Inplanatable method and system for synthetic aperture radar image filtering

Abstract

The invention discloses an interpretable method and system for synthetic aperture radar image filtering. The method comprises the following steps: acquiring an image of a synthetic aperture radar for initial division; carrying out image filtering based on the preliminarily divided structure; and carrying out posterior interpretation on the filtered image to enhance the understanding of the image. According to the invention, through combination of prior guidance and deep learning, an interpretable SAR image filtering method is constructed; for the problem of difficulty in balancing between traditional denoising and detail reservation, robust denoising is carried out on regions with different properties, and meanwhile, detail reservation is balanced; for the black box problem in the traditional denoising process, the prior-guided mathematical interpretation is integrated into the convolution operation of deep learning, and interpretable convolution filtering is constructed. According to the method, through interpretable denoising of the SAR image, the purpose of improving the robustness and precision of downstream tasks is achieved.

Description

Technical Field

[0001] The present invention relates to the field of image filtering, and in particular to an interpretability method and system for synthetic aperture radar image filtering. Background Art

[0002] Effective noise suppression is crucial for subsequent interpretation tasks of SAR images. Traditional SAR image processing techniques often ignore the coherence of noise, resulting in the loss of important details during the filtering process. With the progress of deep learning technology, significant progress has also been made in image processing techniques. However, existing deep learning methods do not fully utilize the imaging mechanism of SAR, resulting in a lack of specificity and interpretability during the filtering process. In order to balance noise reduction and detail preservation, and solve the "black box" problem in filtering.

[0003] Therefore, in order to improve the robustness of SAR image filtering and the interpretability of the filtering process, new products are needed to solve the contradiction between noise suppression and detail retention, while the process is interpretable.

[0004] Currently, to address the key challenge of achieving a balance between speckle noise mitigation and image quality preservation in SAR image filtering, prior guidance combined with deep learning can be used to perform more robust filtering on uniform and non-uniform regions in SAR images. By performing relevant segmentation filtering on the texture and edge contours of different regions, the problems of texture blurring and over-smoothing can be alleviated, thereby improving the image fidelity.

[0005] On the other hand, from the perspective of interpretability: the global nature of traditional prior guidance combined with the locality of deep learning, this integration effectively overcomes the limitation of traditional filters relying too much on local spatial correlation, and has a certain effect on enhancing image details. The filtering process can be explained through deep learning visualization. Summary of the Invention

[0006] To solve the technical problems in the above background, the present invention proposes a robustness and interpretability optimization method for SAR image filtering, aiming to solve the contradiction between noise suppression and detail retention in existing methods by combining prior guidance and deep learning technology. The method distinguishes and processes the uniform and non-uniform regions of the SAR image through prior guidance, and combines the local feature extraction ability of deep learning to perform relevant segmentation filtering on the texture and edge contours of different regions, thereby effectively alleviating the problems of texture blurring and over-smoothing, and improving the image fidelity. At the same time, through the visualization technology of deep learning, the interpretability of the filtering process is enhanced, the limitation of traditional filters relying too much on local spatial correlation is overcome, the balance between noise reduction and detail retention is achieved, and a higher-quality image basis is provided for subsequent interpretation tasks of SAR images.

[0007] To achieve the above object, the present invention provides an interpretability method for synthetic aperture radar image filtering, the steps comprising:

[0008] Collect synthetic aperture radar images for initial segmentation;

[0009] Based on the preliminary divided structure, image filtering is performed;

[0010] Perform a posteriori interpretation on the filtered image to enhance the understanding of the image.

[0011] Preferably, the step of performing the initial division includes: based on the prior guidance of the traditional filtering algorithm, combining the mean shift algorithm with the Markov random field to construct a mean shift-Markov random field filter in the joint space-spectral domain; the MS-MRF filter divides the image into regions through an upward search strategy of the correlation density peak to achieve denoising and detail retention.

[0012] Preferably, the constructed MS-MRF filter expression is as follows:

[0013]

[0014] Among them, M h,Q,D′ (x) represents the expression of MS-MRF in the joint spatial-spectral domain; ρ represents the linear resolution ratio between the two grids; q s represents the sectional function in the space domain; q r represents the profile function in the spectral domain; x i,s represents the i-th pixel in the spatial domain; x i，γ represents the i-th pixel in the spectral domain; h s and h r are the bandwidths used in the spatial domain and the spectral domain respectively; x represents the input data; i represents the i-th pixel; and n represents the total number of pixels.

[0015] Preferably, the method for performing image filtering includes: constructing an interpretable MS-MRF convolution filter based on combining prior guidance and convolutional neural network, integrating the mathematical interpretability of the MS-MRF filter into the convolution operation in deep learning, and designing a 3×3 convolution kernel IMMC:

[0016]

[0017] Among them, w k The total number of kernels used represents the weight of the convolution kernel; k represents the total number of kernels used internally; MS-MRF(x) represents the result of applying the MS-MRF filtering process to the input data x; K represents the total number of kernels used in the convolution operation.

[0018] Preferably, the method for performing a posteriori interpretation on the filtered image comprises: in a deep learning framework, applying the IMMC convolution kernel to the bottleneck module of the ResNet network, replacing the original convolution kernel, and forming a new IMMB residual block; the formed IMMB residual block realizes a posteriori interpretability by visualizing the activation function of the intermediate layer:

[0019] A(x)=IMMB(x)

[0020] Among them, A(x) represents the output activation map of the IMMB module; x represents the input data.

[0021] The present invention also provides an interpretability system for synthetic aperture radar image filtering, the system is used to implement the above method, including: a division module, a filtering module and an interpretation module;

[0022] The division module is used to collect synthetic aperture radar images for initial division;

[0023] The filtering module is used to perform image filtering based on the preliminary divided structure;

[0024] The interpretation module is used to perform a posteriori interpretation on the filtered image to enhance the understanding of the image.

[0025] Preferably, the workflow of the division module includes: based on the prior guidance of the traditional filtering algorithm, the mean shift algorithm is combined with the Markov random field to construct a mean shift-Markov random field filter in the joint space-spectral domain; the MS-MRF filter divides the image into regions through an upward search strategy of the correlation density peak to achieve denoising and detail retention.

[0026] Preferably, the constructed MS-MRF filter expression is as follows:

[0027]

[0028] Among them, M h,Q,D′ (x) represents the expression of MS-MRF in the joint spatial-spectral domain; ρ represents the linear resolution ratio between the two grids; q s represents the sectional function in the space domain; q r represents the profile function in the spectral domain; x i,s represents the i-th pixel in the spatial domain; x i，γ represents the i-th pixel in the spectral domain; h s and h r are the bandwidths used in the spatial domain and the spectral domain respectively; x represents the input data; i represents the i-th pixel; and n represents the total number of pixels.

[0029] Compared with the prior art, the present invention has the following beneficial effects:

[0030] The present invention constructs an interpretable SAR image filtering method through prior guidance combined with deep learning; it can address the difficulty in balancing traditional denoising and detail preservation, perform robust denoising on different property regions, and balance detail preservation at the same time; for the "black box" problem in the traditional denoising process, the mathematical interpretability of prior guidance is incorporated into the convolutional operation of deep learning to construct an interpretable convolutional filter. Through interpretable denoising of SAR images, the present invention aims to improve the robustness and accuracy of downstream tasks. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] To more clearly illustrate the technical solutions of the present invention, the following briefly introduces the drawings required in the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0032] Figure 1 It is a schematic diagram of the method for an embodiment of the present invention;

[0033] Figure 2 It is a schematic diagram of the formation principle of MS-MRF for an embodiment of the present invention;

[0034] Figure 3 It is a schematic diagram of the formation principle of the interpretable SAR residual filter for an embodiment of the present invention;

[0035] Figure 4 It is a schematic diagram of the overall process framework for an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0036] The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the drawings in the embodiments of the present invention. Obviously, the described embodiments are only some, rather than all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts fall within the scope of protection of the present invention.

[0037] To make the above objects, features, and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the drawings and specific embodiments.

[0038] Embodiment 1

[0039] As Figure 1 shown, it is a schematic diagram of the method flow for this embodiment, and the steps include:

[0040] S1. Collect the images of the synthetic aperture radar for initial division.

[0041] Based on the prior guidance of traditional filtering algorithms, a Mean Shift-Markov Random Field (MS-MRF) filter is constructed. The steps include: combining the Mean Shift algorithm with the Markov Random Fields (MRF) to construct an MS-MRF filter in the joint spatial-spectral domain. This filter divides the Synthetic Aperture Radar (SAR) image into regions through an upward search strategy for the peak of correlation density to achieve denoising and detail preservation. The Mean Shift algorithm is used for spatial clustering of the image, while the MRF is used for probability modeling of pixels. By combining the advantages of both, the MS-MRF filter can better retain edge and texture information while removing speckle noise.

[0042] This method performs well in both homogeneous and non-homogeneous regions of SAR images, effectively solving the problem of poor performance of traditional filtering methods in non-homogeneous regions. The specific steps are as follows:

[0043] Suppose \(x\) i is the scattering vector of a p-dimensional original polarized SAR intensity (or amplitude) image, then a series of d-dimensional vectors \(x\) i (where \(d = p + 2\)) can be constructed in the joint spatial-spectral domain. Each vector corresponds to a pixel, where the two-dimensional spatial information existing in the spatial domain is represented by \(x\) s as the spatial domain part of the vector \(x\), and the spectral information existing in the spectral domain is represented by \(x\) r as the spectral domain part of the vector. Suppose the same SAR image

[0044] is composed of a finer resolution (FR) pixel array and a coarser resolution (CR) pixel array. Let be the pixel unit at FR, be the pixel unit at CR, and CR contains \(\rho\) 2 times the number of FR pixels. \(S / I=\rho\) 2 is a linear relationship of \(I\), where \(S\) and \(P\) represent the linear resolutions between the two grids. Assuming that each pixel in the joint spatial-spectral domain is independently and uniformly distributed, the relationship between the pixel and the unit is as follows: \(x\) i \(\in I\) represents a random vector belonging to the fine resolution \(I\) defined in the joint spatial-spectral domain; \(x\) i and \(x\) s \(\in S\) represent the spatial vector \(x\) s in the spatial-spectral joint domain.

[0045] In the context of image analysis, latent variables usually refer to pixel features that cannot be directly observed from the image but can have a significant impact on the results of the filtering process. Suppose \(v\) i represents a latent variable, where \(v\)i ∈I, representing v i Defined in the spatio-spectral joint domain, belonging to the fine-resolution set I. And assume that v i and x i have independent and identically distributed logarithmic ratios.

[0046] The standard variable x i of the latent variable v i The conditional probability p(ν i |x i ) is defined by the following mathematical expression:

[0047]

[0048] where μ log represents the expected value of log(ν i / x i ); σ represents the variance, is the variance of this logarithmic ratio.

[0049] As Figure 2 shown is the formation schematic diagram of the newly constructed MS-MRF, and the prior guidance and mathematical interpretation of the MS-MRF filter:

[0050]

[0051] where M h,Q,D′ (x) represents the expression of the MS-MRF in the spatio-spectral joint domain; ρ represents the linear resolution ratio between two grids; q s represents the dissection function in the spatial domain; q r represents the dissection function in the spectral domain; x i,s represents the i-th pixel in the spatial domain; x i,r represents the i-th pixel in the spectral domain; h s and h r are the bandwidths used in the spatial domain and spectral domain respectively; x represents the input data; i represents the i-th pixel; n represents the total number of pixels.

[0052] S2. Based on the preliminary partitioned structure, perform image filtering.

[0053] Based on the combination of prior guidance and convolutional neural network, construct an interpretable MS-MRF convolution (IMMC) filter. The steps include: integrating the mathematical interpretation of the MS-MRF filter into the convolutional operation in deep learning, and designing a new 3×3 convolutional kernel IMMC. This convolutional kernel combines the density peak search strategy of the MS-MRF with the convolutional operation, enhancing the denoising ability and detail retention ability of the model. The design of the IMMC convolutional kernel is based on the unique imaging mechanism of SAR images and can better adapt to the denoising requirements of different regions.

[0054] As Figure 3 shown is the schematic diagram of the formation principle of the explanatory SAR residual filter, and the mathematical formula of this process can be simply expressed as:

[0055]

[0056] where w k the total number of kernels used represents the weights of the convolutional kernels; k represents the total number of kernels used internally; MS-MRF(x) represents the result of applying the MS-MRF filtering process to the input data x; K represents the total number of kernels used in the convolution operation

[0057] Replace the convolutional kernels in the standard ResNet50 bottleneck layer with IMMC. The mathematical expression of IMMB residual filtering can be expressed as:

[0058] IMMB(x) = ReLU(IMMC(x) + x)

[0059] where ReLU represents the activation function.

[0060] S3. Perform posterior interpretation on the filtered image to enhance the understanding of the image.

[0061] In the deep learning framework, apply the IMMC convolutional kernel to the bottleneck module of the ResNet network, replace the original convolutional kernel, and form a new IMMB residual block. IMMB inherits the advantages of the ResNet residual block, such as easy optimization and strong portability, and realizes the posterior interpretability of the model through the visualization of the activation function. This module shows the balance mechanism between denoising and detail retention of the model by analyzing the activation maps of the intermediate layers.

[0062] ResNet50(IMMB) realizes posterior interpretability by visualizing the activation function of the intermediate layer. This means that although the model may not have built-in interpretability during the training process, its behavior can be explained by analyzing its internal working mechanism. Posterior interpretability is achieved through visualization techniques, aiming to enhance the interpretability of the model by showing its internal operations. The visualization method combines dimensionality reduction techniques to achieve simple and human-readable visualizations. In addition, visualization can also be combined with other techniques to further enhance understanding.

[0063] Ex post facto interpretability can be achieved by visualizing the activation map of the IMMB module, and it can be mathematically described through the following steps:

[0064] Calculation of the activation map: For each input sample, calculate the output activation map A(x) of the IMMB module:

[0065] A(x) = IMMB(x)

[0066] Furthermore, the feature importance and filtering effect can be analyzed based on the visualization of the features by the activation function: Analyze the activation map to determine which features are considered important by the model during the filtering process; Filtering effect evaluation: Evaluate the effectiveness A(x) of the IMMB filtering block in noise suppression and detail preservation by comparing the activation maps before and after filtering. The overall process of the present invention is as Figure 4 shown.

[0067] Embodiment 2

[0068] This embodiment also provides an interpretability system for synthetic aperture radar image filtering, including: a partitioning module, a filtering module, and an interpretation module; the partitioning module is used to collect the image of the synthetic aperture radar for initial partitioning; the filtering module is used to perform image filtering based on the preliminarily partitioned structure; the interpretation module is used to perform posterior interpretation on the filtered image to enhance the understanding of the image.

[0069] Next, in combination with this embodiment, it will be described in detail how the present invention solves the technical problems in real life.

[0070] First, use the partitioning module to collect the image of the synthetic aperture radar for initial partitioning.

[0071] Based on the prior guidance of the traditional filtering algorithm, construct a Mean Shift-Markov Random Field (MS-MRF) filter. The steps include: combining the Mean Shift algorithm with the Markov Random Fields (MRF) to construct an MS-MRF filter in the joint spatial-spectral domain. This filter performs region partitioning on the image through an upward search strategy for the peak of the correlation density to achieve denoising and detail preservation. The Mean Shift algorithm is used for spatial clustering of the image, while the MRF is used for probability modeling of pixels. By combining the advantages of both, the MS-MRF filter can better retain edge and texture information while removing speckle noise.

[0072] This method performs well in both homogeneous and non-homogeneous regions of SAR images, effectively solving the problem of poor performance of traditional filtering methods in non-homogeneous regions. The specific steps are as follows:

[0073] Assume x i is the scattering vector of the p-dimensional original polarized SAR intensity (or amplitude) image, then a series of d-dimensional vectors x i (where d = p + 2) defined in the joint spatial-spectral domain can be constructed. Each vector corresponds to a pixel, where the two-dimensional spatial information existing in the spatial domain is represented by x s as the spatial domain part of the vector x, and the spectral information existing in the spectral domain is represented by x rThe spectral domain part represented as a vector. Assume the same SAR image

[0074] is composed of a finer resolution (FR) pixel array and a coarser resolution (CR) pixel array. Let be the pixel unit at FR, be the pixel unit at CR, and CR contains ρ 2 times the number of FR pixels, S / I = ρ 2 is a linear relationship of I, where S and P represent the linear resolution between two grids. Assume that each pixel in the spatio-spectral joint domain is independently and uniformly distributed, then the relationship between the pixel and the unit is as follows: x i ∈I represents a random vector belonging to the fine resolution I defined in the joint spatio-spectral domain; x i , x s ∈S represents the spatial vector x s .

[0075] In the context of image analysis, latent variables usually refer to pixel features that cannot be directly observed from the image but can have a significant impact on the results of the filtering process. Assume v i represents a latent variable, where v i ∈I, indicating that v i is defined in the spatio-spectral joint domain and belongs to the fine resolution set I. And assume that v i and the log ratio of x i are independent and identically distributed.

[0076] The conditional probability p(ν i ) of the latent variable v i with respect to the standard variable x i |x i ) is defined by the following mathematical expression:

[0077]

[0078] where μ log represents the expected value of log(ν i / x i ); σ represents the variance, is the variance of this log ratio.

[0079] As Figure 2 shown is the formation schematic diagram of the newly constructed MS-MRF, and the prior guidance and mathematical interpretation of the MS-MRF filter:

[0080]

[0081] where, M h,Q,D′(x) represents the expression of MS-MRF in the joint spatial-spectral domain; ρ represents the linear resolution ratio between the two grids; q s represents the sectional function in the space domain; q r represents the profile function in the spectral domain; x i,s represents the i-th pixel in the spatial domain; x i，r represents the i-th pixel in the spectral domain; h s and h r are the bandwidths used in the spatial domain and the spectral domain respectively; x represents the input data; i represents the i-th pixel; and n represents the total number of pixels.

[0082] Then the filtering module filters the image based on the preliminary divided structure.

[0083] Based on the combination of prior guidance and convolutional neural network, an interpretable MS-MRF convolution (IMMC) filter is constructed. The steps include: integrating the mathematical interpretability of the MS-MRF filter into the convolution operation in deep learning, and designing a new 3×3 convolution kernel IMMC. This convolution kernel enhances the denoising and detail retention capabilities of the model by combining the density peak search strategy of MS-MRF with the convolution operation. The design of the IMMC convolution kernel is based on the unique imaging mechanism of SAR images and can better meet the denoising needs of different regions.

[0084] like Figure 3 The figure shows the principle diagram of the formation of the explanatory SAR residual filter. The mathematical formula of this process can be concisely expressed as:

[0085]

[0086] Among them, w k The total number of kernels used represents the weight of the convolution kernel; k represents the total number of kernels used internally; MS-MRF(x) represents the result of applying the MS-MRF filtering process to the input data x; K represents the total number of kernels used in the convolution operation.

[0087] Replace the convolution kernel in the bottleneck layer of the standard ResNet50 with IMMC. The mathematical expression of IMMB residual filtering can be expressed as:

[0088] IMMB(x)=ReLU(IMMC(x)+x)

[0089] Among them, ReLU represents the activation function.

[0090] Finally, the interpretation module performs a posteriori interpretation on the filtered image to enhance the understanding of the image.

[0091] In the deep learning framework, the IMMC convolution kernel is applied to the bottleneck module of the ResNet network to replace the original convolution kernel, forming a new IMMB residual block. The IMMB inherits the advantages of the ResNet residual block, such as easy optimization and strong portability, and realizes the posterior interpretability of the model through the visualization of activation functions. This module analyzes the activation maps of the intermediate layers to demonstrate the balance mechanism between denoising and detail preservation of the model.

[0092] ResNet50(IMMB) realizes posterior interpretability by visualizing the activation functions of the intermediate layers. This means that although the model may not have built-in interpretability during the training process, its behavior can be explained by analyzing its internal working mechanism. Posterior interpretability is achieved through visualization techniques, aiming to enhance the interpretability of the model by showing its internal operations. The visualization method combines dimensionality reduction techniques to achieve simple and human-readable visualizations. In addition, visualization can be combined with other techniques to further enhance understanding.

[0093] The ex-post interpretability is achieved by visualizing the activation maps of the IMMB module, and can be mathematically described through the following steps:

[0094] Calculation of the activation map: For each input sample, calculate the output activation map A(x) of the IMMB module:

[0095] A(x) = IMMB(x)

[0096] Furthermore, the feature importance and filtering effect can be analyzed based on the visualization of the features by the activation function: Analyze the activation map to determine which features are considered important by the model during the filtering process; Filtering effect evaluation: Evaluate the effectiveness of the IMMB filtering block in noise suppression and detail preservation by comparing the activation maps before and after filtering A(x).

[0097] The embodiments described above are only descriptions of the preferred embodiments of the present invention, and do not limit the scope of the present invention. Without departing from the design spirit of the present invention, various deformations and improvements made by those of ordinary skill in the art to the technical solutions of the present invention shall fall within the protection scope determined by the claims of the present invention.

Claims

1. An interpretable method for synthetic aperture radar image filtering, characterized in that the steps including: Collecting the images of synthetic aperture radar for initial division; Performing image filtering based on the initially divided structure; Performing posterior interpretation on the filtered image to enhance the understanding of the image.

2. The interpretability method for synthetic aperture radar image filtering according to claim 1, characterized in that The steps for performing the initial division include: guided by the prior of traditional filtering algorithms, combining the mean shift algorithm and the Markov random field to construct a mean shift-Markov random field filter in the joint spatial-spectral domain; the MS-MRF filter divides the image through an upward search strategy for the peak of correlation density, which is used to achieve denoising and detail retention.

3. The interpretability method for synthetic aperture radar image filtering according to claim 2, characterized in that The expression of the constructed MS-MRF filter is as follows: Among them, M h,Q,D′ (x) represents the expression of the MS-MRF in the spatial-spectral joint domain; ρ represents the linear resolution ratio between two grids; q s represents the partitioning function in the spatial domain; q r represents the partitioning function in the spectral domain; x i,s represents the i-th pixel in the spatial domain; x i,γ represents the i-th pixel in the spectral domain; h s and h r are the bandwidths used in the spatial domain and the spectral domain respectively; x represents the input data; i represents the i-th pixel; n represents the total number of pixels.

4. The interpretability method for synthetic aperture radar image filtering according to claim 1, characterized in that The method for performing image filtering includes: constructing an interpretable MS-MRF convolutional filter based on the combination of prior guidance and convolutional neural network, integrating the mathematical interpretability of the MS-MRF filter into the convolutional operation in deep learning, and designing a 3×3 convolutional kernel IMMC: where, w k The total number of kernels used represents the weights of the convolutional kernels; MS-MRF(x) represents the result of applying the MS-MRF filtering process to the input data x; K represents the total number of kernels used in the convolution operation.

5. The interpretability method for synthetic aperture radar image filtering according to claim 1, characterized in that The method for performing posterior interpretation on the filtered image includes: in the deep learning framework, applying the IMMC convolutional kernel to the bottleneck module of the ResNet network to replace the original convolutional kernel, forming a new IMMB residual block; the formed IMMB residual block realizes posterior interpretability by visualizing the activation function of the intermediate layer: A(x) = IMMB(x) where A(x) represents the output activation map of the IMMB module; x represents the input data.

6. An interpretable system for synthetic aperture radar image filtering, the system being used to implement the method according to any one of claims 1-5, characterized in that, including: a division module, a filtering module, and an interpretation module; the division module is used to collect the images of synthetic aperture radar for initial division; the filtering module is used to perform image filtering based on the initially divided structure; the interpretation module is used to perform posterior interpretation on the filtered image to enhance the understanding of the image.

7. The interpretability system for synthetic aperture radar image filtering according to claim 6, characterized in that, The working process of the division module includes: guided by the prior of traditional filtering algorithms, combining the mean shift algorithm and the Markov random field to construct a mean shift-Markov random field filter in the joint spatial-spectral domain; the MS-MRF filter divides the image through an upward search strategy for the peak of correlation density, which is used to achieve denoising and detail retention.

8. The interpretability system for synthetic aperture radar image filtering according to claim 7, wherein, The expression of the constructed MS-MRF filter is as follows: Among them, M h,Q,D′ (x) represents the expression of the MS-MRF in the spatial-spectral joint domain; ρ represents the linear resolution ratio between two grids; q s represents the segmentation function in the spatial domain; q r represents the segmentation function in the spectral domain; x i,s represents the i-th pixel in the spatial domain; x i，γ represents the i-th pixel in the spectral domain; h s and h r are the bandwidths used in the spatial domain and the spectral domain respectively; x represents the input data; i represents the i-th pixel; n represents the total number of pixels.

Images