A motion blur removal method for infrared images based on generative adversarial networks

Abstract

The present invention discloses a method for deblurring infrared images based on a generative adversarial network, belonging to the field of computer vision. The method comprises the following steps: S1, constructing an infrared image dataset, simulating motion-blurred infrared images, and using blurred and clear infrared image pairs as training and test data; S2, constructing an infrared image deblurring model based on a generative adversarial network, inputting the training data into the model for training, and testing it with test data to obtain a trained generator network; the generator network adopts a generative network based on longitudinal channels and dynamic convolution; S3, inputting the infrared image to be deblurred into the trained generator network to obtain a deblurred infrared image. The method can retain more detailed information in the image, avoid detail loss, and enhance the robustness of image processing under different blur scenarios.

Description

Technical Field

[0001] The present invention relates to the field of computer vision, and in particular to a method for de-motion blurring of infrared images based on a generative adversarial network. Background Art

[0002] Infrared imaging systems form images by detecting the varying intensities of thermal radiation emitted by different objects. This imaging process is minimally affected by environmental factors and exhibits strong anti-interference capabilities, making it widely applicable. However, in everyday life, infrared imaging systems often capture moving objects. This causes relative movement between the infrared imaging system and the object being imaged during the integration time, resulting in motion blur in the infrared image. Therefore, post-processing of blurred infrared images to remove motion blur caused by object movement is crucial.

[0003] Traditional image deblurring methods mainly estimate blur based on prior knowledge of image statistics. With the development of deep learning, it has also begun to be applied in the field of image deblurring. Early deep learning methods continued the ideas of traditional image deblurring methods and mainly focused on estimating the blur kernel. With the introduction of end-to-end network structures, image deblurring work has gradually broken away from the limitation of estimating the blur kernel. However, existing methods often ignore the multi-level feature extraction of blurred images, resulting in the loss of image detail information. In addition, most of the proposed methods are trained on specific datasets, and the trained network structure has fixed network parameters. When the input is blurred images with different blur levels, it cannot achieve good deblurring results. Therefore, it is necessary to develop a new deblurring method to enhance the network's deblurring effect. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to propose an infrared image deblurring method based on a generative adversarial network, which can retain more detail information in the infrared image, thereby removing the motion blur caused by the movement of objects in the infrared image.

[0005] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0006] A method for de-motion blurring of infrared images based on a generative adversarial network comprises the following steps:

[0007] S1. Construct an infrared image dataset, simulate and generate motion blurred images of infrared images, and use infrared image blurred-clear image pairs as training data and test data;

[0008] S2. Construct an infrared image motion blur removal model based on a generative adversarial network, input the training data into the model for training, and test it with the test data to obtain a trained generator network; the generator network adopts a generation network based on longitudinal channels and dynamic convolution, including an encoder network based on longitudinal channels, a residual feature extraction network based on dynamic convolution, and a decoder network based on biquadratic linear difference plus convolution;

[0009] S3. Input the infrared image to be deblurred into the trained generator network to obtain a deblurred infrared image.

[0010] A further improvement of the technical solution of the present invention is that: in S2, the encoder network based on the longitudinal channel includes a forward channel and a longitudinal channel with the same input; the forward channel includes a first convolution block, a second convolution block, a third convolution block and a fourth convolution block; the longitudinal channel includes a first longitudinal channel, a second longitudinal channel and a third longitudinal channel; the output of the first convolution block and the second convolution block after sequential connection is spliced with the output of the first longitudinal channel as the input of the third convolution block, the output of the third convolution block and the output of the second longitudinal channel are spliced as the input of the fourth convolution block, and the output of the fourth convolution block and the output of the third longitudinal channel are spliced as the output of the encoder network.

[0011] A further improvement of the technical solution of the present invention is that: in S2, the residual feature extraction network based on dynamic convolution includes 6 identical residual modules; the residual module includes a first dynamic convolution module, a first instance normalization layer, an activation function layer, a second dynamic convolution module and a second instance normalization layer connected in sequence; the first dynamic convolution module and the second dynamic convolution module both include a global pooling layer, a convolution layer, a sub-activation function layer, a fully connected layer and a softmax layer connected in sequence; the 4 different parameter values π1~π4 output by the softmax layer are multiplied with the weight matrices W1~W4 of corresponding sizes and then added to generate a convolution kernel.

[0012] A further improvement of the technical solution of the present invention is that a Dropout layer is added after the activation function layer of the residual module.

[0013] A further improvement of the technical solution of the present invention is that: in S2, the decoder network based on biquadratic linear difference plus convolution includes three biquadratic linear interpolation plus convolution modules, a convolution layer and a Tanh activation function layer connected in sequence; the biquadratic linear interpolation plus convolution module includes a biquadratic linear interpolation layer, a convolution layer and an activation function layer connected in sequence.

[0014] A further improvement of the technical solution of the present invention is that in S2, the activation function applied by the generator is the IR-ReLU function:

[0015]

[0016] The derivative of the activation function applied by the network during backpropagation is:

[0017]

[0018] A further improvement of the technical solution of the present invention is that: S2 specifically includes the following steps:

[0019] S2.1 build the generator network;

[0020] S2.2 build the discriminator network;

[0021] S2.3 train the generator network and the discriminator network and optimize them through the loss function;

[0022] S2.4 Test the trained generator network using test data.

[0023] A further improvement of the technical solution of the present invention is that in S2.2, the discriminator network adopts a PatchGAN structure.

[0024] A further improvement of the technical solution of the present invention is that in S2.3, the loss function includes adversarial loss and content loss, and the loss function is:

[0025] L=L GAN +λ1L x

[0026] Where, L GAN Denotes the adversarial loss, L x represents the content loss, λ1 represents the regularization coefficient of the content loss;

[0027] The expression of adversarial loss is:

[0028]

[0029] Content loss is expressed as a new structural coefficient l str Multiplying with the L2 loss, the expression of the function is:

[0030] L x =l str L2

[0031] In the formula, the expression of L2 loss is:

[0032]

[0033] Where, I S x,y Represents the pixel value of the clear image, Represents the pixel value of the restored image.

[0034] Structural coefficient l str The expression is:

[0035]

[0036] Where, represents the variance of the restored image, represents the variance of the clear image, Represents the covariance between the restored image and the clear image.

[0037] Due to the adoption of the above technical solution, the technical advancements achieved by the present invention are:

[0038] 1. The present invention optimizes the generator of the generative adversarial network and adds a vertical channel between the original image and each convolution module during the feature extraction process of the encoder, thereby retaining more detail information in the blurred image and effectively solving the problem of detail information loss caused by the difficulty in feature extraction of blurred infrared images.

[0039] 2. The present invention replaces the convolution layer in the residual network with a dynamic convolutional network, so that the network can dynamically generate the convolution kernel parameters required by the residual network according to different blurred images, thereby enhancing the robustness of the network in processing motion blur caused by different moving objects to infrared images.

[0040] 3. The present invention proposes a new IR-ReLU activation function, which is applied to the activation function layer in the generator network. The zero output value of the negative half-axis of the ReLU activation function is replaced by an inverse proportional function, which effectively solves the problem of neuronal necrosis in the negative half-axis. The activation function is linear at the zero point and the output mean is approximately zero, which accelerates the convergence speed of the network in processing infrared images.

[0041] 4. This paper proposes a new structure coefficient l for the overall similarity of brightness and structure between the restored and clear infrared images. str , multiplied by the pixel-level MSE loss function in the content loss in the loss function, so that the loss function can better reflect the structural similarity between the restored image and the clear image, and improve the visual effect of the restored infrared image. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 This is the overall architecture diagram of the infrared image motion blur removal based on the generative adversarial network of the present invention;

[0043] Figure 2 It is a diagram of the generator network structure in the present invention;

[0044] Figure 3 1 is a diagram of the encoder network structure based on the longitudinal channel in the present invention;

[0045] Figure 4 It is a structural diagram of the dynamic convolution module in the present invention;

[0046] Figure 5(a) and Figure 5(b) are images of the IR-ReLU activation function and the derivative of the activation function in the present invention;

[0047] Figures 6(a), 6(b), 6(c), 6(d), 6(e), and 6(f) are comparison diagrams of a blurred image, a deblurred image using the DeblurGAN method, a deblurred image using the DeblurGAN-v2 method, a deblurred image using the DeepDeblur method, a deblurred image using the image deblurring method of the present invention, and a clear image;

[0048] FIG7( a ) and FIG7 ( b ) are comparison diagrams of the effects of the present invention before and after deblurring in a real blurred infrared image. DETAILED DESCRIPTION

[0049] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments:

[0050] like Figure 1 As shown, a method for de-motion blurring of infrared images based on a generative adversarial network includes the following steps:

[0051] S1. Construct an infrared image dataset, simulate and generate motion blurred images of infrared images, and use infrared image blurred-clear image pairs as training data and test data;

[0052] The infrared image dataset FLIR is selected, and the average frame synthesis method is used to simulate the motion blurred image caused by the movement of the object. The blurred and clear image pairs of infrared images are obtained and stored in the FLIR1 dataset. The FLIR1 dataset includes a training set for storing training data and a test set for storing test data.

[0053] S2. Build an infrared image motion blur removal model based on a generative adversarial network, input the training data into the model for training, and obtain a trained generator network;

[0054] The infrared image motion blur removal model based on the generative adversarial network includes a generator network and a discriminator network.

[0055] S2.1 build the generator network;

[0056] like Figure 2 As shown in the figure, the generator network adopts a generation network based on vertical channels and dynamic convolution, including an encoder network based on vertical channels, a residual feature extraction network based on dynamic convolution, and a decoder network based on bi-quadratic linear difference plus convolution.

[0057] The encoder network structure based on the longitudinal channel is as follows Figure 3 As shown, it includes a forward channel and a longitudinal channel with the same input; the forward channel includes a first convolution block, a second convolution block, a third convolution block and a fourth convolution block; the longitudinal channel includes a first longitudinal channel, a second longitudinal channel and a third longitudinal channel; the output of the first convolution block and the second convolution block after sequential connection is spliced with the output of the first longitudinal channel as the input of the third convolution block, the output of the third convolution block and the output of the second longitudinal channel are spliced as the input of the fourth convolution block, and the output of the fourth convolution block and the output of the third longitudinal channel are spliced as the output of the encoder network.

[0058] The input image x0 is processed by the first convolution block and the second convolution block to obtain the feature sequence x 10 , x0 is processed by the first longitudinal channel to obtain the characteristic sequence x 11 , change x 10 with x 11 The first feature sequence x1 is obtained by splicing; the feature sequence x1 is processed by the third convolution block to obtain the feature sequence x 20 , x0 is processed by the second longitudinal channel to obtain the feature sequence x 21 , x 20 and x 21 The second feature sequence x2 is obtained by splicing; the feature sequence x2 is processed by the fourth convolution block to obtain the feature sequence x 30 , x0 is processed through the third longitudinal channel to obtain the characteristic sequence x 31 , change x 30 and x 31 The third feature sequence x3 is obtained by splicing.

[0059] The convolutional layer in the first convolutional block has a kernel size of 7×7 and a stride of 1. The convolutional layers in the second, third, and fourth convolutional blocks have the same settings: a kernel size of 3×3 and a stride of 1. The number of channels in the four convolutional layers is 64, 64, 128, and 256, respectively. The convolutional layers in the longitudinal channels all have a kernel size of 3×3 and a stride of 2, with the number of channels being 64, 128, and 256, respectively. The average pooling layer sizes in the second and third longitudinal channels are set to 2 and 4, respectively. After processing by the longitudinal channel-based encoder network, the network generates a feature sequence of size 512×32×32.

[0060] The residual feature extraction network based on dynamic convolution includes six identical residual modules; the residual modules include a first dynamic convolution module, a first instance normalization layer, an activation function layer, a second dynamic convolution module, and a second instance normalization layer, which are connected in sequence. The purpose of the first and second dynamic convolution modules is to generate dynamic convolution kernels for convolution, and both include a global pooling layer, a convolution layer, a sub-activation function layer, a fully connected layer, and a softmax layer, which are connected in sequence. The four different parameter values π1 to π4 output by the softmax layer are multiplied by weight matrices W1 to W4 of corresponding sizes and then added to generate the convolution kernel. The activation function layer is the same as the sub-activation function layer.

[0061] The decoder network based on biquadratic linear difference plus convolution includes three biquadratic linear interpolation plus convolution modules connected in sequence, a convolution layer and a Tanh activation function layer; the biquadratic linear interpolation plus convolution module includes a biquadratic linear interpolation layer, a convolution layer and an activation function layer connected in sequence.

[0062] In addition, in order to speed up the training of the network and prevent overfitting, a Dropout layer is added after the activation function layer of each residual block to randomly reduce half of the training parameters; and a horizontal channel is added to the feature sequences with the same dimension of the encoder and decoder.

[0063] Unless otherwise specified, the activation function of the generator network uses the IR-ReLU activation function.

[0064] The IR-ReLU activation function and its derivative used in the generator network are shown in Figure 5. The positive axis of the function outputs y = x, the negative axis of the function is an inverse proportional function, the horizontal asymptote is y = -1, and it is approximately linear near zero. Therefore, the activation function is:

[0065]

[0066] The derivative of the activation function applied by the network during backpropagation is:

[0067]

[0068] As shown in Figure 5(a), the output function value of the negative half-axis of the activation function is not zero, which can effectively solve the problem of neuronal necrosis in the negative half-axis. Compared with the ReLU function, the curve of the activation function is smoother, and has the advantage of an output mean of 0, which can effectively avoid the occurrence of output offset. As shown in Figure 5(b), the activation function retains the advantage of the positive half-axis derivative of the ReLU activation function being 1, which can effectively solve the problem of gradient disappearance. The derivative of the activation function is not zero in the negative half-axis area, and can also effectively activate the input value, making the gradient update more accurate. When the input value is zero, the derivative of the activation function is continuous, that is, the activation function is a linear relationship here, which can make the network converge faster.

[0069] S2.2 build the discriminator network;

[0070] The discriminator network uses the PatchGAN architecture, whose primary task is to distinguish between generated and legible images. The network divides the input image into N×N blocks, generating an N×N matrix. The mean of the matrix is used as the output value, which is then used to determine whether the input image is real.

[0071] S2.3 train the generator network and the discriminator network and optimize them through the loss function;

[0072] During the training of the generator and discriminator networks and the optimization of the loss function, a 300-cycle training is performed on all images in the training set. During each training cycle, the discriminator parameters are updated once for each image trained, and the generator parameters are updated once for every five images trained. This cycle repeats until the training is complete.

[0073] The loss function includes adversarial loss and content loss, and the expression of the loss function is:

[0074] L=L GAN +λ1L x

[0075] Where, L GAN Denotes the adversarial loss, L x represents the content loss, and λ1 represents the regularization coefficient of the content loss.

[0076] The expression of adversarial loss is:

[0077]

[0078] Content loss is expressed as a new structural coefficient l str Multiplying with the L2 loss, the expression of the function is:

[0079] L x =l str L2

[0080] In the formula, the expression of L2 loss is:

[0081]

[0082] Where, I S x,y Represents the pixel value of the clear image, Represents the pixel value of the restored image.

[0083] Structural coefficient l str The expression is:

[0084]

[0085] Where, represents the variance of the restored image, represents the variance of the clear image, Represents the covariance between the restored image and the clear image.

[0086] In order to illustrate the effectiveness of the structural coefficient, the present invention introduces the correlation coefficient of two infrared images It can represent the correlation between two images. According to probability theory, the more similar the structures of two images are, the greater the correlation coefficient is. The relationship between the correlation coefficient and the covariance function is:

[0087]

[0088] but:

[0089]

[0090] because Therefore And when Closer to When, function Get a smaller value, that is, when the contrast of the restored image is closer to the contrast of the real image, the function Get a smaller value. In addition, when the structures between the two images are closer, Get a larger value.

[0091] In summary, minimizing l str This allows the network to obtain a restored image that is closer to the contrast and structure of the clear image.

[0092] S2.4 uses the test data to test the trained generator network, and generates a trained generator network after the test passes.

[0093] S3. Input the infrared image to be deblurred into the trained generator network to obtain a deblurred infrared image.

[0094] Example

[0095] A method for de-motion blurring of infrared images based on a generative adversarial network comprises the following steps:

[0096] S1. Construct an infrared image dataset and use infrared image blur and clear image pairs as training data;

[0097] The FLIR infrared image dataset, which contains 14,452 infrared images, was used as the dataset for this example. 8,860 of these images were selected as the dataset for this example. Using the average frame synthesis method to simulate motion blur caused by object motion, blurry and sharp infrared image pairs were obtained for training and testing. These blurry and sharp infrared image pairs are referred to as dataset FLIR1. 7,090 of these image pairs were selected as the training set, and 1,770 as the testing set.

[0098] S2. Build an infrared image motion blur removal model based on a generative adversarial network, input the training data into the model for training, and test it with the test data to obtain a trained generator network;

[0099] The infrared image motion blur removal model based on the generative adversarial network includes a generator network and a discriminator network.

[0100] S2.1 build the generator network;

[0101] The generator network adopts a generation network based on vertical channels and dynamic convolution, including an encoder network based on vertical channels, a residual feature extraction network based on dynamic convolution, and a decoder network based on bi-quadratic linear difference plus convolution.

[0102] The convolutional blocks in the longitudinal channel-based encoder network include convolutional layers, instance normalization layers, and activation function layers. The first longitudinal channel includes convolutional layers, instance normalization layers, and activation function layers; the second and third longitudinal channels include convolutional layers, average pooling layers, instance normalization layers, and activation function layers. The convolution kernel size of the convolutional layer in the first convolutional block is set to 7×7, and the convolution step size is set to 1. The convolution layers in the second, third, and fourth convolutional blocks have the same settings: the convolution kernel size is set to 3×3, the convolution step size is set to 1, and the number of channels in the four convolutional layers is 64, 64, 128, and 256, respectively. The convolution kernel size of the convolutional layers in the longitudinal channel is all 3×3, the convolution step size is all 2, and the number of channels is 64, 128, and 256, respectively. The average pooling layer sizes in the second and third longitudinal channels are set to 2 and 4, respectively. After processing by the longitudinal channel-based encoder network, the network can obtain a feature sequence of size 512×32×32.

[0103] The residual feature extraction network based on dynamic convolution includes six identical residual modules; the residual modules include a first dynamic convolution module, a first instance normalization layer, an activation function layer, a second dynamic convolution module, and a second instance normalization layer, which are connected in sequence. The purpose of the first and second dynamic convolution modules is to generate dynamic convolution kernels for convolution, and both include a global pooling layer, a convolution layer, a sub-activation function layer, a fully connected layer, and a softmax layer, which are connected in sequence. The four different parameter values π1 to π4 output by the softmax layer are multiplied by weight matrices W1 to W4 of corresponding sizes and then added to generate the convolution kernel. The activation function layer is the same as the sub-activation function layer.

[0104] The decoder network based on biquadratic linear difference plus convolution includes three biquadratic linear interpolation plus convolution modules connected in sequence, a convolution layer and a Tanh activation function layer; the biquadratic linear interpolation plus convolution module includes a biquadratic linear interpolation layer, a convolution layer and an activation function layer connected in sequence.

[0105] S2.2 build the discriminator network;

[0106] The discriminator network adopts the PatchGAN network structure.

[0107] S2.3 train the generator network and the discriminator network and optimize them through the loss function;

[0108] The training data consists of 7,090 image pairs randomly cropped to 256×256 size as input. Training was performed for 300 epochs, with the learning rate set to 0.0001 for the first 150 epochs and then gradually decayed to 0 for the next 150 epochs. The training environment was built using the PyTorch framework. The test was conducted on a workstation running Ubuntu 18.04, equipped with an Intel Core i9-10900X CPU and an NVIDIA GeForce RTX 3060 12GB GPU.

[0109] S2.4 Test the trained generator network using test data;

[0110] The 1770 image pairs of test data were input into the trained generator network. As shown in Figure 6, compared with the existing different image deblurring algorithms, the infrared image motion deblurring method based on the generative adversarial network generated clearer visual effects, more realistic and natural image structures, and closer to clear images. The test passed and a trained generator network was generated.

[0111] S3. Input the infrared image to be deblurred into the trained generator network to obtain a deblurred infrared image.

[0112] FIG7 shows the deblurring effect of the image deblurring method of the present invention on a real blurred infrared image. The image generated by the image deblurring method of the present invention has fewer ringing artifacts, a clear image structure, and a deblurring effect with good robustness.

[0113] Finally, it should be noted that the above preferred examples are only used to illustrate the technical solutions of the present invention and are not limiting. Although the present invention has been described in detail through the above specific examples, those skilled in the art should understand that various changes can be made in form and details without departing from the scope defined in the claims of the present invention.

[0114] In summary, the present invention retains more detail information in blurred images, effectively solves the problem of detail information loss caused by the difficulty in feature extraction of blurred infrared images, enhances the robustness of the network in processing motion blur caused by different moving objects in infrared images, accelerates the convergence speed of the network in processing infrared images, and improves the visual effect of restored infrared images.

Claims

1. A method for deblurring infrared images based on a generative adversarial network, characterized by: The following steps are involved: S1. Construct an infrared image dataset, simulate and generate motion blurred images of infrared images, and use infrared image blurred-clear image pairs as training data and test data; S2. Construct an infrared image motion blur removal model based on a generative adversarial network, input the training data into the model for training, and test it with the test data to obtain a trained generator network; the generator network adopts a generation network based on longitudinal channels and dynamic convolution, including an encoder network based on longitudinal channels, a residual feature extraction network based on dynamic convolution, and a decoder network based on biquadratic linear difference plus convolution; The encoder network based on the longitudinal channel includes a forward channel and a longitudinal channel with the same input; the forward channel includes a first convolution block, a second convolution block, a third convolution block, and a fourth convolution block; the longitudinal channel includes a first longitudinal channel, a second longitudinal channel, and a third longitudinal channel; the output of the first convolution block and the second convolution block after sequential connection is spliced with the output of the first longitudinal channel as the input of the third convolution block, the output of the third convolution block is spliced with the output of the second longitudinal channel as the input of the fourth convolution block, and the output of the fourth convolution block is spliced with the output of the third longitudinal channel as the output of the encoder network; The residual feature extraction network based on dynamic convolution includes 6 identical residual modules; the residual module includes a first dynamic convolution module, a first instance normalization layer, an activation function layer, a second dynamic convolution module and a second instance normalization layer connected in sequence; the first dynamic convolution module and the second dynamic convolution module each include a global pooling layer, a convolution layer, a sub-activation function layer, a fully connected layer and a softmax layer connected in sequence; the four different parameter values π1 to π4 output by the softmax layer are respectively multiplied with weight matrices W1 to W4 of corresponding sizes and then added to generate a convolution kernel; S3. Input the infrared image to be deblurred into the trained generator network to obtain a deblurred infrared image.

2. The infrared image motion deblurring method based on a generative adversarial network according to claim 1, characterized in that: A Dropout layer is added after the activation function layer of the residual module.

3. The infrared image motion deblurring method based on a generative adversarial network according to claim 1, characterized in that: In S2, the decoder network based on biquadratic linear difference plus convolution includes three biquadratic linear interpolation plus convolution modules, a convolution layer and a Tanh activation function layer connected in sequence; the biquadratic linear interpolation plus convolution module includes a biquadratic linear interpolation layer, a convolution layer and an activation function layer connected in sequence.

4. The infrared image motion blur removal method based on a generative adversarial network according to claim 1, characterized in that: In S2, the activation function applied by the generator is the IR-ReLU function: The derivative of the activation function applied by the network during backpropagation is:

5. The infrared image motion deblurring method based on a generative adversarial network according to claim 1, characterized in that: S2 specifically includes the following steps: S2.1 build the generator network; S2.2 build the discriminator network; S2.3 train the generator network and the discriminator network and optimize them through the loss function; S2.4 Test the trained generator network using test data.

6. The infrared image motion blur removal method based on generative adversarial network according to claim 5, characterized in that: In S2.2, the discriminator network adopts the PatchGAN structure.

7. The infrared image motion blur removal method based on generative adversarial network according to claim 5, characterized in that: In S2.3, the loss function includes adversarial loss and content loss, and the loss function is: L=L GAN +λ1L x Where, L GAN Denotes the adversarial loss, L x represents the content loss, λ1 represents the regularization coefficient of the content loss; The expression of adversarial loss is: Content loss is expressed as a new structural coefficient l str Multiplying with the L2 loss, the expression of the function is: L x =l str ·L2 In the formula, the expression of L2 loss is: Where, I S x,y Represents the pixel value of the clear image, Represents the pixel value of the restored image; Structural coefficient l str The expression is: Where, represents the variance of the restored image, represents the variance of the clear image, Represents the covariance between the restored image and the clear image.

Images