well even when the appearances of the foreground and the background are similar and the foreground occupies most of images.

Description

Saliency target detection method and system based on weighted low-rank matrix recovery model
Technical Field
The invention relates to the technical field of computer graphic image processing, in particular to a method and a system for detecting a salient object through weighted low-rank matrix recovery.
Background
The detection of saliency targets is to detect and segment important regions in natural scenes, which has become a very popular topic in the past half century. It is useful for many applications such as object detection and recognition, image classification and retrieval, object collaborative segmentation, and content-based image editing.
Over the past few decades, many successful salient object detection models have been proposed, which can be roughly divided into two categories: top down approach and bottom up approach. The top-down approach is based on the user's task or goal, typically requiring manual labeling of truth values for supervised learning. The bottom-up approach utilizes low-level image attributes, such as color, gradient, edge, to construct a saliency map, regardless of the internal state of the user.
In a bottom-up salient object detection model, a representative set of methods is based on low-order matrix recovery (LRR) theory. This is mainly because in most cases the background of an image is usually located in a low-dimensional subspace, whereas salient regions that usually deviate from this subspace can be considered sparse noise or errors.
Notably, the LRR model can be used directly to ensure that there is not a high degree of consistency between the underlying low-rank matrix and the sparse matrix, without cluttering the background of the scene and with high contrast. However, in practice, many backgrounds are cluttered and the foreground may have a similar appearance to the background. In addition, the foreground sometimes occupies most of the image, and this does not satisfy the assumption of sparseness. In these cases, previous LRR-based methods have difficulty extracting salient objects from the background.
Disclosure of Invention
In order to overcome the technical problem, the invention provides a background priority low-level matrix recovery model (WLRR) for highlighting object detection. The background weighting matrix is applied to let the LR model know the likelihood that each image region belongs to the background. A similar weighted projection function was introduced in the paper "Robust principal component analysis with missing data" by f.shang, y.liu, j.cheng, and h.cheng to recover low order and sparse matrices from missing and severely corrupted observations.
The technical scheme adopted by the invention for solving the technical problems is as follows: an image saliency target detection method based on a weighted low-rank matrix recovery model is constructed, and the method comprises the following steps:
s1, for a given input image, dividing it into N non-overlapping superpixel regions p by superpixel division method1、p2、K、pN(ii) a N is a positive integer greater than 1;
s2, for each super pixel region piExtracting a D-dimensional feature vector expressed as fi∈RDThen, an image feature matrix F ═ F is obtained by integrating all the feature vectors1,f2,ΛfN]∈RD(ii) a Wherein D is a positive integer;
s3, applying the position, color and boundary connectivity attributes to each superpixel region piFusing the position, color and boundary connectivity attributes to generate a high-level background prior image, and then converting all the high-level background prior images into a weighting matrix W;
s4, decomposing the image characteristic matrix F into a low-order matrix L representing redundant background information and a sparse matrix S representing a significant part, wherein the objective function in the decomposition is
Figure GDA0003176701060000021
The constraint conditions are as follows:
Figure GDA0003176701060000022
wherein | · | purple sweet*Represents the nuclear norm of the matrix, which is the convex relaxation of the rank function, defined as the sum of the matrix singular values, | · | > u1L representing a matrix1The norm of the number of the first-order-of-arrival,
Figure GDA0003176701060000034
represents the multiplication of the elements of the two matrices, λ is a trade-off parameter that balances L and S, and is greater than zero;
s5, restoring and mapping the significance value of each super pixel area to the position of the super pixel area in the input image and smoothing to obtain a final significance map; wherein the super pixel PiIs l of the ith column of the sparse matrix S1And (4) norm.
Preferably, in the image saliency target detection method of the present invention, in step S3, for any super pixel region piThe specific steps for generating the high-level background prior map are as follows:
s31, position processing step: for each super pixel region piCalculating the distance between its average position and the center position c of the input image, denoted as d (p)iC) then obtaining a super-pixel region piPosition prior value of (d) lp (i) ═ 1-exp (-d (p)i,c)/σ2) σ is the standard deviation of the control gaussian kernel width;
s32, color processing step: super-pixel region p is obtained by adopting significant target detection method based on low-rank matrix recoveryiColor C ofiThen, the corresponding background color prior value CP (i) ═ 1-C is obtainedi；
S33, boundary connectivity processing step: using super-pixel regions piAnd the length of the intersection between the image boundary superpixels to quantify the degree of connection with the image border
Figure GDA0003176701060000031
Wherein, | - | represents the length of the intersection, and B represents the set of boundary superpixels;
Figure GDA0003176701060000032
representing a super pixel region piThe number of pixels contained;
s34, calculating each super pixel area piFinal high-level background prior map of (a): w (i) · lp (i) · cp (i) · bp (i), and then combines the calculated W (i) into a weighting matrix W:
Figure GDA0003176701060000033
preferably, in the image saliency target detection method of the present invention, the specific steps of decomposing the feature matrix F in step S4 are:
s41, introducing a Lagrangian multiplier Y, and transferring a solver of the model to minimize the following augmented Lagrangian function kappa:
Figure GDA0003176701060000041
and iteratively executing steps S42-S44 by using an ADMM model until the optimal S, L and Y are searched, wherein mu is a normal number, | | · |. luminance |FRepresents the F range;
s42, updating L: when S and Y are fixed, L is in (k +1) iterations(k+1)The solution of (a) is obtained by solving an optimal solution of the following formula:
Figure GDA0003176701060000042
s43, updating S: updating S with fixed L and Y(k+1)Solving for S by the following formula(k+1)：
Figure GDA0003176701060000043
S44, updating Y by adopting the following formula to obtain Yk+1：
Figure GDA0003176701060000044
According to another aspect of the present invention, to solve the technical problem, there is provided an image saliency target detection system based on a weighted low rank matrix recovery model, including the following modules:
a super-pixel processing module for dividing a given input image into N non-overlapping super-pixel regions p by using a super-pixel division method1、p2、K、pN(ii) a N is a positive integer greater than 1;
a feature matrix processing module for processing each super pixel region piExtracting a D-dimensional feature vector expressed as fi∈RDThen, an image feature matrix F ═ F is obtained by integrating all the feature vectors1,f2,ΛfN]∈RD(ii) a Wherein D is a positive integer;
a weighting matrix processing module for applying the position, color and boundary connectivity attributes to each superpixel region piFusing the position, color and boundary connectivity attributes to generate a high-level background prior image, and then converting all the high-level background prior images into a weighting matrix W;
a feature matrix processing module for decomposing the image feature matrix F into a low-order matrix L representing redundant background information and a sparse matrix S representing a significant portion, the objective function during decomposition being
Figure GDA0003176701060000051
The constraint conditions are as follows:
Figure GDA0003176701060000054
wherein | · | purple sweet*Represents the nuclear norm of the matrix, which is the convex relaxation of the rank function, defined as the sum of the matrix singular values, | · | > u1L representing a matrix1The norm of the number of the first-order-of-arrival,
Figure GDA0003176701060000055
representing multiplication of elements of two matrices, λ being a trade-off parameter balancing L and SAnd is greater than zero;
the final result processing module is used for restoring and mapping the significance value of each super pixel area to the position of the super pixel area in the input image and performing smoothing processing to obtain a final significance map; wherein the super pixel region piIs l of the ith column of the sparse matrix S1And (4) norm.
Preferably, in the image saliency target detection system of the present invention, in the weighting matrix processing module, for any super pixel region piGenerating a high-level background prior map by adopting the following sub-modules:
a position processing step submodule for processing each of the super-pixel regions piCalculating the distance between its average position and the center position c of the input image, denoted as d (p)iC) then obtaining a super-pixel region piPosition prior value of (d) lp (i) ═ 1-exp (-d (p)i,c)/σ2) σ is the standard deviation of the control Gaussian clock width;
a color processing step submodule for obtaining the super-pixel region p by adopting a significant object detection method based on low-rank matrix recoveryiColor C ofiThen, the corresponding background color prior value CP (i) ═ 1-C is obtainedi；
A boundary connectivity processing submodule for using the superpixel region piAnd the length of the intersection between the image boundary superpixels to quantify the degree of connection with the image border
Figure GDA0003176701060000052
Wherein, | - | represents the length of the intersection, and B represents the set of boundary superpixels;
Figure GDA0003176701060000053
representing a super pixel region piThe number of pixels contained;
a weight matrix synthesis submodule for calculating the respective superpixel regions piFinal high-level background prior map of (a): w (i) · lp (i) · cp (i) · bp (i), and then combines the calculated W (i) into a weighting matrix W:
Figure GDA0003176701060000061
preferably, in the image saliency target detection system of the present invention, the feature matrix processing module decomposes the feature matrix F using the following sub-modules:
a lagrangian processing submodule for introducing a lagrangian multiplier Y to transfer the solver of the model to minimize the following augmented lagrangian function κ:
Figure GDA0003176701060000062
and using an ADMM model iteration calling step to update the L submodule, the S submodule and the Y submodule until the optimal S, L and Y are searched, wherein mu is a normal number, | | | · |. the neutral cellsFRepresents the F range;
updating L submodule for L in (k +1) iterations when S and Y are fixed(k+1)The solution of (a) is obtained by solving an optimal solution of the following formula:
Figure GDA0003176701060000063
update S submodule for updating S with fixed L and Y(k+1)Solving for S by the following formula(k+1)：
Figure GDA0003176701060000064
Updating Y submodule for obtaining Y by adopting the following formulak+1：
Figure GDA0003176701060000065
By implementing the salient object detection method and the salient object detection system restored by the weighted low-rank matrix, the salient objects can be well extracted from the background when the appearances of the foreground and the background are similar and the foreground occupies most of images.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
fig. 1 is a flowchart of a significant object detection model recovered by a weighted low-rank matrix according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of the steps of a WLRR solution algorithm using an ADMM;
FIG. 3 is a visual comparison of the performance of the present invention on different data sets with some of the recently proposed methods;
FIG. 4 is a PR curve and F-measure curve of WLRR and other recently proposed methods on ECSSD dataset;
FIG. 5 is a PR curve and F-measure curve of WLRR and other recently proposed methods on a SOD data set;
FIG. 6 shows PR curves and F-measure curves on the MSRA10K data set for WLRR and other recently proposed methods.
Detailed Description
For a more clear understanding of the technical features, objects and effects of the present invention, embodiments of the present invention will now be described in detail with reference to the accompanying drawings.
Referring to fig. 1, it is a flowchart of a salient object detection model recovered by a weighted low rank matrix according to an embodiment of the present invention. In the image saliency target detection method based on the weighted low-rank matrix recovery model of the embodiment, the method comprises the following steps:
s1, for a given input image, dividing it into N non-overlapping superpixel regions p by superpixel division method1、p2、K、pN(ii) a N is a positive integer greater than 1.
S2, for each super pixel region piExtracting a D-dimensional feature vector expressed as fi∈RDThen, an image feature matrix F ═ F is obtained by integrating all the feature vectors1,f2,ΛfN]∈RD(ii) a Wherein D is a positive integer.
S3, applying the position, color and boundary connectivity attributes to each superpixel region piThe position, color and boundary connectivity attributes of the image are fused together to generate a high-level background prior image, and then all the high-level background prior images are converted into a weighting matrix W together.
For any super pixel region piThe specific steps for generating the high-level background prior map are as follows:
s31, position processing step: generally, objects near the center of the image are more attractive to humans. On objects that are far from the center of the image, there is a greater likelihood of belonging to the background; for each super pixel region piThe distance between its average position and the center position c of the input image is calculated and denoted as d (P)iC) then obtaining a super-pixel region piPosition prior value of (d) lp (i) ═ 1-exp (-d (p)i,c)/σ2) σ is the standard deviation of the control Gaussian clock width;
s32, color processing step: in our daily life, warm colors such as red and yellow tend to be more pronounced, where the background color prior is obtained directly using the method proposed in "a unified approach to a liquid object detection view low matrix recovery" published in 2012 by x.shen and y.wu: adopting a significant object detection method (namely, A uniform approach to spatial object detection view low rank matrix recovery) based on low rank matrix recovery to obtain a super-pixel region piColor C ofiThen, the corresponding background color CP (i) ═ 1-C is obtainedi；
S33, boundary connectivity processing step: the probability that the background region is connected to the image border is very high, and few salient regions are connected to the image border; here, a super pixel region p is usediAnd the length of the intersection between the image boundary superpixels to quantify the degree of connection with the image border
Figure GDA0003176701060000081
Wherein, | - | represents the length of the intersection, and B represents the set of boundary superpixels;
Figure GDA0003176701060000082
representing a super pixel region piThe number of pixels contained;
s34, obtaining the three background priorities, the super pixel area piThe final background of (a) may be w (i) ═ lp (i) · cp (i) · bp (i). For convenience of representation and calculation, the invention combines W (i) by copying D rows to obtain a weight matrix W, and then combines the calculated W (i) into the weight matrix W:
Figure GDA0003176701060000091
s4, the image feature matrix F is decomposed into a low-order matrix L representing redundant background information and a sparse matrix S representing significant portions. The invention provides a novel background priority WLRR model:
Figure GDA0003176701060000092
wherein | · | purple sweet*Represents the nuclear norm of the matrix, which is the convex relaxation of the rank function, defined as the sum of the matrix singular values, | · | > u1L representing a matrix1The norm of the number of the first-order-of-arrival,
Figure GDA0003176701060000094
which represents the multiplication of the elements of the two matrices, is a trade-off parameter that balances L and S and is greater than zero.
The WLRR model of the invention expands the classical matrix recovery model by taking into account the possibility that each region of the image belongs to the background, thereby enhancing the low level and sparsity of L and S. When the weighting matrix assigns a smaller weight in the feature matrix F, the corresponding vector l in the recovered sparse matrix S1The norm tends to be smaller. Thus, the foreground portion can be more effectively highlighted.
The WLRR model is apparently a convex optimization problem that can be solved efficiently by the multiplier (ADMM) approach to the direction of intersection. Referring to fig. 2, the lagrangian multiplier Y is first introduced, and then the solver of the model can be shifted to minimize the following augmented lagrangian function:
Figure GDA0003176701060000093
then the ADMM model is used to alternately search for the best S, L and Y, where μ is a normal number, | · | | computationally |, in turnFRepresents the F range.
The ADMM model is proposed by "Linear alternating orientation method with adaptive dependency for low-link prediction" published in Proc.Adv.neural Inf. Process.Syst.24,2011, pp.612-620, by Z.Lin, R.Liu, and Z.Su i.
And L is updated: when S and Y are fixed, L is in (k +1) iterations(k+1)The solution of (2) can be obtained by solving the following problems:
Figure GDA0003176701060000101
and (4) updating S: updating S with fixed L and Y(k+1)The following minimization problem is obtained:
Figure GDA0003176701060000102
S(k+1)the solution of (a) has an approximate form:
Figure GDA0003176701060000103
here, shrink (X, t) ═ sign (X) max (abs (X) -t,0)
Updating Y:
Figure GDA0003176701060000104
s5, restoring and mapping the significance value of each super pixel to the super pixel in the input imagePerforming smoothing treatment on the positions to obtain a final saliency map; wherein the super pixel region piIs the significance of1And (4) norm.
The main computational cost of algorithm 1 in fig. 2 is the Singular Value Decomposition (SVD) of the D × N matrix in the step of updating L when using the linearized ADMM method. Its computational complexity is o (D)2N+DN2+N3). Fortunately, D and N are not very large in algorithm 1 (specifically, D ═ 53, N ≈ 200). Thus, the algorithm of the present invention is also very efficient, requiring about 1.51s to compute an image with a resolution of 400 x 300 on the MSRA10K dataset.
The method of the present invention is compared with other various recently proposed significance detection methods on three published data sets with reference to fig. 4-6. In particular, they used images with only one salient object in the MSRA10K dataset, images with multiple salient objects in the SOD dataset, and images with complex scenes in the ECSSD dataset. In the experiments of the present invention, the present invention extracted the same features as the ULR model and set λ to 0.6.
The ULR model is proposed by X.Shen and Y.Wu in Proc.IEEE Compout.Vis.Pattern Recognit.2012, pp.853-860 paper "influenced assessment to medical object detection via low rank matrix recovery".
The effectiveness of the model is illustrated by comparison with some advanced technology methods, as shown in fig. 3. It can be seen that our method can accommodate different scenes and image processing with complex foreground and background.
On the basis of quantitative evaluation, the invention compares two general performance indexes: PR curve and F-measure curve.
The F-measure curve is defined as:
Figure GDA0003176701060000111
where set to beta20.3 to improve accuracy.
On the MSRA10K dataset, the PR curve of our method is almost the same as the best method, while the region under the F-measure curve of our model is almost the best.
On the SOD data set, the method of the invention is also competitive on PR curves, and it is noted that the WLRR model of the invention is better on RR curves than all other methods, and the area under the F-measure curve of the model is also the best.
On the ECSSD dataset, the PR of the model of the invention is somewhat lower than the best method, but better than all other methods. For the area under the F-measure curve, our model is also only a little smaller than the best method. These results verify that our WLRR model is able to handle images of complex scenes very well.
While the present invention has been described with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, which are illustrative and not restrictive, and it will be apparent to those skilled in the art that various changes and modifications can be made therein without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (4)

1. An image saliency target detection method based on a weighted low-rank matrix recovery model is characterized by comprising the following steps:
s1, for a given input image, dividing it into N non-overlapping superpixel regions p by superpixel division method1、p2、K、pN(ii) a N is a positive integer greater than 1;
s2, for each super pixel region piExtracting a D-dimensional feature vector expressed as fi∈RDThen, an image feature matrix F ═ F is obtained by integrating all the feature vectors1,f2,ΛfN]∈RD(ii) a Wherein D is a positive integer;
s3, applying the position, color and boundary connectivity attributes to each superpixel region piFusing the position, color and boundary connectivity attributes to generate a high-level background prior map,then converting all the high-level background prior images into a weighting matrix W;
s4, decomposing the image characteristic matrix F into a low-order matrix L representing redundant background information and a sparse matrix S representing a significant part, wherein the objective function in the decomposition is
Figure FDA0003176701050000011
The constraint conditions are as follows:
Figure FDA0003176701050000012
wherein | · | purple sweet*Represents the nuclear norm of the matrix, which is the convex relaxation of the rank function, defined as the sum of the matrix singular values, | · | > u1L representing a matrix1The norm of the number of the first-order-of-arrival,
Figure FDA0003176701050000013
represents the multiplication of the elements of the two matrices, λ is a trade-off parameter that balances L and S, and is greater than zero;
s5, restoring and mapping the significance value of each super pixel area to the position of the super pixel area in the input image and smoothing to obtain a final significance map; wherein the super pixel PiIs l of the ith column of the sparse matrix S1A norm;
wherein in step S3, for any super pixel region piThe specific steps for generating the high-level background prior map are as follows:
s31, position processing step: for each super pixel region piCalculating the distance between its average position and the center position c of the input image, denoted as d (p)iC) then obtaining a super-pixel region piPosition prior value of (d) lp (i) ═ 1-exp (-d (p)i,c)/σ2) σ is the standard deviation of the control gaussian kernel width;
s32, color processing step: super-pixel region p is obtained by adopting significant target detection method based on low-rank matrix recoveryiColor C ofiThen, the corresponding background color prior value CP (i) ═ 1-C is obtainedi；
S33, boundary connectivity processing step: using super-pixel regions piAnd the length of the intersection between the image boundary superpixels to quantify the degree of connection with the image border
Figure FDA0003176701050000021
Wherein, | - | represents the length of the intersection, and B represents the set of boundary superpixels;
Figure FDA0003176701050000022
representing a super pixel region piThe number of pixels contained;
s34, calculating each super pixel area piFinal high-level background prior map of (a): w (i) · lp (i) · cp (i) · bp (i), and then combines the calculated W (i) into a weighting matrix W:
Figure FDA0003176701050000023
2. the image salient object detection method according to claim 1, wherein the specific steps of decomposing the feature matrix F in step S4 are as follows:
s41, introducing a Lagrangian multiplier Y, and transferring a solver of the model to minimize the following augmented Lagrangian function kappa:
Figure FDA0003176701050000024
and iteratively executing steps S42-S44 by using an ADMM model until the optimal S, L and Y are searched, wherein mu is a normal number, | | · |. luminance |FExpressing the F range, wherein lambda is a balanced parameter of L and S and is larger than zero;
s42, updating L: when S and Y are fixed, L is in (k +1) iterations(k+1)The solution of (a) is obtained by solving an optimal solution of the following formula:
Figure FDA0003176701050000031
s43, updating S: updating S with fixed L and Y(k+1)Solving for S by the following formula(k+1)：
Figure FDA0003176701050000032
S44, updating Y by adopting the following formula to obtain Yk+1：
Figure FDA0003176701050000033
3. An image saliency target detection system based on a weighted low-rank matrix recovery model is characterized by comprising the following modules:
a super-pixel processing module for dividing a given input image into N non-overlapping super-pixel regions p by using a super-pixel division method1、p2、K、pN(ii) a N is a positive integer greater than 1;
a feature matrix processing module for processing each super pixel region piExtracting a D-dimensional feature vector expressed as fi∈RDThen, an image feature matrix F ═ F is obtained by integrating all the feature vectors1,f2,ΛfN]∈RD(ii) a Wherein D is a positive integer;
a weighting matrix processing module for applying the position, color and boundary connectivity attributes to each superpixel region piThe position, color and boundary connectivity attributes of the super-pixel region are fused together to generate a high-level background prior map, all the high-level background prior maps are converted into a weighting matrix W, and in a weighting matrix processing module, for any super-pixel region piGenerating a high-level background prior map by adopting the following sub-modules:
a position processing sub-module for each super-pixel regionpiCalculating the distance between its average position and the center position c of the input image, denoted as d (p)iC) then obtaining a super-pixel region piPosition prior value of (d) lp (i) ═ 1-exp (-d (p)i,c)/σ2) σ is the standard deviation of the control Gaussian clock width;
a color processing step submodule for obtaining the super-pixel region p by adopting a significant object detection method based on low-rank matrix recoveryiColor C ofiThen, the corresponding background color CP (i) ═ 1-C is obtainedi；
A boundary connectivity processing submodule for using the superpixel region piAnd the length of the intersection between the image boundary superpixels to quantify the degree of connection with the image border
Figure FDA0003176701050000041
Wherein, | - | represents the length of the intersection, and B represents the set of boundary superpixels;
Figure FDA0003176701050000042
representing a super pixel region piThe number of pixels contained;
a weight matrix synthesis submodule for calculating the respective superpixel regions piFinal high-level background prior map of (a): w (i) · lp (i) · cp (i) · bp (i), and then combines the calculated W (i) into a weighting matrix W:
Figure FDA0003176701050000043
a feature matrix processing module for decomposing the image feature matrix F into a low-order matrix L representing redundant background information and a sparse matrix S representing a significant portion, the objective function during decomposition being
Figure FDA0003176701050000044
The constraint conditions are as follows:
Figure FDA0003176701050000045
wherein | · | purple sweet*Representing the kernel of a matrixNorm, which is the convex relaxation of the rank function, defined as the sum of matrix singular values, | · | | survival1L representing a matrix1The norm of the number of the first-order-of-arrival,
Figure FDA0003176701050000046
represents the multiplication of the elements of the two matrices, λ is a trade-off parameter that balances L and S, and is greater than zero;
the final result processing module is used for restoring and mapping the significance value of each super pixel area to the position of the super pixel area in the input image and performing smoothing processing to obtain a final significance map; wherein the super pixel region piIs l of the ith column of the sparse matrix S1And (4) norm.
4. The image salient object detecting system of claim 3, wherein the feature matrix processing module decomposes the feature matrix F using the following sub-modules:
a lagrangian processing submodule for introducing a lagrangian multiplier Y to transfer the solver of the model to minimize the following augmented lagrangian function κ:
Figure FDA0003176701050000051
and using an ADMM model iteration calling step to update the L submodule, the S submodule and the Y submodule until the optimal S, L and Y are searched, wherein mu is a normal number, | | | · |. the neutral cellsFRepresents the F range;
updating L submodule for L in (k +1) iterations when S and Y are fixed(k+1)The solution of (a) is obtained by solving an optimal solution of the following formula:
Figure FDA0003176701050000052
update S submodule for updating S with fixed L and Y(k+1)Solving for S by the following formula(k+1)：
Figure FDA0003176701050000053
λ is a trade-off parameter that balances L and S, and is greater than zero;
updating Y submodule for obtaining Y by adopting the following formulak+1：
Figure FDA0003176701050000054
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