A hyperspectral image anomaly detection method based on coefficient analysis collaborative representation

Abstract

The application provides a hyperspectral image anomaly detection method based on coefficient analysis collaborative representation, comprising: acquiring a hyperspectral image and performing normalization processing; setting inner and outer window sizes and sequentially traversing each spatial position of the hyperspectral image; judging whether the traversal is completed; if the traversal is not completed, performing a collaborative representation algorithm on a center test pixel by using a pixel set of a local region, calculating a representation coefficient vector; estimating maximum and minimum background coefficient thresholds, eliminating abnormal points and updating the pixel set of the local region; performing the collaborative representation algorithm on the center test pixel again by using the updated pixel set of the local region, calculating the representation coefficient vector; obtaining a representation error of the center test pixel; if the traversal is completed, performing normalization processing on the representation error of a global pixel; comparing with a preset threshold value to obtain an anomaly detection result of the whole hyperspectral image. The application can greatly improve the accuracy of hyperspectral image anomaly detection.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of hyperspectral image processing, and particularly relates to a hyperspectral image anomaly detection method based on coefficient analysis collaborative representation. BACKGROUND

[0002] The hyperspectral image presents a three-dimensional cubic data structure of "graph and spectrum integration", and provides high-dimensional spectral information reflecting the inherent properties of various ground objects while providing two-dimensional spatial information about the distribution of ground objects. The spectral resolution of the hyperspectral image can reach 10 -2 λ order of magnitude, which provides key information for ground object identification and analysis. Hyperspectral image anomaly detection aims to identify pixels / sub-pixels or objects with obvious spectral differences from the surrounding background. Compared with target detection which relies on prior spectral information, it has stronger practicability and has become a research hotspot in the field of hyperspectral image processing.

[0003] As the most representative classical method in hyperspectral image anomaly detection, the RX algorithm proposed by Reed and Xiaoli is a constant false alarm rate detection model based on generalized likelihood ratio test. The method assumes that the background components in the hyperspectral image obey a multivariate Gaussian distribution model, and is finally formulated as the Mahalanobis distance between the pixel spectrum and the background mean spectrum. However, a single multivariate Gaussian distribution model is usually difficult to describe real hyperspectral images with complex composition. In addition, the anomaly pollution problem in the background mean vector and the background covariance matrix also becomes a key factor restricting the RX detection performance. Therefore, a series of improved RX versions are proposed. The local RX method adopts a sliding double-window strategy, assumes that the local area surrounded by the inner and outer windows obeys a multivariate Gaussian distribution, and performs RX detection in it in turn. The kernel RX method proposed by Kown uses kernel theory to project the original hyperspectral image into a high-dimensional feature space to better fit the Gaussian distribution and mine high-dimensional nonlinear information. Professor Zhang Liangpei's team of Wuhan University uses the GoDec algorithm to decompose the hyperspectral image into a low-rank background matrix, a sparse anomaly matrix and an additive noise matrix, and calculates the background mean vector and the background covariance matrix from the low-rank background matrix, thereby alleviating the anomaly pollution problem in the background statistics.

[0004] With the rapid development of signal representation theory in the field of computer vision, the hyperspectral image anomaly detection method based on representation theory has gradually attracted widespread attention. Professor Li Wei's team of Beijing University of Technology first applied the collaborative representation model to hyperspectral image anomaly detection in 2015. The method adopts a sliding double window strategy, assumes that all background pixels can be approximately represented as a linear combination of their surrounding neighborhood pixels, while anomalies cannot, and thus realizes anomaly detection according to the representation error. However, due to the lack of prior labels, the existence of abnormal pollution in the local region will weaken its representation ability of background information, leading to the missed detection of real anomalies, which is the key factor restricting the performance of the collaborative representation method. Therefore, Vafadar et al. assumed that the background and abnormal signals have significantly different spectral intensity values, and removed potential anomalies from the local region by measuring the spectral intensity information. However, the different spectral distribution of the background and the anomaly cannot represent the significant difference in spectral intensity, which to some extent limits the effectiveness of the method. Therefore, it is urgent to solve the problem of abnormal pollution in the local region faced by the traditional collaborative representation method and realize accurate detection of abnormal targets in the hyperspectral image. SUMMARY

[0005] The application provides a hyperspectral image anomaly detection method based on coefficient analysis collaborative representation, which aims to solve the problem of abnormal pollution in the local region faced by the traditional collaborative representation method and improve the accuracy of hyperspectral image anomaly detection.

[0006] The technical scheme of the application is as follows: the application provides a hyperspectral image anomaly detection method based on coefficient analysis collaborative representation, which comprises the following steps:

[0007] Step 1: Obtain the hyperspectral image and perform normalization processing;

[0008] Step 2: Adopt a sliding double window strategy, use the pixel set of the local region to perform a collaborative representation algorithm on the center test pixel, and calculate the representation coefficient vector;

[0009] Step 3: Estimate the maximum and minimum background coefficient threshold, remove abnormal points, and update the pixel set of the local region;

[0010] Step 4: Use the updated pixel set of the local region to perform a collaborative representation algorithm on the center test pixel again, calculate the representation coefficient vector, and obtain the representation error of the center test pixel;

[0011] Step 5: Perform normalization processing on the representation error of the global pixel to obtain the anomaly detection result.

[0012] Further, in step 1, the normalization processing is specifically:

[0013] Step (11): Convert the acquired hyperspectral image X into a two-dimensional hyperspectral matrix X of size S×N. 2D Where S is the number of bands in X and N is the number of pixels in X;

[0014] Step (12): Calculate X 2D The maximum value of each row element corresponding to each band in the middle band. i and minimum value MIN i and for X 2D The elements in each row of the table are normalized using the following formula (1):

[0015]

[0016] Among them, X 2D (i,:) represents X 2D The element in the i-th row, i = 1, 2, ..., S;

[0017] Step (13): The normalized two-dimensional hyperspectral matrix X 2D It is then converted back into a three-dimensional hyperspectral matrix X.

[0018] Furthermore, step two specifically involves:

[0019] Step (21): Set the inner window size a×a and the outer window size b×b to test the pixel x. j,k Concentric rectangular double windows are constructed around the center, where j = 1, 2, ..., H, k = 1, 2, ..., W, and H and W represent the spatial length and width dimensions of the hyperspectral image X, respectively;

[0020] Step (22): Utilize the pixel set S of the local region enclosed by the concentric rectangular double windows j,k For the center test pixel x j,k Execute the collaborative representation algorithm to compute the representation coefficient vector α j,k The objective function of this process is:

[0021]

[0022] Where λ is a Lagrange multiplier, the above equation applies to α j,k Find the partial derivative and set the result to 0 to obtain α. j,k for:

[0023]

[0024] Where I is the identity matrix with all diagonal elements being 1.

[0025] Furthermore, step three specifically includes:

[0026] Step (31): Based on the coefficient vector α obtained in Step 2j,k , compute its mean mean and standard deviation std, estimate the maximum, minimum background coefficient threshold value according to formula (4):

[0027] MAX α = mean + n * std (4)

[0028] MIN α = mean - n * std

[0029] where n is a positive integer;

[0030] Step (32): According to the representation coefficient vector a j,k , eliminate those pixels whose representation coefficients are greater than MAX j,k from the pixel set S α of the local region, and those whose representation coefficients are less than MIN α , thereby updating the pixel set S

[0031] Further, the fourth step is specifically:

[0032] Step (41): Using the updated pixel set S of the local region, again perform the collaborative representation algorithm on the center test pixel x j,k , and calculate the updated representation coefficient vector a The objective function of this process is:

[0033]

[0034] where λ is the Lagrange multiplier, and the above formula is taken with respect to , and the result is 0, so that is:

[0035]

[0036] Step (42): Calculate the representation error r j,k of the center test pixel x j,k :

[0037]

[0038] Further, the fifth step is specifically:

[0039] Step (51): Move the concentric rectangular double window forward by one pixel distance, and take the common center of the concentric rectangular double window as the center test pixel x j+1,k+1 , and again perform steps two to four; with the traversal of the concentric rectangular double window, this process is repeatedly repeated, and finally the representation error r∈R H×W of the global pixel is obtained.

[0040] Step (52): calculate the maximum value r of all elements in the global pixel representation error r max and the minimum value r min , and perform the following normalization processing formula (8) on the global pixel representation error r:

[0041]

[0042] Wherein r j,k is the jth row and kth column element in r, j=1,2,...,H, k=1,2,...,W;

[0043] Step (53): compare the global pixel representation error r with a preset threshold t, mark the pixels with error values greater than t as abnormal points, and mark the pixels with error values less than t as background points, thereby obtaining the abnormal detection result of the entire hyperspectral image, wherein 0≤t≤1.

[0044] Compared with the prior art, the beneficial effects of the present application are:

[0045] When the pixel set in the local area surrounded by the inner and outer windows is used to perform collaborative representation on the center test pixel, it can be found that the background pixels occupying the dominant position in the local area present a smooth and regular coefficient distribution, and the coefficients of the rarely existing abnormal pixels seriously deviate from the background coefficient distribution. Accordingly, the present application adopts a sliding double window strategy, uses the pixel set in the local area to perform a collaborative representation algorithm on the center test pixel, and estimates the maximum and minimum background coefficient thresholds according to the different coefficient distribution rules of the background and abnormal pixels, thereby eliminating potential abnormal pixels and retaining background pixels, thereby solving the problem of abnormal pollution in the local area. After eliminating the abnormal pollution in the local area, the remaining pixels are used to perform collaborative representation on the center test pixel again to obtain the representation error of the center test pixel. With the continuous traversal of the concentric rectangular double window, the representation error of the global pixel can be obtained, and then the abnormal detection result of the entire hyperspectral image can be obtained.

[0046] The present application discloses the different coefficient distribution rules of background and abnormal pixels in collaborative representation, estimates the maximum and minimum background coefficient thresholds accordingly, eliminates potential abnormal pixels in the local area, solves the problem of abnormal pollution in the local area, improves the traditional collaborative representation method, realizes accurate abnormal detection result, and the background and abnormality present strong separability.

[0047] The above description is only a summary of the technical solutions of the present application, in order to more clearly understand the technical means of the present application, and can be implemented according to the content of the specification, the following preferred embodiments of the present application are described in detail with the help of the drawings. BRIEF DESCRIPTION OF DRAWINGS

[0048] In order to more clearly illustrate the technical solutions of the embodiments of the present application or the prior art, the drawings required to be used in the embodiments will be briefly introduced as follows. Obviously, the drawings in the following description only constitute some embodiments of the present application, and for those skilled in the art, other drawings can also be obtained without creative labor on the basis of these drawings.

[0049] Figure 1 for the flowchart of the present application;

[0050] Figure 2 for the schematic diagram of different coefficient distribution rules of background and abnormal pixels in collaborative representation;

[0051] Figure 3 for the comparison diagram of abnormal detection results of four different methods;

[0052] Figure 4 for the comparison diagram of AUC values of four different methods;

[0053] Figure 3 In FIG. 1, the scene schematic diagram and the abnormal true value diagram of the Urban hyperspectral dataset are respectively shown, and the comparison diagram of abnormal detection results obtained by using the traditional RX method, the local RX method, the traditional collaborative representation method and the method of the present application on the Urban hyperspectral dataset is shown.

[0054] Figure 4 In FIG. 2, the comparison of AUC values obtained by using the traditional RX method, the local RX method, the traditional collaborative representation method and the method of the present application on the Urban hyperspectral dataset is respectively shown. DETAILED DESCRIPTION

[0055] The present application will be described in detail below in combination with the embodiments shown in the drawings, but it should be noted that these embodiments are not a limitation on the present application, and equivalent transformations or substitutions of function, method or structure made by those skilled in the art on the basis of these embodiments are all within the protection scope of the present application.

[0056] Embodiment 1

[0057] As shown in FIG. 1, the present application provides a hyperspectral image anomaly detection method based on coefficient analysis collaborative representation, which comprises the following steps: Figure 1

[0058] Step 1: Obtain the hyperspectral image and perform normalization processing;

[0059] Step 2: Use the sliding double-window strategy, use the pixel set of the local region to perform the collaborative representation algorithm on the center test pixel, and calculate the representation coefficient vector;

[0060] ​Step three: estimate the maximum and minimum background coefficient threshold, eliminate abnormal points, and update the pixel set of the local region;

[0061] Step four: use the updated pixel set of the local region to perform the collaborative representation algorithm on the center test pixel again, calculate the representation coefficient vector, and obtain the representation error of the center test pixel;

[0062] Step five: perform normalization processing on the representation error of the global pixel to obtain the abnormal detection result.

[0063] Further, in step five, the representation error of the global pixel is normalized, and the abnormal detection result is obtained by comparing with a preset threshold, which is generally between 0 and 1.

[0064] Further, the abnormal detection result is the abnormal detection result of the entire hyperspectral image.

[0065] The embodiment of the application adopts a sliding double window strategy, uses the pixel set of the local region to perform the collaborative representation algorithm on the center test pixel, estimates the maximum and minimum background coefficient threshold according to the different coefficient distribution rules of the background and abnormal pixels, eliminates the potential abnormal pixels, and retains the background pixels, thereby solving the problem of abnormal pollution in the local region. After eliminating the abnormal pollution in the local region, the collaborative representation algorithm is performed on the center test pixel again using the remaining pixels to obtain the representation error of the center test pixel. With the continuous traversal of the concentric rectangular double window, the representation error of the global pixel can be obtained, and then the abnormal detection result of the entire hyperspectral image can be obtained.

[0066] Compared with the existing hyperspectral image anomaly detection method, the application discloses the different coefficient distribution rules of the background and abnormal pixels in the collaborative representation, estimates the maximum and minimum background coefficient threshold, eliminates the potential abnormal pixels in the local region, solves the problem of abnormal pollution in the local region, improves the traditional collaborative representation method, realizes accurate abnormal detection result, and the background and abnormality show strong separability.

[0067] Embodiment 2

[0068] On the basis of embodiment 1, in step one, the normalization processing is specifically:

[0069] Step (11): convert the obtained hyperspectral image X into a two-dimensional hyperspectral matrix X with a size of SxN 2D , wherein S is the number of bands of X, and N is the number of pixels in X;

[0070] Step (12): calculate the maximum value MAX 2D and the minimum value MIN i of each row element corresponding to each band in X iand each row element in X 2D is normalized by the following formula (1):

[0071]

[0072] where X 2D (i,:) denotes the i-th row element of X 2D , i = 1, 2, …, S;

[0073] Step (13): The normalized two-dimensional hyperspectral matrix X 2D is converted into a three-dimensional hyperspectral matrix X again.

[0074] Further, the step two is specifically:

[0075] Step (21): Set the inner window size a x a and the outer window size b x b to test the pixel x j,k as the center, and construct a concentric rectangular double window, where j = 1, 2, …, H, k = 1, 2, …, W, and H and W represent the spatial length and width dimensions of the hyperspectral image X respectively;

[0076] Step (22): Use the pixel set S j,k surrounded by the local area of the concentric rectangular double window to perform the collaborative representation algorithm on the center test pixel x j,k , and calculate the representation coefficient vector a j,k , the objective function of the process is:

[0077]

[0078] where λ is the Lagrange multiplier, and the partial derivative of the above formula with respect to a j,k is taken and set to 0 to obtain a j,k :

[0079]

[0080] where I is a unit matrix with diagonal elements being 1.

[0081] Further, the step three is specifically:

[0082] Step (31): According to the representation coefficient vector a j,k obtained in step two, calculate the mean and standard deviation std, and estimate the maximum and minimum background coefficient threshold values according to formula (4):

[0083] MAX α = mean + n*std (4)

[0084] MIN α = mean - n*std

[0085] where n is a positive integer;

[0086] Step (32): According to the representation coefficient vector a j,k , remove those pixels whose representation coefficients are greater than MAX j,k and those pixels whose representation coefficients are less than MIN α from the pixel set S α of the local region, thereby updating the pixel set S

[0087] Further, the fourth step is specifically:

[0088] Step (41): Using the updated pixel set S of the local region, again perform the collaborative representation algorithm on the center test pixel x j,k to calculate the updated representation coefficient vector a The objective function of this process is:

[0089]

[0090] where λ is the Lagrange multiplier, and the above formula is differentiated with respect to and set the result to 0 to obtain :

[0091]

[0092] Step (42): Calculate the representation error r j,k of the center test pixel x j,k :

[0093]

[0094] Further, the fifth step is specifically:

[0095] Step (51): Move the concentric rectangular double window forward by a distance of one pixel point, with the common center of the concentric rectangular double window as the center test pixel x j+1,k+1 , and again perform steps two to four for analysis; with the traversal of the concentric rectangular double window, this process is repeatedly performed, and finally the representation error r ∈ R H×W of the global pixel is obtained.

[0096] Step (52): Calculate the maximum value r max and the minimum value r min of all elements in the representation error r of the global pixel, and perform the following normalization processing formula (8) on the representation error r of the global pixel:

[0097]

[0098] wherein r j,k is the element in the jth row and the kth column in r, j = 1, 2, …, H, k = 1, 2, …, W;

[0099] Step (53): comparing the representation error r of the global pixel with a preset threshold t, marking the pixel with an error value greater than t as an anomaly point, and marking the pixel with an error value less than t as a background point, so as to obtain the anomaly detection result of the entire hyperspectral image, wherein 0≤t≤1.

[0100] Embodiment 3

[0101] In this embodiment, the different coefficient distribution rules of the background pixels and the anomaly pixels in a local region when performing collaborative representation on the center test pixel are analyzed, please refer to Figure 2 For the Urban hyperspectral data set, Figure 2 (a) shows the coefficient distribution curve of the background pixels (represented as light-colored parts) and the anomaly pixels (represented as dark-colored parts) in a local region when performing collaborative representation on the center background pixel, and arranges them in descending order. Figure 2 (b) shows the coefficient distribution curve of the background pixels (represented as light-colored parts) and the anomaly pixels (represented as dark-colored parts) in a local region when performing collaborative representation on the center anomaly pixel, and arranges them in descending order. It can be found that the dominant background pixels in the local region present a smooth and regular coefficient distribution, while the coefficients of the rarely occurring anomaly pixels obviously deviate from the background coefficient distribution. Therefore, potential anomaly pixels can be removed by removing those pixels deviating from the main distribution, and this embodiment proves the theoretical rationality of the method of the present application.

[0102] Embodiment 4

[0103] The experimental hyperspectral data set used in the embodiment of the present application is taken by a hyperspectral digital image collection experiment instrument HYDICE, covering a residential area in the suburbs of the United States, with a spatial length and width of 80 and 100 respectively, and containing 160 spectral bands. This data set is called Urban hyperspectral data set.

[0104] In order to prove the effectiveness and superiority of the method of the present application, the Urban hyperspectral data set is subjected to anomaly detection by means of the MATLAB simulation platform, and the method of the present application is compared with a series of classic methods, and the simulation experiment and result analysis are as follows:

[0105] Simulation experiment 1: Figure 3This is a comparison chart of anomaly detection results obtained using the traditional RX method, the local RX method, the traditional collaborative representation method, and the method of this invention on the Urban hyperspectral dataset.

[0106] in Figure 3 (a) is a scene illustration of the Urban hyperspectral dataset; Figure 3 (b) is the anomaly truth map for this scene, where black represents the background and white represents anomalies; Figure 3 (c) is a graph showing the anomaly detection results of the traditional RX method on this dataset; Figure 3 (d) is a graph showing the anomaly detection results of the local RX method on this dataset; Figure 3 (e) is a graph showing the anomaly detection results of the traditional collaborative representation method on this dataset; Figure 3 (f) is a diagram showing the anomaly detection results of the method of the present invention on this dataset;

[0107] for Figure 3 The anomaly detection results shown in (c)-(f) indicate that brighter pixels correspond to stronger anomaly response values, while darker pixels correspond to weaker anomaly response values. Compared to the other three classic methods, the method of this invention can accurately detect the vast majority of anomalous pixels with fewer false alarms and has strong background-anomaly differentiation. In contrast, the RX method and traditional collaborative representation methods suffer from excessively high false alarm rates, while the local RX method struggles to detect a sufficient number of true anomalies.

[0108] Simulation Experiment 2: Figure 4 This is a comparison chart of AUC values ​​obtained using the traditional RX method, the local RX method, the traditional collaborative representation method, and the method of this invention on the Urban hyperspectral dataset.

[0109] Figure 4 As can be seen, the AUC value obtained by the method of the present invention is the highest, close to 1, which quantitatively proves the superiority of the method of the present invention compared with the other three classical methods.

[0110] Simulation results show that the method of the present invention can effectively improve the traditional collaborative representation method and can achieve accurate detection of abnormal targets in hyperspectral images with strong background-anomaly discrimination.

[0111] Those skilled in the art will understand that the above embodiments are specific examples of implementing the present invention, and in practical applications, various changes in form and detail can be made without departing from the spirit and scope of the present invention. The steps or processes not described in detail in this embodiment are common and well-known methods in the industry, and will not be described here.

Claims

1. A method for hyperspectral image anomaly detection based on coefficient analysis collaborative representation, characterized in that: The method comprises the following steps: Step 1: obtaining a hyperspectral image and performing normalization processing; Step 2: using a sliding double-window strategy, performing a collaborative representation algorithm on a center test pixel by using a pixel set of a local region to calculate a representation coefficient vector; Step 3: estimating maximum and minimum background coefficient thresholds, eliminating abnormal points, and updating the pixel set of the local region; Step 4: performing the collaborative representation algorithm on the center test pixel again by using the updated pixel set of the local region to calculate the representation coefficient vector and obtain a representation error of the center test pixel; Step 5: performing normalization processing on the representation error of the global pixel to obtain an abnormality detection result; The step 2 is specifically as follows: Step (21): Set inner window size a x a and outer window size b x b to test pixel x j,k For the center, construct a concentric rectangular dual window, where j = 1, 2, …, H, k = 1, 2, …, W, H and W represent the spatial length and width dimensions of the hyperspectral image X, respectively; Step (22): Using the pixel set S of the local region surrounded by the concentric rectangular double window j,k For the center test pixel x j,k Performing the collaborative representation algorithm, compute the representation coefficient vector a j,k The objective function of this process is: where λ is the Lagrange multiplier, the above equation for α j,k Taking partial derivatives and setting the result to zero, we obtain α j,k is: wherein I is a unit matrix with diagonal elements being 1; The step 5 is specifically as follows: Step (51): move the concentric rectangular double window forward by a distance of one pixel point, and test the pixel x with the common center of the concentric rectangular double window as the center j+1,k+1 Again, analyze according to steps two to four; with the traversal of the concentric rectangular double window, the process is repeatedly, and finally the representation error r∈R of the global pixel is obtained H×W ; Step (52): Calculate the maximum value r of all elements in the global pixel representation error r max and the minimum value r min And perform the following normalization processing formula (8) on the global pixel representation error r: where r j,k is the element in the jth row and kth column of r, j = 1, 2,..., H, k = 1, 2,..., W; Step 53: comparing the representation error r of the global pixel with a preset threshold t, marking the pixels with an error value greater than t as abnormal points, marking the pixels with an error value less than t as background points, and thus obtaining an abnormality detection result of the whole hyperspectral image, wherein 0≤t≤1.

2. The hyperspectral image anomaly detection method based on coefficient analysis collaborative representation according to claim 1, characterized in that, In the step 1, the normalization processing is specifically as follows: Step (11): converting the acquired hyperspectral image X into a two-dimensional hyperspectral matrix X of size S x N 2D where S is the number of bands of X and N is the number of pixels in X; Step (12): Calculate X 2D the maximum value MAX and the minimum value MIN of each row element corresponding to each wave band in the matrix X i i and perform the following normalization processing formula (1) on each row element in X 2D ​​ where X 2D (i, :) denotes the i-th row element of X 2D , i = 1, 2,..., S; Step (13): The normalized two-dimensional hyperspectral matrix X is converted into a three-dimensional hyperspectral matrix X again. 2D Step (13): The normalized two-dimensional hyperspectral matrix X is converted into a three-dimensional hyperspectral matrix X again.

3. The hyperspectral image anomaly detection method based on coefficient analysis collaborative representation according to claim 1, characterized in that, The step 3 is specifically as follows: Step (31): Compute the mean and standard deviation of the vector of the representation coefficients α j,k , and estimate the maximum and minimum background coefficient thresholds according to formula (4): wherein n is a positive integer; Step (32): updating the pixel set S of the local region according to the representation coefficient vector α j,k , removing those pixels whose representation coefficients are greater than MAX j,k , and those pixels whose representation coefficients are less than MIN α , thereby updating the pixel set S of the local region α ​ 4. The hyperspectral image anomaly detection method based on coefficient analysis collaborative representation according to claim 1, characterized in that, The step 4 is specifically as follows: Step (41): Utilizing the updated local region of pixel set Again, the center test pixel x j,k Performing the collaborative representation algorithm, compute the updated representation coefficient vector The objective function of this process is: where λ is the Lagrange multiplier, the above equation is solved for the partial derivative and setting the result to zero, we obtain is Step (42): Calculate the center test pixel x j,k The representation error r j,k is:

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