Hyperspectral abnormal target detection method based on low rank and sparse decomposition

Abstract

The invention provides a hyperspectral abnormal target detection method based on low rank and sparse decomposition, and the method comprises the steps: improving an existing superpixel segmentation method through combining the space and spectrum information of a hyperspectral image when constructing a background dictionary, and obtaining a superpixel region with high space and spectrum similarity, the pollution of an abnormal target to a background dictionary is suppressed by abandoning isolated pixels; for each superpixel region, a sparse representation model is adopted to describe, in this way, representative background features are obtained, and the influence of abnormal features on background dictionary construction is suppressed. According to the hyperspectral abnormal target detection method based on low rank and sparse decomposition, the precision and robustness of target detection in the hyperspectral image are improved, and the hyperspectral abnormal target detection method has great application prospects in the fields of precision agriculture, environment monitoring, military target investigation and the like.

Description

Technical Field

[0001] The present invention belongs to the technical field of hyperspectral image target detection, and specifically relates to a hyperspectral abnormal target detection method based on low rank and sparse decomposition. Background Art

[0002] Hyperspectral remote sensing images have the characteristics of "image-spectrum integration" and "fingerprint recognition" effect, which give them unique advantages in target detection-related applications. For example, in the fields of precision agriculture, environmental monitoring, military target reconnaissance, etc. Depending on whether the target spectral information is known, it can be divided into target matching detection with prior information and abnormal target detection without prior information. Target matching detection is to match the spectrum of the target to be detected with the spectral database to achieve target detection and identification. Due to the limitations of hyperspectral imaging technology, economy and other factors, the spectral database cannot achieve a wide coverage range, and there is a problem of missing spectral information of some targets. Compared with the target matching detection method, abnormal target detection can detect pixels or areas that are very different from the local or global background distribution without any prior information, and realize the positioning of low-probability, small-size targets, which has a wider range of applications and fields.

[0003] In recent years, a large number of hyperspectral anomaly detection methods have been proposed. The most classic anomaly detection algorithm is the RX detection algorithm proposed by Reed and Xiaoli Yu. This method assumes that the background follows a multivariate Gaussian distribution and determines the abnormal target by calculating the Mahalanobis distance between the test pixel and the background distribution. Inspired by the idea of ​​the RX method, algorithms such as local RX detection and kernel RX detection have been proposed. However, the background of hyperspectral images is often complex, and using multivariate Gaussian distribution to estimate its distribution is often inaccurate.

[0004] In order to avoid estimating the distribution of the background, representation-based theory is applied to the field of hyperspectral abnormal target detection. Representation-based methods can be divided into two categories: anomaly detection methods based on sparse representation and low-rank representation. The sparse representation-based method assumes that background pixels can be linearly represented by their domain pixels, while abnormal targets cannot be represented. Among them, one of the representative algorithms is the detector CRD based on collaborative representation. Then, this type of method needs to use a sliding double window for detection. If the size of the double window is not appropriate, the abnormal pixels will be mixed into the representation dictionary of the background, thereby affecting the separation performance of the background and abnormal targets.

[0005] The method based on low-rank representation transforms the hyperspectral anomaly detection problem into a low-rank matrix decomposition problem. Due to the redundant characteristics of the background and the sparse characteristics of the target, the method based on the low-rank representation (LRR) model imposes different constraints on the background and the target, and uses convex optimization theory to solve the constrained optimization problem to decompose the original image into two parts: the background and the target. However, there is often sparse strip noise in the image, which is usually misdetected as a target, that is, there is a problem that image noise reduces the accuracy of target detection. In addition, the method based on the low-rank representation model needs to construct a background dictionary and describe the background component in a linear representation. There is a problem that the background dictionary is contaminated by abnormal targets, that is, there are abnormal targets used to describe the background features, which affects the separation of the background and the target and reduces the detection performance of the algorithm.

[0006] Therefore, how to construct a robust background dictionary and suppress the influence of noise on detection, and provide a hyperspectral detection method that effectively improves the anomaly detection performance of hyperspectral images are technical problems that technicians in this field need to solve urgently. Summary of the invention

[0007] The purpose of the present invention is to provide a hyperspectral abnormal target detection method based on low rank and sparse decomposition to address the problems in the prior art.

[0008] To this end, the above-mentioned purpose of the present invention is achieved through the following technical solutions:

[0009] A method for detecting abnormal hyperspectral targets based on low-rank and sparse decomposition includes the following steps:

[0010] S1, build background dictionary,

[0011] S1.1, perform superpixel segmentation on the hyperspectral image to be detected, obtain superpixel segmentation areas, and discard isolated pixels that cannot be clustered;

[0012] S1.2, using sparse representation model to describe superpixel regions;

[0013] S1.3, calculating the sum of the column vectors in the background coefficient matrix, and arranging them in descending order, selecting the atoms corresponding to the first p% of the results, and using them to construct the background dictionary of the entire image;

[0014] S2, decomposes the hyperspectral image,

[0015] S2.1, the hyperspectral image to be detected is represented as three parts: background, target, and noise. By constraining the coefficient matrix of the background to be low-rank, the target to be sparse, and the noise to be sparse and zero-valued, the original image decomposition problem is converted into a constrained optimization problem;

[0016] S2.2, introduce two auxiliary variables, construct the augmented Lagrangian function, and transform the constrained optimization problem into an unconstrained optimization problem;

[0017] S2.3, using the alternating direction multiplier optimization method to perform convex optimization on the problem in step S2.2, when the optimization problem converges, three components of background, target and noise are obtained;

[0018] S3, locate the abnormal target, set a threshold, compare the target component obtained in step S2 with the threshold, and determine whether the pixel in the target component is the target. If it is greater than the threshold, it is the target.

[0019] While adopting the above technical solutions, the present invention may also adopt or combine the following technical solutions: As the preferred technical solution of the present invention: In step S1.1, the hyperspectral image to be detected is subjected to superpixel segmentation, and the superpixel region with high spatial and spectral similarity is obtained by the superpixel segmentation method for the input hyperspectral image to be detected, and isolated pixels that cannot be clustered are discarded, including the following steps: The input hyperspectral image to be detected is represented as Where w is the image width, h is the image height, and b is the number of image bands

[0020] S1.1.1, determine the initial segmentation step size S, the image can be divided into K grids;

[0021] S1.1.2, the central pixel of each grid is determined as the initial cluster center c i , i = 1, 2, ..., K;

[0022] S1.1.3, for each cluster center, the search range is 2S×2S, and the pixel d in this range is calculated. i With cluster center d c The distance is composed of spectral distance and spatial distance, which is expressed as

[0023] D i，c =αD spectral +(1-α)D spatial

[0024] Among them, α is the weight coefficient for balancing the spectral and spatial terms, D spectral represents the spectral angle mapping metric, D spatial represents the normalized spatial Euclidean clustering. The spectral distance and spatial distance can be specifically expressed as

[0025]

[0026]

[0027] Where (dx, dy) is the pixel d iThe spatial coordinates of , (cx, cy) are the cluster centers d c The spatial coordinates of , l is the diagonal length of the search neighborhood;

[0028] After completing this search and clustering, the original cluster center is updated to the average vector of all pixels belonging to the cluster;

[0029] S1.1.4, when the number of iterations of step S1.1.3 reaches the set threshold iter, the final superpixel segmentation result is obtained, and pixels that are not successfully clustered are discarded.

[0030] As a preferred technical solution of the present invention: in step S1.2.1, a sparse representation model is used to describe the superpixel segmentation area, for each superpixel area, background atoms are selected according to the spatial Euclidean distance, and the superpixel area is described using the sparse representation model, including the following steps:

[0031] S1.2.2 For superpixels Randomly select a pixel as the initial atom, and then use the spatial Euclidean distance to select M atoms to construct an overcomplete dictionary Where N is the number of pixels in the superpixel region, M is the number of pixels in the overcomplete dictionary,

[0032] S1.2.3 uses a sparse representation model to represent the superpixel area, that is, Y = Bα + R, where is the coefficient matrix of the sparse representation, is the error term.

[0033] As a preferred technical solution of the present invention: Step S2.1 includes the following steps:

[0034] S2.1.1, treat the hyperspectral image to be detected as a two-dimensional matrix Where n represents the number of pixels, and the hyperspectral image to be detected is decomposed into three parts: background, target, and noise, which can be expressed as

[0035] X=DZ+A+E

[0036] Where DZ is the background component, Represented by atom d i , 1≤i≤m, the background dictionary, represents the coefficient matrix corresponding to the background dictionary, is the abnormal quantity, is the noise component;

[0037] S2.1.2, respectively, using the coefficient matrix of the nuclear norm constraint background, L1 norm constraint target, L 2,1 The optimization problem of constrained noise,image decomposition is defined as

[0038]

[0039] stX=DZ+A+E

[0040] Among them, λ>0 and β>0 are the balance coefficients,

[0041] By introducing two auxiliary variables J and L to represent the variables Z and A in the objective function, the optimization problem of image decomposition can be transformed into:

[0042]

[0043] stX=DZ+A+Z,Z=J,A=L.

[0044] As a preferred technical solution of the present invention: Step S2.2 includes the following steps: constructing an augmented Lagrangian function to convert the constrained optimization problem into an unconstrained optimization problem, that is,

[0045]

[0046] Among them, Y1, Y2, Y3 are Lagrange multipliers, and μ>0 is the penalty parameter.

[0047] As a preferred technical solution of the present invention: in step S2.3, an alternating direction multiplier is used to perform convex optimization on the unconstrained optimization problem.

[0048] S2.3.1, fix variables Z, A, L and E, update variable J

[0049]

[0050] S2.3.2, fix variables Z, J, A and L, update variable E

[0051]

[0052] S2.3.3, fix variables Z, J, A and E, update variable L

[0053]

[0054] S2.3.4, fix variables J, A, E and L, update variable Z

[0055]

[0056] S2.3.5, fix variables Z, J, L and E, update variable A

[0057]

[0058] S2.3.6, Update Lagrange multiplier Y1

[0059] Y1=Y1+μ(X-DZ-AE)S2.3.7, update the Lagrange multiplier Y2

[0060] Y2=Y2+μ(ZJ)

[0061] S2.3.8, Update Lagrange multiplier Y3

[0062] S2.3.9, set the error threshold ε, calculate the decomposition error, and determine whether the following conditions are met

[0063] ||X-DZ-AE|| ∞ <ε,‖ZJ‖ ∞ <ε,,and ‖AL‖ ∞ <ε

[0064] If it is satisfied, stop the iteration and obtain the abnormal component A. If it is not satisfied, continue to repeat steps S2.3.1 to S2.3.9.

[0065] As a preferred technical solution of the present invention: Step S3 includes the following steps: By calculating the sum of each column in the abnormal component, the abnormal response of the pixel T = [t1, t2, ..., t n ].

[0066] Set the threshold τ to locate abnormal targets, expressed as

[0067]

[0068] Compared with the prior art, the method for detecting abnormal hyperspectral targets based on low-rank and sparse decomposition in the present invention combines the spatial and spectral information of hyperspectral images when constructing a background dictionary, designs a criterion suitable for measuring the similarity of hyperspectral pixels to improve the existing superpixel segmentation method, realizes superpixel segmentation of hyperspectral images, and suppresses the pollution of the background dictionary by abnormal targets by discarding isolated pixels; for each superpixel area, a sparse representation model is used to describe it, and representative background features are obtained in this way, and the influence of abnormal features on the construction of the background dictionary is suppressed. The present invention adopts superpixel segmentation and sparse representation model to effectively suppress abnormal targets, solves the problem of background dictionary being polluted by abnormal targets, and thus constructs a robust background dictionary; a hyperspectral abnormal target detection method based on low-rank and sparse decomposition of the present invention, when the hyperspectral image to be detected is decomposed, since the background has low-rank characteristics, the target has sparse characteristics, and the noise has low-rank and zero-value characteristics, the nuclear norm constrained background coefficient matrix, L1 norm constrained target, and L2,1 constrained noise are respectively used to simultaneously consider the low-rank and zero-value characteristics of the noise to help separate the noise component from the data, realize the separation of the target and the noise, and solve the problem of noise reducing the detection accuracy, thereby improving the accuracy and robustness of target detection in hyperspectral images, making it have great application prospects in the fields of precision agriculture, environmental monitoring, military target reconnaissance, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Figure 1 It is a flow chart of a method for detecting abnormal hyperspectral targets based on low-rank and sparse decomposition in the present invention;

[0070] Figure 2 : is the algorithm detection result diagram, where (a) is the input hyperspectral image, from top to bottom are TexasCoast, Los Angeles, San Diego data sets, (b) is the reference ground truth, (c) is the result of the GRX algorithm, (d) is the result of the LRX algorithm, (e) is the result of the CRD algorithm, (f) is the result of the LRASR algorithm, (g) is the result of the ADLR algorithm, and (h) is the result of the method of the present invention;

[0071] Figure 3 is the ROC curve of the algorithm, where (a) is the result of the Texas Coast dataset, (b) is the result of the Los Angeles dataset, and (c) is the result of the AVIRIS dataset;

[0072] Figure 4 is the AUC value of the algorithm. DETAILED DESCRIPTION

[0073] The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments.

[0074] Example 1

[0075] like Figure 1 As shown, a method for detecting abnormal hyperspectral targets based on low-rank and sparse decomposition in the present invention comprises the following steps:

[0076] Step S1. Build background dictionary:

[0077] The input hyperspectral image is represented as Where w is the image width, h is the image height, and b is the number of image bands.

[0078] S1.1, image superpixel segmentation:

[0079] S1.1.1, determine the initial segmentation step S, the image can be divided into K grids

[0080] S1.1.2, the central pixel of each grid is determined as the initial cluster center c i (i=1, 2, ..., K)

[0081] S1.1.3, for each cluster center, the search range is 2S×2S, and the distance between the pixels in this range and the cluster center is calculated. The distance consists of two parts: spatial distance and spectral distance, expressed as

[0082] D i，c =αD spectral +(1-α)D spatial

[0083] Here, α is the weight coefficient to balance the spectral and spatial terms.

[0084] S1.1.4, after reaching a certain number of iterations iter, the final superpixel segmentation result is obtained, and pixels that fail to be clustered are discarded.

[0085] In step S1.2.1, a sparse representation model is used to describe the superpixel segmentation area.

[0086] a. For superpixels Randomly select a pixel as the initial atom, and then use the spatial Euclidean distance to select M atoms to construct an overcomplete dictionary

[0087] b. Use a sparse representation model to represent the superpixel area, that is, Y=Bα+R.

[0088] in, is the coefficient matrix of the sparse representation, is the error term.

[0089] S1.3, build background dictionary:

[0090] According to the column vector α i The sum of the atoms determines the frequency of use of the atoms and sorts them in descending order. The first 3% of atoms are selected to construct the image background dictionary.

[0091] Step S2. Decompose the original image:

[0092] Step S2.1.1, treat the hyperspectral image to be detected as a two-dimensional matrix Where n represents the number of pixels. The image is decomposed into three parts: background, target, and noise, which can be expressed as

[0093] X=DZ+A+E

[0094] Where DZ is the background component, Represented by atom d i (1≤i≤m, and m<n), represents the coefficient matrix corresponding to the background dictionary, is the abnormal quantity, is the noise component

[0095] In step S2.1.2, since the background has low rank characteristics, the target has sparse characteristics, and the noise has low rank and zero-valued characteristics, the nuclear norm is used to constrain the background coefficient matrix, the L1 norm is used to constrain the target, and the L 2,1 Constrained noise. Therefore, the optimization problem of image decomposition is defined as

[0096]

[0097] stX=DZ+A+E

[0098] Two auxiliary variables J and L are introduced to represent the variables in the objective function. The optimization problem can be transformed into

[0099]

[0100] stX=DZ+A+Z,Z=J,A=L

[0101] In step S2.2, an augmented Lagrangian function is constructed to transform the constrained optimization problem into an unconstrained problem, that is,

[0102]

[0103] In step S2.3, alternating direction multipliers are used to perform convex optimization on the unconstrained optimization problem.

[0104] S2.3.1, fix variables Z, A, L and E, update variable J

[0105]

[0106] S2.3.2, fix variables Z, J, A and L, update variable E

[0107]

[0108] S2.3.3, fix variables Z, J, A and E, update variable L

[0109]

[0110] S2.3.4, fix variables J, A, E and L, update variable Z

[0111]

[0112] S2.3.5, fix variables Z, J, L and E, update variable A

[0113]

[0114] S2.3.6, Update Lagrange multiplier Y1

[0115] Y1=Y1+μ(X-DZ-AE)

[0116] S2.3.7, Update Lagrange multiplier Y2

[0117] Y2=Y2+μ(ZJ)

[0118] S2.3.8, Update Lagrange multiplier Y3

[0119] Y3=Y3+μ(AL)

[0120] S2.3.9, set the error threshold ε, calculate the decomposition error, and determine whether the following conditions are met

[0121] ||X-DZ-AE|| ∞ <ε,||ZJ|| ∞ <ε, and ||AL|| ∞ <ε

[0122] If satisfied, stop the iteration and get the abnormal component A. If not satisfied, continue to repeat steps a to h

[0123] Step S3: locate abnormal targets

[0124] By calculating the sum of each column in the abnormal component, the abnormal response of the pixel T = [t1, t2, ..., t n ].

[0125] Set the threshold τ to locate abnormal targets, expressed as

[0126]

[0127] In order to verify the detection performance of the algorithm of the present invention, the public data sets of Texas Coast, Los Angeles, and San Diego were used for verification, and compared with five detection methods including GRX, LRX, CRD, LRASR, and ADLR. Figure 2 It is the detection result diagram of the algorithm. From the detection results, the algorithm of the present invention can effectively suppress image noise, that is, reduce the influence of noise on the detection results. In addition, the algorithm of the present invention can effectively suppress background information, that is, can effectively separate the background and the target.

[0128] In order to quantitatively evaluate the performance of the detection algorithm, ROC curve and AUC value are used for evaluation. The horizontal axis of ROC is the false alarm rate, and the vertical axis is the detection rate. The AUC value is the area under the ROC curve. Figure 3 is the ROC curve result, Figure 4 is the corresponding AUC value result. From the quantitative results, compared with other detection algorithms, the algorithm of the present invention can obtain better detection effect.

[0129] The above-mentioned specific implementation methods are used to explain the present invention and are only preferred embodiments of the present invention, rather than limiting the present invention. Any modifications, equivalent substitutions, improvements, etc. made to the present invention within the spirit of the present invention and the protection scope of the claims shall fall within the protection scope of the present invention.

Claims

1. A method for detecting abnormal hyperspectral targets based on low-rank and sparse decomposition, comprising the following steps: S1, build background dictionary, S1.1, perform superpixel segmentation on the hyperspectral image to be detected, obtain superpixel segmentation areas, and discard isolated pixels that cannot be clustered; S1.2, using sparse representation model to describe superpixel regions; S1.3, calculating the sum of the column vectors in the background coefficient matrix, and arranging them in descending order, selecting the atoms corresponding to the first p% of the results, and using them to construct the background dictionary of the entire image; S2, decomposes the hyperspectral image, S2.1, the hyperspectral image to be detected is represented as three parts: background, target, and noise. By constraining the coefficient matrix of the background to be low-rank, the target to be sparse, and the noise to be sparse and zero-valued, the original image decomposition problem is converted into a constrained optimization problem; S2.2, introduce two auxiliary variables, construct the augmented Lagrangian function, and transform the constrained optimization problem into an unconstrained optimization problem; S2.3, using the alternating direction multiplier optimization method to perform convex optimization on the problem in step S2.2, when the optimization problem converges, three components of background, target and noise are obtained; S3, locate the abnormal target, set a threshold, compare the target component obtained in step S2 with the threshold, and determine whether the pixel in the target component is the target. If it is greater than the threshold, it is the target.

2. The method for detecting abnormal hyperspectral targets based on low-rank and sparse decomposition according to claim 1, characterized in that: In step S1.1, the hyperspectral image to be detected is segmented into superpixels. The superpixel segmentation method is used to obtain superpixel regions with high spatial and spectral similarities for the input hyperspectral image to be detected, and isolated pixels that cannot be clustered are discarded. The steps include: the input hyperspectral image to be detected is represented as Where w is the image width, h is the image height, and b is the number of image bands S1.1.1, determine the initial segmentation step size S, the image can be divided into K grids; S1.1.2, the central pixel of each grid is determined as the initial cluster center c i , i = 1, 2, ..., K; S1.1.3, for each cluster center, the search range is 2S×2S, and the pixel d in this range is calculated. i With cluster center d c The distance is composed of spectral distance and spatial distance, which is expressed as D i，c =αD spectral +(1-α)D spatial Among them, α is the weight coefficient for balancing the spectral and spatial terms, D spectral represents the spectral angle mapping metric, D spatial represents the normalized spatial Euclidean clustering. The spectral distance and spatial distance can be specifically expressed as Where (dx, dy) is the pixel d i The spatial coordinates of , (cx, cy) are the cluster centers d c The spatial coordinates of , l is the diagonal length of the search neighborhood; After completing this search and clustering, the original cluster center is updated to the average vector of all pixels belonging to the cluster; S1.1.4, when the number of iterations of step S1.1.3 reaches the set threshold iter, the final superpixel segmentation result is obtained, and pixels that fail to be clustered are discarded.

3. The method for detecting abnormal hyperspectral targets based on low-rank and sparse decomposition as claimed in claim 1, characterized in that: In step S1.2.1, a sparse representation model is used to describe the superpixel segmentation region. For each superpixel region, background atoms are selected according to the spatial Euclidean distance, and the superpixel region is described using the sparse representation model, including the following steps: S1.2.2 For superpixels Randomly select a pixel as the initial atom, and then use the spatial Euclidean distance to select M atoms to construct an overcomplete dictionary Where N is the number of pixels in the superpixel region, M is the number of pixels in the overcomplete dictionary, S1.2.3 uses a sparse representation model to represent the superpixel area, that is, Y = Bα + R, in, is the coefficient matrix of the sparse representation, is the error term.

4. The method for detecting abnormal hyperspectral targets based on low-rank and sparse decomposition according to claim 1, characterized in that: Step S2.1 includes the following steps: S2.1.1, treat the hyperspectral image to be detected as a two-dimensional matrix Where n represents the number of pixels, and the hyperspectral image to be detected is decomposed into three parts: background, target, and noise, which can be expressed as X=DZ+A+E Where DZ is the background component, Represented by atom d i , 1≤i≤m, the background dictionary, Represents the coefficient matrix corresponding to the background dictionary, is the abnormal quantity, is the noise component; S2.1.2, respectively, using the coefficient matrix of the nuclear norm constraint background, L1 norm constraint target, L 2，1 The optimization problem of constrained noise,image decomposition is defined as stX=DZ+A+E Among them, λ>0 and β>0 are the balance coefficients, By introducing two auxiliary variables J and L to represent the variables Z and A in the objective function, the optimization problem of image decomposition can be transformed into: stX=DZ+A+Z, Z=J, A=L.

5. The method for detecting abnormal hyperspectral targets based on low-rank and sparse decomposition as claimed in claim 3, characterized in that: Step S2.2 includes the following steps: constructing an augmented Lagrangian function to transform the constrained optimization problem into an unconstrained optimization problem, that is, Among them, Y1, Y2, Y3 are Lagrange multipliers, and μ>0 is the penalty parameter.

6. The method for detecting abnormal hyperspectral targets based on low-rank and sparse decomposition as claimed in claim 4, characterized in that: In step S2.3, the unconstrained optimization problem is convexly optimized using alternating direction multipliers. S2.3.1, fix variables Z, A, L and E, update variable J S2.3.2, fix variables Z, J, A and L, update variable E S2.3.3, fix variables Z, J, A and E, update variable L S2.3.4, fix variables J, A, E and L, update variable Z S2.3.5, fix variables Z, J, L and E, update variable A S2.3.6, Update Lagrange multiplier Y1 Y1=Y1+μ(X-DZ-AE) S2.3.7, Update Lagrange multiplier Y2 Y2=Y2+μ(ZJ) S2.3.8, Update Lagrange multiplier Y3 Y3=Y3+μ(AL) S2.3.9, set the error threshold ε, calculate the decomposition error, and determine whether the following conditions are met ||X-Dz-A-E|| ∞ <ε, ||Z-J|| ∞ <ε, and ||A-L|| ∞ <ε If it is satisfied, stop the iteration and obtain the abnormal component A. If it is not satisfied, continue to repeat steps S2.3.1 to S2.3.

9.

7. The method for detecting abnormal hyperspectral targets based on low-rank and sparse decomposition according to claim 1, characterized in that: Step S3 includes the following steps: By calculating the sum of each column in the abnormal component, the abnormal response of the pixel T = [t1, t2, ..., t n ], Set the threshold τ to locate abnormal targets, expressed as

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