CN110930331A - Noise blurred image non-blind restoration method, system and storage medium - Google Patents

Noise blurred image non-blind restoration method, system and storage medium Download PDF

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CN110930331A
CN110930331A CN201911156703.2A CN201911156703A CN110930331A CN 110930331 A CN110930331 A CN 110930331A CN 201911156703 A CN201911156703 A CN 201911156703A CN 110930331 A CN110930331 A CN 110930331A
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CN110930331B (en
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廖帆
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Nanjing College of Information Technology
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Abstract

The invention discloses a noise blurred image non-blind restoration method, a system and a storage medium, wherein the method comprises the following steps: performing iterative computation based on a pre-established noise blurred image restoration model, wherein the iterative computation comprises the following steps: acquiring a noise blurred image correlation matrix pair by adopting a gradient projection algorithm, and adaptively generating horizontal and vertical regularization weighting parameters of pixels according to the difference of gray values between adjacent pixels in the image; and acquiring a restored image according to the matrix pair and a matrix group constructed by the pixel transverse and longitudinal regularization weighting parameters. The method can remove noise and blur of the noise blurred image and improve the restoration effect of the texture part and the detail part in the image.

Description

Noise blurred image non-blind restoration method, system and storage medium
Technical Field
The invention relates to a noise blurred image non-blind restoration method, a noise blurred image non-blind restoration system and a storage medium, and belongs to the technical field of image processing.
Background
In the process of acquiring, transmitting and maintaining the image, the phenomena of noise, blurring and the like are inevitably caused due to various reasons such as optical conditions, camera shooting technology, transmission channels, natural environment, artificial damage and the like, so that the image quality is reduced, and the visual effect is obviously reduced. Therefore, it is necessary to restore an image including noise and having a blur, to improve image quality, and to improve the visual effect of the image.
Disclosure of Invention
The embodiment of the invention aims to overcome the defects in the prior art, and provides a noise blurred image non-blind restoration method, a noise blurred image non-blind restoration system and a storage medium, which can improve the restoration effect of texture parts and detail parts in an image while denoising and deblurring a noise blurred image.
In order to achieve the above purpose, the embodiment of the invention is realized by adopting the following technical scheme:
in a first aspect, an embodiment of the present invention provides a noise-blurred image non-blind restoration method, where the method includes the following steps:
performing the following iterative operations based on a pre-established noise blurred image restoration model:
according to xk-1Generating a pair of matrices (p, q) from a pair of pre-constructed matrices (p, q)k-1,qk-1)；xk-1Represents the value of the digital image x after the k-1 iteration; p represents the difference of the gray value between the pixel in the digital image x and the adjacent pixel below the pixel, and q represents the difference of the gray value between the pixel in the digital image x and the adjacent pixel on the right of the pixel; p is a radical ofk-1Representing the digital image x after the k-1 iterationk-1Difference of gray value between middle pixel and its lower adjacent pixel, qk-1Representing the digital image x after the k-1 iterationk-1K is more than or equal to 1 and less than or equal to N, and N represents the total iteration times of the algorithm;
according to matrix pair (p)k-1,qk-1) Generating a longitudinal adaptive weighting parameter ak-1And a transverse adaptive weighting parameter bk-1；
According to matrix pair (p)k-1,qk-1)、ak-1And bk-1And a pre-constructed matrix set (p, q, a, b) resulting in a matrix set (p)k -1,qk-1,ak-1,bk-1) (ii) a a represents the longitudinal adaptive weighting parameters of the pixels in the digital image x, b represents the transverse adaptive weighting parameters of the pixels in the digital image x; a isk-1Representing a digital image xk-1Longitudinal adaptive weighting parameter of middle pixel, bk-1Representing a digital image xk-1A horizontal adaptive weighting parameter of the middle pixel;
according to a matrix set (p)k-1,qk-1,ak-1,bk-1) Obtaining matrix pairs (p) by using gradient projection algorithmk,qk)；pkRepresenting the value of p after the kth iteration, qkRepresents the value of q after the kth iteration;
according to matrix pair (p)k,qk) Generating a longitudinal adaptive weighting parameter akAnd a transverse adaptive weighting parameter bkObtaining a matrix set (p)k,qk,ak,bk)；akRepresenting the digital image x after the kth iterationkLongitudinal adaptive weighting parameter of middle pixel, bkRepresenting the digital image x after the kth iterationkA horizontal adaptive weighting parameter of the middle pixel;
according to a matrix set (p)k,qk,ak,bk) Obtaining a digital image x after the kth iterationk；
If the iteration times are less than the set iteration times N, adding 1 to the iteration times to execute the iteration step again, otherwise, finishing the iteration and obtaining the final restored image xN；
Wherein the noise blurred image restoration model comprises a fidelity term and an adaptive weighted total variation regularization term.
Further, the method for establishing the noise blurred image restoration model comprises the following steps:
establishing a mathematical model of the noise blurred image f:
f＝Axoriginal+nadditive(1)
in the formula (1), f is ∈ Rm×nFor blurred images containing noise, Rm×nRepresenting a matrix of m rows and n columns, xoriginal∈Rm×nFor sharp images, A is a blurring operator, nadditive∈Rm×nTo be added to the blurred image AxoriginalAdditive noise in (1);
establishing a self-adaptive weighted total variation noise blurred image restoration model as follows:
Figure BDA0002285001770000031
in the formula (2), the reaction mixture is,
Figure BDA0002285001770000032
is a fidelity term, 2 λ AWTV (x) is an adaptive weighted total variationThe regularization term, AWTV (x), represents the adaptive weighted total variation of the digital image x, | · | | survival2Representing the vector 2 norm, λ>0 is a regularization parameter;
establishing the following self-adaptive weighted total variation regular term model:
Figure BDA0002285001770000033
in formula (3), i and j represent the row and column coordinates of the elements in the matrix, xi,jRepresents the pixel value at image pixel coordinate (i, j) in the digital image x; x is the number ofi+1,jRepresents the pixel value at image pixel coordinate (i +1, j) in the digital image x; x is the number ofi,j+1Represents the pixel value at image pixel coordinate (i, j +1) in the digital image x; x is the number ofi,nRepresents the pixel value at image pixel coordinates (i, n) in the digital image x; x is the number ofi+1,nRepresents the pixel value at image pixel coordinate (i +1, n) in the digital image x; x is the number ofm,jRepresents the pixel value at image pixel coordinates (m, j) in the digital image x; x is the number ofm,j+1Represents the pixel value at image pixel coordinate (m, j +1) in the digital image x; a isi,jRepresents the value of the longitudinal adaptive weighting parameter at image pixel coordinates (i, j) in the digital image x; bi,jRepresents the value of the laterally adaptive weighting parameter at image pixel coordinates (i, j) in the digital image x; a isi,nRepresents the value of the longitudinal adaptive weighting parameter at image pixel coordinates (i, n) in the digital image x; bm,jRepresents the value of the laterally adaptive weighting parameter at the image pixel coordinates (m, j) in the digital image x; i-0, 1, …, m-1, j-0, 1, …, n-1; the value range of the longitudinal self-adaptive weighting parameter is more than or equal to 0 and less than or equal to ai,jNot more than 1, and the value range of the transverse self-adaptive weighting parameter is not less than 0 and not more than bi,j≤1。
Further, the matrix pair (p, q) is constructed as follows:
constructing a digital matrix with the size of (m +1) x n as p according to the size of the noise blurred image f, wherein the size of p belongs to the range of R(m+1)×nConstructing a number matrix of size mx (n +1) as q, q ∈ Rm×(n+1)；
Let p bei,jRepresenting the position of the element coordinate (i, j) in the number matrix pValue of element, qi,jRepresenting the value of an element at the element coordinate (i, j) in the number matrix q, i and j representing the row and column coordinates of the element in the matrix, i being 0,1, …, m, j being 0,1, …, n, respectively, then:
Figure BDA0002285001770000041
and constructing a matrix pair (p, q) according to the constructed number matrixes p and q.
Further, the matrix set (p, q, a, b) is constructed as follows:
constructing a digital matrix with the size of (m-1) x (n-1) as a according to the size of m x n of the noise blurred image f, wherein a belongs to R(m -1)×(n-1)，ai,jRepresenting the value of an element at element coordinate (i, j) in the number matrix a, then:
Figure BDA0002285001770000042
in the formula: i and j respectively represent the row and column coordinates of the elements in the matrix, i is 0,1, …, m-1, j is 0,1, …, n-1; ω is a constant greater than 0;
constructing a number matrix of size (m-1) × (n-1) as b, b ∈ R(m-1)×(n-1)，bi,jRepresenting the value of an element at element coordinate (i, j) in the number matrix b, then:
Figure BDA0002285001770000043
and constructing a matrix group (p, q, a, b) according to the constructed number matrix p, q, a, b.
Further, x isk-1The initial assignment formula of (a) is as follows:
x0＝0m×n(7)
in the formula (7), x0Represents the start x of the 1 st iterationk-1Value of (1), 0m×nRepresenting a zero matrix of size m x n.
Further, a longitudinal adaptive weighting parameter ak-1The generation formula of (c) is as follows:
Figure BDA0002285001770000051
transverse adaptive weighting parameter bk-1The generation formula of (c) is as follows:
Figure BDA0002285001770000052
in the formula: a isk-1∈R(m-1)×(n-1)Is a numerical matrix of size (m-1) × (n-1), ω is a constant greater than 0; bk-1∈R(m-1)×(n-1)Is a number matrix of size (m-1) × (n-1).
Further, the matrix pair (p)k,qk) The calculation method of (2) is as follows:
a function g (-) is established, mathematically defined as:
Figure BDA0002285001770000053
the formula for the first derivative g' (x) of x for g (x) is as follows:
g'(x)＝2AT(Ax-f) (11)
in the formula (11), ATTranspose of fuzzy operator A;
the second derivative g "(x) for x is found by the equation:
g”(x)＝2ATA (12)
the mathematical definition of the Lipschitz constant L of equation (10) is:
Figure BDA0002285001770000054
the Tylor formula is used for expanding the formula (10), and the formula (10) is in xk-1The Tylor expansion of (A) is expressed in the following mathematical form:
Figure BDA0002285001770000061
in the formula (14), 0! Represents a hierarchy of 0, 1! Represents a hierarchy of 1, 2! Represents a hierarchy level of 2, t! Represents the hierarchy of t, (t + 1)! Represents a hierarchy of t +1, g' (x)k-1) Represents g (x) in xk-1The value of the first derivative, g ″ (x)k-1) Represents g (x) in xk-1Value of the second derivative of (g)(t)(xk-1) Represents g (x) in xk-1T derivative value of g(t+1)(xk-1+θ(x-xk-1) Is g (x) at xk-1+θ(x-xk-1) T +1 derivative value of (0 ≦ t, 0)<θ<1；
Combining equations (11), (12), (13), and (14), taking the first three terms of equation (14), equation (14) is expressed in mathematical form as follows:
Figure BDA0002285001770000062
in the formula (15), the first three terms of the formula (14) are omitted, and the formula (15) is expressed in the following mathematical form:
Figure BDA0002285001770000063
a function Γ (·) is established, mathematically defined as:
Γ(pi,j,qi,j,ai,j,bi,j)＝ai,j(pi,j-pi-1,j)+bi,j(qi,j-qi,j-1) (17)
combining equations (2), (3), (16), and (17), equation (2) after the k-1 th iteration is expressed in mathematical form as follows:
Figure BDA0002285001770000064
a function Φ (·) is established, mathematically defined as:
Figure BDA0002285001770000065
in the formula (19), M represents the gray level digit of x pixels of the digital image, and M is more than or equal to 1;
combining equations (18) and (19), expressing equation (18) in the mathematical form:
Figure BDA0002285001770000071
a function h (-) is established, mathematically defined as:
Figure BDA0002285001770000072
a function Λ (·) is established, mathematically defined as:
Λ(x)＝(p,q) (22)
formula (21) is as in (p)k-1,qk-1,ak-1,bk-1) A first derivative value h' (p) ofk-1,qk-1,ak-1,bk-1) The formula is as follows:
Figure BDA0002285001770000073
a function P (-) is established, mathematically defined as:
Figure BDA0002285001770000074
combining the equations (23) and (24), and calculating the matrix pair (p) after the k iteration by using a gradient projection algorithmk,qk) The calculation is as follows:
Figure BDA0002285001770000075
in the formula (25), the reaction mixture,
Figure BDA0002285001770000081
is the gradient of the gradient projection algorithm,
Figure BDA0002285001770000082
is the projection step size of the gradient projection algorithm.
Further, the k-th iterationPost-generation restoration image xkThe values are calculated as follows:
Figure BDA0002285001770000083
in a second aspect, an embodiment of the present invention provides a noise-blurred image non-blind restoration system, including a processor and a storage medium;
the storage medium is used for storing instructions;
the processor is configured to operate in accordance with the instructions to perform the steps of the method according to any one of the preceding claims.
In a third aspect, the present invention provides a computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, the computer program implements the steps of any one of the methods described above.
Compared with the prior art, the noise-blurred image non-blind restoration method, the noise-blurred image non-blind restoration system and the storage medium have the beneficial effects that:
according to the embodiment of the invention, the transverse and longitudinal regularization weighting parameters of the pixel can be generated in a self-adaptive manner according to the difference of the gray values between adjacent pixels in the image; the non-blind restoration method for the noise blurred image provided by the embodiment of the invention can improve the restoration capability of the texture part and the detail part of the image and improve the restoration effect of the noise blurred image.
Drawings
Fig. 1 is a flowchart of an implementation of a noise-blurred image non-blind restoration method according to an embodiment of the present invention;
FIG. 2a is a "Peppers" sharp image;
FIG. 2b is a "Peppers" noise motion blurred image;
FIG. 2c is a restored image obtained by applying a total variation method to the noise motion blurred image of "Peppers";
FIG. 2d is a restored image obtained by applying the method of the present invention to the "Peppers" noise motion blurred image;
FIG. 2e is a "Peppers" noise Gaussian blur image;
FIG. 2f is a restored image obtained by a total variation method for a Peppers noise Gaussian blurred image;
FIG. 2g is a restored image obtained by the method of the present invention for a "Peppers" noise Gaussian blur image;
FIG. 2h is a "Peppers" noise-averaged blurred image;
FIG. 2i is a restored image obtained by a total variation method for the noise average blurred image of "Peppers";
FIG. 2j is a restored image obtained by applying the method of the present invention to the "Peppers" noise-averaged blurred image;
FIG. 3a is a clear image of "Plane";
FIG. 3b is a motion blurred image for "Plane" noise;
FIG. 3c is a restored image obtained by applying a total variation method to a "Plane" noise motion-blurred image;
FIG. 3d is a restored image obtained by the method of the present invention for a "Plane" noise motion-blurred image;
FIG. 3e is a "Plane" noise Gaussian blur image;
FIG. 3f is a restored image obtained by a total variation method for a Gaussian blurred image with "Plane" noise;
FIG. 3g is a restored image obtained by applying the method of the present invention to a Gaussian blurred image with "Plane" noise;
FIG. 3h is a "Plane" noise averaged blurred image;
FIG. 3i is a restored image obtained by a total variation method for the "Plane" noise average blurred image;
FIG. 3j is a restored image obtained by the method of the present invention for the "Plane" noise-averaged blurred image;
FIG. 4a is a "Milkdrop" sharp image;
FIG. 4b is a "Milkdrop" noise motion blurred image;
FIG. 4c is a restored image obtained by applying a total variation method to the "Milkdrop" noise motion blurred image;
FIG. 4d is a restored image obtained by applying the method of the present invention to a "Milkdrop" noise motion blurred image;
FIG. 4e is a "Milkdrop" noise Gaussian blur image;
FIG. 4f is a restored image obtained by applying a total variation method to a "Milkdrop" noise Gaussian blur image;
FIG. 4g is a restored image obtained by applying the method of the present invention to a "Milkdrop" noise Gaussian blur image;
FIG. 4h is a "Milkdrop" noise averaged blurred image;
FIG. 4i is a restored image obtained by applying a total variation method to the "Milkdrop" noise-averaged blurred image;
fig. 4j is a restored image obtained by applying the method of the present invention to a "Milkdrop" noise-averaged blurred image.
Detailed Description
The invention is further described below with reference to the accompanying drawings. The following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present invention is not limited thereby.
As shown in fig. 1, it is a flowchart of an implementation method of a noise-blurred image non-blind restoration method provided in an embodiment of the present invention, and the implementation method includes the following steps:
step 1: the method for establishing the self-adaptive weighted total variation noise blurred image restoration model specifically comprises the following steps:
a mathematical model of a digital noise blurred image f of size m × n is established as follows:
f＝Axoriginal+nadditive(1)
in the formula (1), f is ∈ Rm×nFor blurred images containing noise, Rm×nRepresenting a matrix of m rows and n columns, xoriginal∈Rm×nFor sharp images, A is a blurring operator, nadditive∈Rm×nTo be added to the blurred image AxoriginalAdditive noise in (1);
establishing a self-adaptive weighted total variation noise blurred image restoration model as follows:
Figure BDA0002285001770000101
in the formula (2), the reaction mixture is,
Figure BDA0002285001770000102
is a fidelity term, 2 λ AWTV (x) is an adaptive weighted total variation regularization term, AWTV (x) represents the adaptive weighted total variation of the digital image x, | | · | (|) survival2Representing the vector 2 norm, λ>0 is a regularization parameter;
establishing the following self-adaptive weighted total variation regular term model:
Figure BDA0002285001770000111
in equation (3), i and j represent the row and column coordinates, respectively, of the elements in the matrix (the digital image x is also a matrix), xi,jRepresenting the pixel value at the image pixel coordinate (i, j) in the digital image x, xi+1,jRepresenting the pixel value at the image pixel coordinate (i +1, j) in the digital image x, xi,j+1Representing the pixel value at the image pixel coordinate (i, j +1) in the digital image x, xi,nRepresenting the pixel value at image pixel coordinate (i, n) in image x, xi+1,nRepresenting the pixel value at the image pixel coordinate (i +1, n) in the digital image x, xm,jRepresenting the pixel value at the image pixel coordinate (m, j) in the digital image x, xm,j+1Representing the pixel value, a, at the image pixel coordinate (m, j +1) in the digital image xi,jRepresenting the value of a longitudinal adaptive weighting parameter at the image pixel coordinates (i, j) in the digital image x, bi,jRepresenting the value of a laterally adaptive weighting parameter at the image pixel coordinates (i, j) in the digital image x, ai,nRepresenting the value of a longitudinal adaptive weighting parameter at image pixel coordinates (i, n) in a digital image x, bm,jRepresents the value of the horizontal adaptive weighting parameter at the image pixel coordinate (m, j) in the digital image x, i is 0,1, …, m-1, j is 0,1, …, n-1, the value range of the vertical adaptive weighting parameter is 0 ≦ ai,jNot more than 1, and the value range of the transverse self-adaptive weighting parameter is not less than 0 and not more than bi,j≤1。
Step 2: constructing a matrix pair (p, q), a set of matrices(p, q, a, b), and for xk-1Performing an initial assignment, wherein: x is the number ofk-1Representing the digital image after the k-1 iteration; p represents the difference of the gray value between the pixel in the digital image x and the adjacent pixel below the pixel, and q represents the difference of the gray value between the pixel in the digital image x and the adjacent pixel on the right of the pixel; p is a radical ofk-1Representing the digital image x after the k-1 iterationk-1Difference of gray value between middle pixel and its lower adjacent pixel, qk-1Representing the digital image x after the k-1 iterationk-1The difference value of the gray values between the middle pixel and the right adjacent pixel, k is more than or equal to 1 and less than or equal to N, N represents the total iteration times of the algorithm, and the method specifically comprises the following steps:
step 201. establish a matrix pair (p, q) with the mathematical definition as follows:
Figure BDA0002285001770000121
in the formula (4), i and j represent the row and column coordinates of the elements in the matrix respectively, and p ∈ R(m+1)×nIs a number matrix of size (m +1) x n, pi,jRepresenting the element value at the element coordinate (i, j) in the digital matrix p, representing the difference of the gray value between the pixel at the coordinate (i, j) and the pixel at the coordinate (i +1, j) in the digital image x (due to the limitation of the image x boundary, the 0 th row and the m th row in the digital matrix p cannot be calculated, all the values are assigned 0), q ∈ Rm×(n+1)Is a number matrix of size mx (n +1), qi,jRepresenting the value of an element at the element coordinate (i, j) in the number matrix q, representing the difference in gray value between the pixel at coordinate (i, j) and coordinate (i, j +1) in the digital image x (due to the limitation of the image x boundary, the 0 th and nth columns in the number matrix q cannot be computed, all values are assigned 0), i is 0,1, …, m, j is 0,1, …, n;
step202, establishing a matrix group (p, q, a, b), adaptively generating values a and b of longitudinal and transverse weighting parameters according to the values of the matrix pair (p, q), wherein a represents the longitudinal adaptive weighting parameter of the pixel in the digital image x and is used for representing a comparison degree between the difference value of the gray value between the pixel in the digital image x and the adjacent pixel below the pixel and the difference value of the gray value between the pixel on the right side of the pixel; b represents the lateral self of a pixel in the digital image xAn adaptive weighting parameter characterizing a degree of comparison between the difference in gray value between a pixel in the digital image x and its right-hand neighboring pixel and the difference in gray value between its lower-hand neighboring pixel, a longitudinal adaptive weighting parameter ai,jThe values of (c) were calculated as follows:
Figure BDA0002285001770000122
in the formula (5), a is R(m-1)×(n-1)Is a number matrix with the size of (m-1) x (n-1), i and j respectively represent the row and column coordinates of the elements in the matrix, ai,jRepresenting the value of the longitudinal adaptive weighting parameter at the image pixel coordinate (i, j) in the digital image x (i.e. the element value at the element coordinate (i, j) in the digital matrix a) for characterizing a degree of comparison between the difference in gray-scale values between the coordinate (i, j) and the pixel at the coordinate (i +1, j) in the digital image x and the difference in gray-scale values between the coordinate (i, j) and the pixel at the coordinate (i, j +1), i ═ 1, …, m-1, j ═ 1, …, n-1, ω is a constant slightly larger than 0 (avoiding denominator of 0);
step203, transverse adaptive weighting parameter bi,jThe values of (c) were calculated as follows:
Figure BDA0002285001770000131
in the formula (6), b ∈ R(m-1)×(n-1)Is a number matrix with size (m-1) x (n-1), i and j represent the row and column coordinates of the elements in the matrix, respectively, bi,jRepresenting the value of the transversal adaptive weighting parameter at the image pixel coordinate (i, j) in the digital image x (i.e. the element value at the element coordinate (i, j) in the digital matrix b) for characterizing a degree of comparison between the difference in gray-scale values between the coordinate (i, j) and the pixel at coordinate (i, j +1) in the digital image x and the difference in gray-scale values between the coordinate (i, j) and the pixel at coordinate (i +1, j), i ═ 1, …, m-1, j ═ 1, …, n-1, ω is a constant slightly larger than 0 (avoiding denominator of 0);
for xk-1The initial assignment is made, as shown below:
x0＝0m×n(7)
in the formula (7), x0Represents the start x of the 1 st iterationk-1Value of (1), 0m×nRepresenting a zero matrix of size m x n.
And step 3: start the k-th iteration according to xk-1Value of (d) to generate a matrix pair (p)k-1,qk-1) The value of (c).
The k-1 th iteration is followed by the matrix pair (p) according to equation (4)k-1,qk-1) The values of (c) were calculated as follows:
Figure BDA0002285001770000141
in the formula (8), pk-1Representing the digital image x after the k-1 iterationk-1Difference of gray value between middle pixel and its lower adjacent pixel, qk-1Representing the digital image x after the k-1 iterationk-1The difference between the gray values of the middle pixel and the pixel adjacent to the right of the middle pixel, k is 1, …, N, i and j represent the row and column coordinates of the matrix element, pk-1∈R(m+1)×nIs a digital matrix of size (m +1) x n,
Figure BDA0002285001770000142
representing a digital matrix pk-1The element value at the middle element coordinate (i, j) represents the digital image x after the k-1 iterationk-1Difference in gray value between the pixel at coordinate (i, j) and coordinate (i +1, j) (due to image x)k-1Limitation of boundaries, number matrix pk-1Row 0 and row m cannot be calculated, all values are assigned 0), qk-1∈Rm×(n+1)Is a matrix of numbers of size mx (n +1),
Figure BDA0002285001770000143
representing a matrix of numbers qk-1The value of the element at the medium element coordinate (i, j), representing the digital image xk-1Difference in gray value between the pixel at coordinate (i, j) and coordinate (i, j +1) (due to image x)k-1Limitation of boundaries, digit matrix qk-1The 0 th row and the n th row in the sequence are not calculated, all values are assigned 0),i＝0,1,…,m，j＝0,1,…,n。
and 4, step 4: according to matrix pair (p)k-1,qk-1) Adaptively generating values a of longitudinal and lateral weighting parametersk-1And bk -1Obtaining a matrix set (p)k-1,qk-1,ak-1,bk-1) A value of (d);
ak-1representing a digital image xk-1Longitudinal adaptive weighting parameter of middle pixel for characterizing digital image xk-1A degree of comparison between the difference in gray value between the middle pixel and its lower adjacent pixel and the difference in gray value between its right adjacent pixel; bk-1Representing a digital image xk-1Transversal adaptive weighting parameter of medium pixel for characterizing digital image xk-1A degree of comparison between the difference in gray value between the middle pixel and its right adjacent pixel and the difference in gray value between its lower adjacent pixels;
step 401: combining equations (5) and (8), longitudinal adaptive weighting parameter a after k-1 iterationk-1The values of (c) were calculated as follows:
Figure BDA0002285001770000151
in the formula (9), ak-1Represents the basis matrix pair (p) after the k-1 iterationk-1,qk-1) Is adaptively generated, k is 1, …, N, ak-1∈R(m-1)×(n-1)Is a number matrix with the size of (m-1) x (n-1), i and j respectively represent the row and column coordinates of the elements in the matrix,
Figure BDA0002285001770000152
representing a digital matrix ak-1The element value at the middle element coordinate (i, j), i-1, …, m-1, j-1, …, n-1, ω is a constant slightly larger than 0 (avoiding a denominator of 0);
step 402: combining equations (6) and (8), transversal adaptive weighting parameter b after k-1 iterationk-1The values of (c) were calculated as follows:
Figure BDA0002285001770000153
in the formula (10), bk-1Represents the basis matrix pair (p) after the k-1 iterationk-1,qk-1) Is adaptively generated, k is 1, …, N, bk-1∈R(m-1)×(n-1)Is a number matrix with the size of (m-1) x (n-1), i and j respectively represent the row and column coordinates of the elements in the matrix,
Figure BDA0002285001770000154
representing a digital matrix bk-1The element value at the middle element coordinate (i, j), i-1, …, m-1, j-1, …, n-1, ω is a constant slightly larger than 0 (avoiding a denominator of 0);
step 403: by combining equations (8), (9) and (10), the k-1 th iteration matrix set (p) can be obtainedk-1,qk-1,ak-1,bk-1) The value of (c).
And 5: according to a matrix set (p)k-1,qk-1,ak-1,bk-1) Using a gradient projection algorithm to obtain a matrix pair (p)k,qk) A value of (d);
step 501: a function g (-) is established, mathematically defined as:
Figure BDA0002285001770000155
the first derivative g' (x) of equation (11) for x is shown below:
g'(x)＝2AT(Ax-f) (12)
in the formula (12), ATTranspose of fuzzy operator A;
the second derivative g "(x) of equation (11) for x is given by:
g”(x)＝2ATA (13)
the mathematical definition of the Lipschitz constant L of equation (11) is:
Figure BDA0002285001770000161
step 502: the Tylor formula is used for developing the formula (11), and the formula (11) is in xk-1The Tylor expansion of (A) is expressed in the following mathematical form:
Figure BDA0002285001770000162
in the formula (15), 0! Represents a hierarchy of 0, 1! Represents a hierarchy of 1, 2! Represents a hierarchy level of 2, t! Represents the hierarchy of t, (t + 1)! Represents a hierarchy of t +1, g' (x)k-1) Represents g (x) in xk-1The value of the first derivative, g ″ (x)k-1) Represents g (x) in xk-1Value of the second derivative of (g)(t)(xk-1) Represents g (x) in xk-1T derivative value of g(t+1)(xk-1+θ(x-xk-1) Is g (x) at xk-1+θ(x-xk-1) T +1 derivative value of (0 ≦ t, 0)<θ<1；
Step 503: combining equations (12), (13), (14), and (15), taking the first three terms of equation (15), equation (15) is expressed in mathematical form as follows:
Figure BDA0002285001770000163
in the formula (16), since the noise-blurred image restoration is an engineering problem, omitting the high-order infinitesimal values after the first three terms of the formula (15) is completely acceptable in the actual restoration of the digital noise-blurred image;
equation (16) can be expressed in mathematical form as follows:
Figure BDA0002285001770000171
step 504: a function Γ () is established, mathematically defined as:
Γ(pi,j,qi,j,ai,j,bi,j)＝ai,j(pi,j-pi-1,j)+bi,j(qi,j-qi,j-1) (18)
combining equations (2), (3), (17), and (18), equation (2) after the k-1 th iteration can be expressed in mathematical form as follows:
Figure BDA0002285001770000172
step 505: a function phi (·) is established, mathematically defined as:
Figure BDA0002285001770000173
in the formula (20), M represents the gray level digit of x pixels of the digital image, and M is more than or equal to 1;
combining equations (19) and (20), equation (19) can be expressed in mathematical form as follows:
Figure BDA0002285001770000174
step 506: a function h (-) is established, mathematically defined as:
Figure BDA0002285001770000175
step 507: a function Λ (·) is established, mathematically defined as:
Λ(x)＝(p,q) (23)
formula (22) is as in (p)k-1,qk-1,ak-1,bk-1) A first derivative value h' (p) ofk-1,qk-1,ak-1,bk-1) The formula is as follows:
Figure BDA0002285001770000181
step 508: a function P (-) is established, mathematically defined as:
Figure BDA0002285001770000182
combining equations (24) and (25), the k-th iteration matrix pair (p) can be calculated by using gradient projection algorithmk,qk) The calculation is as follows:
Figure BDA0002285001770000183
in the formula (26), the reaction mixture is,
Figure BDA0002285001770000184
is the gradient of the gradient projection algorithm,
Figure BDA0002285001770000185
is the projection step size, p, of the gradient projection algorithmkAnd q iskAnd represents the values of p and q after the kth iteration, respectively, k being 1, …, N.
Step 6: according to matrix pair (p)k,qk) Adaptively generating values a of longitudinal and lateral weighting parameterskAnd bkObtaining a matrix set (p)k,qk,ak,bk) A value of (d);
step 601: combining equations (5) and (26), longitudinal adaptive weighting parameter a after k-th iterationkThe values of (c) were calculated as follows:
Figure BDA0002285001770000186
in the formula (27), akRepresents the basis matrix pair (p) after the k-th iterationk,qk) Is adaptively generated, k is 1, …, N, ak∈R(m-1)×(n-1)Is a number matrix with the size of (m-1) x (n-1), i and j respectively represent the row and column coordinates of the elements in the matrix,
Figure BDA0002285001770000191
representing a digital matrix akThe value of the element at the medium element coordinate (i, j), representing the digital image xkA degree of comparison between the difference in gray value between the pixel at coordinate (i, j) and the pixel at coordinate (i +1, j) and the difference in gray value between the pixel at coordinate (i, j) and the pixel at coordinate (i, j +1), i 1, …, m-1, j 1, …, n-1, ω being a constant value slightly greater than 0Number (avoid denominator 0);
step 602: combining equations (6) and (26), the transverse adaptive weighting parameter b after the k-th iterationkThe values of (c) were calculated as follows:
Figure BDA0002285001770000192
in the formula (28), bkRepresents the basis matrix pair (p) after the k-th iterationk,qk) Is adaptively generated, k is 1, …, N, bk∈R(m-1)×(n-1)Is a number matrix with the size of (m-1) x (n-1), i and j respectively represent the row and column coordinates of the elements in the matrix,
Figure BDA0002285001770000193
representing a digital matrix bkThe value of the element at the medium element coordinate (i, j), representing the digital image xkA degree of comparison between the difference in gray value between the pixel at coordinate (i, j) and the pixel at coordinate (i, j +1) and the difference in gray value between the pixel at coordinate (i, j) and the pixel at coordinate (i +1, j), i 1, …, m-1, j 1, …, n-1, ω being a constant slightly greater than 0 (avoiding denominator of 0);
step 603: combining equations (26), (27) and (28) to obtain the k-th iteration matrix set (p)k,qk,ak,bk) The value of (c).
And 7: according to a matrix set (p)k,qk,ak,bk) The value of (a) is obtained as a restored image x after the kth iterationk。
Combinations of formulae (19) and (p)k,qk,ak,bk) Value of (c), restoring image x after the kth iterationkThe values are calculated as follows:
Figure BDA0002285001770000194
and 8: judging whether the iteration number k is less than the set iteration number N, if k<N, let k equal to k +1, and re-enter step 3; if k is N, the iteration endsTo obtain the final restored image xN。
According to the embodiment of the invention, the transverse and longitudinal regularization weighting parameters of the pixel can be generated in a self-adaptive manner according to the difference of the gray values between adjacent pixels in the image; the non-blind restoration method for the noise blurred image provided by the embodiment of the invention can improve the restoration capability of the texture part and the detail part of the image and improve the restoration effect of the noise blurred image.
To further explain the non-blind restoration method of a noise-blurred image provided by the embodiment of the present invention, PSNR (peak signal-to-noise ratio) is used to compare the image restoration capabilities of the non-blind restoration method of a total variation noise-blurred image and the non-blind restoration method of a noise-blurred image provided by the embodiment of the present invention. It should be understood that the following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present invention is not limited thereby.
The mathematical definition of PSNR is as follows:
Figure BDA0002285001770000201
in the formula (30), M ≧ 1 represents the number of gray-scale bits of the digital image pixel, 2M-1 is the maximum value of the gray value of the M-bit gray image, the size of the image being M × n, Xi,jRepresenting the pixel value at the image pixel coordinate (i, j) in a sharp image, Yi,jRepresenting the pixel value at image pixel coordinate (i, j) in the inspection image.
PSNR is an objective evaluation index of an image that is most widely used in image processing, and is used to measure the degree of distortion between a sharp image and a detected image, and the higher the PSNR is, the smaller the degree of distortion between the sharp image and the detected image is.
The blurring operator a is respectively set as a motion blurring operator (motion displacement is 12 pixels, motion angle is 6 degrees), a gaussian blurring operator (template size is 9 × 9, standard value is 6), and an average blurring operator (template size is 7 × 7). The sharp images "Peppers", "Plane" and "Milkdrop" are respectively transformed into a "Peppers" motion blurred image, "Peppers" gaussian blurred image, "Peppers" average blurred image, "Plane" motion blurred image, "Plane" gaussian blurred image, "Plane" average blurred image, "Plane" motion blurred image, "Plane" average blurred image, "Milkdrop" motion blurred image, "Milkdrop" gaussian blurred image, "Milkdrop" average blurred image by the blurring operator a.
Adding normally distributed Gaussian white noise with the average value of 0.255 into a 'Peppers' motion blurred image, a 'Peppers' Gaussian blurred image, a 'Peppers' average blurred image, a 'Plane' motion blurred image, a 'Plane' Gaussian blurred image, a 'Plane' average blurred image, a 'Milldrp' motion blurred image, a 'Milldrp' Gaussian blurred image and a 'Milldrp' average blurred image to respectively obtain a 'Peppers' noise motion blurred image, a 'Peppers' noise Gaussian blurred image, a 'Peppers' noise average blurred image, a 'Plane' noise motion blurred image, a 'Plane' noise Gaussian blurred image, a 'Plane' noise average blurred image, a 'Milldrp' noise motion blurred image, a 'Milldrp' noise Gaussian blurred image and a 'Milldrp' noise average blurred image.
The regularization parameter lambda is 0.0001, the Lipschitz constant L is 2, the constant omega is 0.00001, the iteration number N is 3000, and a total variation method and the method are respectively used for restoring a Peppers noise motion blurred image, a Peppers noise Gaussian blurred image, a Peppers noise average blurred image, a Plane noise motion blurred image, a Plane noise Gaussian blurred image, a Plane noise average blurred image, a Milldrop noise motion blurred image, a Milldrop noise Gaussian blurred image and a Milldrop noise average blurred image.
FIG. 2a is a "Peppers" sharp image; FIG. 2b is a "Peppers" noise motion blurred image; FIG. 2c is a diagram of a "Peppers" noise motion blurred image restored by a total variation method; FIG. 2d is a diagram of a "Peppers" noise motion blurred image restored by the method of the present invention; FIG. 2e is a "Peppers" noise Gaussian blur image; FIG. 2f is a diagram of a full variation method for restoring an image of a noise Gaussian blurred image of Peppers; FIG. 2g is a 'Peppers' noise Gaussian blur image, and an image is restored by adopting the method of the invention; FIG. 2h is a "Peppers" noise-averaged blurred image; FIG. 2i is a 'Peppers' noise average blurred image restored by a total variation method; FIG. 2j is a diagram of an image restored by the method of the present invention for "Peppers" noise-averaged blurred images.
FIG. 3a is a clear image of "Plane"; FIG. 3b is a "Plane" noise motion blurred image; FIG. 3c is a view of a "Plane" noise motion-blurred image restored by a total variation method; FIG. 3d is a diagram of a "Plane" noise motion-blurred image restored by the method of the present invention; FIG. 3e is a "Plane" noise Gaussian blur image; FIG. 3f is a view of a "Plane" noise Gaussian blur image restored by a total variation method; FIG. 3g is a Gaussian blur image with "Plane" noise restored by the method of the present invention; FIG. 3h is a "Plane" noise averaged blurred image; FIG. 3i is a view showing that the "Plane" noise average blurred image is restored by a total variation method; FIG. 3j is a diagram of an image restored by the method of the present invention for a "Plane" noise-averaged blurred image.
FIG. 4a is a "Milkdrop" sharp image; FIG. 4b is a "Milkdrop" noise motion blurred image; FIG. 4c is a view of a "Milkdrop" noise motion blur image restored by a total variation method; FIG. 4d is a diagram of a "Milkdrop" noise motion blurred image restored by the method of the present invention; FIG. 4e is a "Milkdrop" noise Gaussian blur image; FIG. 4f is a diagram of a "Milkdrop" noise Gaussian blur image restored by a total variation method; FIG. 4g is a "Milkdrop" noise Gaussian blur image which is restored by the method of the present invention; FIG. 4h is a "Milkdrop" noise averaged blurred image; FIG. 4i is a view of a "Milkdrop" noise-averaged blurred image restored by a total variation method; FIG. 4j is a diagram of an image restored by the method of the present invention using a "Milkdrop" noise-averaged blurred image. The results of the experiment are shown in table 1:
Figure BDA0002285001770000221
Figure BDA0002285001770000231
table 1 shows PSNR values of images restored by restoring "Peppers" noise motion-blurred images, "Peppers" noise gaussian-blurred images, "Peppers" noise mean-blurred images, "Plane" noise motion-blurred images, "Plane" noise gaussian-blurred images, "mikrdrop" noise motion-blurred images, "mikrdrop" noise gaussian-blurred images, "and" mikrdrop "noise mean-blurred images, respectively, by using the total variation method and the method provided by the embodiment of the present invention. It can be seen that in all images subjected to experiments, the method provided by the embodiment of the invention improves the PSNR of the total variation method to different degrees, and therefore, the method provided by the embodiment of the invention has a better restoration effect on noise blurred images compared with the total variation method.
The embodiment of the invention also provides a noise blurred image non-blind restoration system, which comprises a processor and a storage medium;
the storage medium is used for storing instructions;
the processor is configured to operate in accordance with the instructions to perform the steps of the method according to any one of the preceding claims.
Embodiments of the present invention also provide a computer-readable storage medium, on which a computer program is stored, which when executed by a processor implements the steps of any of the methods described above.
As will be appreciated by one skilled in the art, embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
The above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, several modifications and variations can be made without departing from the technical principle of the present invention, and these modifications and variations should also be regarded as the protection scope of the present invention.

Claims (10)

1. A noise-blurred image non-blind restoration method is characterized by comprising the following steps:
performing the following iterative operations based on a pre-established noise blurred image restoration model:
according to xk-1Generating a pair of matrices (p, q) from a pair of pre-constructed matrices (p, q)k-1,qk-1)；xk-1Represents the value of the digital image x after the k-1 iteration; p represents the difference of the gray value between the pixel in the digital image x and the adjacent pixel below the pixel, and q represents the difference of the gray value between the pixel in the digital image x and the adjacent pixel on the right of the pixel; p is a radical ofk-1Representing the digital image x after the k-1 iterationk-1Difference of gray value between middle pixel and its lower adjacent pixel, qk-1Representing the digital image x after the k-1 iterationk-1K is more than or equal to 1 and less than or equal to N, and N represents the total iteration times of the algorithm;
according to matrix pair (p)k-1,qk-1) Generating a longitudinal adaptive weighting parameter ak-1And a transverse adaptive weighting parameter bk-1；
According to matrix pair (p)k-1,qk-1)、ak-1And bk-1And a pre-constructed matrix set (p, q, a, b) resulting in a matrix set (p)k-1,qk -1,ak-1,bk-1) (ii) a a represents the longitudinal adaptive weighting parameters of the pixels in the digital image x, b represents the transverse adaptive weighting parameters of the pixels in the digital image x; a isk-1Representing a digital image xk-1Longitudinal adaptive weighting parameter of middle pixel, bk-1Representing a digital image xk-1A horizontal adaptive weighting parameter of the middle pixel;
according to a matrix set (p)k-1,qk-1,ak-1,bk-1) Obtaining matrix pairs (p) by using gradient projection algorithmk,qk)；pkRepresenting the value of p after the kth iteration, qkRepresents the value of q after the kth iteration;
according to matrix pair (p)k,qk) Generating a longitudinal adaptive weighting parameter akAnd a transverse adaptive weighting parameter bkObtaining a matrix set (p)k,qk,ak,bk)；akRepresenting the digital image x after the kth iterationkLongitudinal adaptive weighting parameter of middle pixel, bkRepresenting the digital image x after the kth iterationkA horizontal adaptive weighting parameter of the middle pixel;
according to a matrix set (pk,qk,ak,bk) Obtaining a digital image x after the kth iterationk；
If the iteration times are less than the set iteration times N, adding 1 to the iteration times to execute the iteration step again, otherwise, finishing the iteration and obtaining the final restored image xN；
Wherein the noise blurred image restoration model comprises a fidelity term and an adaptive weighted total variation regularization term.
2. The method for non-blind restoration of a noise-blurred image according to claim 1, wherein the method for establishing the noise-blurred image restoration model comprises the following steps:
establishing a mathematical model of the noise blurred image f:
f＝Axoriginal+nadditive(1)
in the formula (1), f is ∈ Rm×nFor blurred images containing noise, Rm×nRepresenting a matrix of m rows and n columns, xoriginal∈Rm×nFor sharp images, A is a blurring operator, nadditive∈Rm×nTo be added to the blurred image AxoriginalAdditive noise in (1);
establishing a self-adaptive weighted total variation noise blurred image restoration model as follows:
Figure FDA0002285001760000021
in the formula (2), the reaction mixture is,
Figure FDA0002285001760000022
is a fidelity term, 2 λ AWTV (x) is an adaptive weighted total variation regularization term, AWTV (x) represents the adaptive weighted total variation of the digital image x, | | · | (|) survival2Representing the vector 2 norm, λ>0 is a regularization parameter;
establishing the following self-adaptive weighted total variation regular term model:
Figure FDA0002285001760000023
in formula (3), i and j represent the row and column coordinates of the elements in the matrix, xi,jRepresents the pixel value at image pixel coordinate (i, j) in the digital image x; x is the number ofi+1,jRepresents the pixel value at image pixel coordinate (i +1, j) in the digital image x; x is the number ofi,j+1Represents the pixel value at image pixel coordinate (i, j +1) in the digital image x; x is the number ofi,nRepresents the pixel value at image pixel coordinates (i, n) in the digital image x; x is the number ofi+1,nRepresents the pixel value at image pixel coordinate (i +1, n) in the digital image x; x is the number ofm, jRepresents the pixel value at image pixel coordinates (m, j) in the digital image x; x is the number ofm,j+1Represents the pixel value at image pixel coordinate (m, j +1) in the digital image x; a isi,jRepresents the value of the longitudinal adaptive weighting parameter at image pixel coordinates (i, j) in the digital image x; bi,jRepresents the value of the laterally adaptive weighting parameter at image pixel coordinates (i, j) in the digital image x; a isi,nRepresents the value of the longitudinal adaptive weighting parameter at image pixel coordinates (i, n) in the digital image x; bm,jRepresents the value of the laterally adaptive weighting parameter at the image pixel coordinates (m, j) in the digital image x; i-0, 1, …, m-1, j-0, 1, …, n-1; the value range of the longitudinal self-adaptive weighting parameter is more than or equal to 0 and less than or equal to ai,jNot more than 1, and the value range of the transverse self-adaptive weighting parameter is not less than 0 and not more than bi,j≤1。
3. The non-blind restoration method of a noise-blurred image according to claim 2, wherein the matrix pair (p, q) is constructed by the following method:
constructing a digital matrix with the size of (m +1) x n as p according to the size of the noise blurred image f, wherein the size of p belongs to the range of R(m+1)×nConstructing a number matrix of size mx (n +1) as q, q ∈ Rm×(n+1)；
Let p bei,jRepresenting the value of an element at the element coordinate (i, j) in the number matrix p, qi,jRepresenting the value of an element at the element coordinate (i, j) in the number matrix q, i and j representing the row and column coordinates of the element in the matrix, i being 0,1, …, m, j being 0,1, …, n, respectively, then:
Figure FDA0002285001760000031
and constructing a matrix pair (p, q) according to the constructed number matrixes p and q.
4. The non-blind restoration method of a noise-blurred image according to claim 3, wherein the matrix set (p, q, a, b) is constructed by the following method:
constructing a digital matrix with the size of (m-1) x (n-1) as a according to the size of m x n of the noise blurred image f, wherein a belongs to R(m -1)×(n-1)，ai,jRepresenting the value of an element at element coordinate (i, j) in the number matrix a, then:
Figure FDA0002285001760000041
in the formula: i and j respectively represent the row and column coordinates of the elements in the matrix, i is 0,1, …, m-1, j is 0,1, …, n-1; ω is a constant greater than 0;
constructing a number matrix of size (m-1) × (n-1) as b, b ∈ R(m-1)×(n-1)，bi,jRepresenting the value of an element at element coordinate (i, j) in the number matrix b, then:
Figure FDA0002285001760000042
and constructing a matrix group (p, q, a, b) according to the constructed number matrix p, q, a, b.
5. The method of non-blind restoration of a noise-blurred image according to claim 1, wherein x is the number of pixels in the imagek-1The initial assignment formula of (a) is as follows:
x0＝0m×n(7)
in the formula (7), x0Represents the start x of the 1 st iterationk-1Value of (1), 0m×nRepresenting a zero matrix of size m x n.
6. The method for non-blind restoration of a noise-blurred image according to claim 1, wherein the longitudinal adaptive weighting parameter ak-1The generation formula of (c) is as follows:
Figure FDA0002285001760000043
the transverse adaptive weighting parameter bk-1The generation formula of (c) is as follows:
Figure FDA0002285001760000044
in the formula: a isk-1∈R(m-1)×(n-1)Is a numerical matrix of size (m-1) × (n-1), ω is a constant greater than 0; bk-1∈R(m -1)×(n-1)Is a number matrix of size (m-1) × (n-1).
7. Method for non-blind restoration of a noise-blurred image according to claim 2, wherein said matrix pair (p) isk,qk) The calculation method of (2) is as follows:
a function g (-) is established, mathematically defined as:
Figure FDA0002285001760000051
the formula for the first derivative g' (x) of x for g (x) is as follows:
g'(x)＝2AT(Ax-f) (11)
in the formula (11), ATTranspose of fuzzy operator A;
the second derivative g "(x) for x is found by the equation:
g”(x)＝2ATA (12)
the mathematical definition of the Lipschitz constant L of equation (10) is:
Figure FDA0002285001760000052
the Tylor formula is used for expanding the formula (10), and the formula (10) is in xk-1The Tylor expansion of (A) is expressed in the following mathematical form:
Figure FDA0002285001760000053
in the formula (14), 0! Represents a hierarchy of 0, 1! Represents a hierarchy of 1, 2! Represents a hierarchy level of 2, t! Represents the hierarchy of t, (t + 1)! Represents a hierarchy of t +1, g' (x)k-1) Represents g (x) in xk-1The value of the first derivative, g ″ (x)k-1) Represents g (x) in xk-1Value of the second derivative of (g)(t)(xk-1) Represents g (x) in xk-1T derivative value of g(t+1)(xk-1+θ(x-xk-1) Is g (x) at xk-1+θ(x-xk -1) T +1 derivative value of (0 ≦ t, 0)<θ<1；
Combining equations (11), (12), (13), and (14), taking the first three terms of equation (14), equation (14) is expressed in mathematical form as follows:
Figure FDA0002285001760000054
in the formula (15), the first three terms of the formula (14) are omitted, and the formula (15) is expressed in the following mathematical form:
Figure FDA0002285001760000061
a function Γ (·) is established, mathematically defined as:
Γ(pi,j,qi,j,ai,j,bi,j)＝ai,j(pi,j-pi-1,j)+bi,j(qi,j-qi,j-1) (17)
combining equations (2), (3), (16), and (17), equation (2) after the k-1 th iteration is expressed in mathematical form as follows:
Figure FDA0002285001760000062
a function Φ (·) is established, mathematically defined as:
Figure FDA0002285001760000063
in the formula (19), M represents the gray level digit of x pixels of the digital image, and M is more than or equal to 1;
combining equations (18) and (19), expressing equation (18) in the mathematical form:
Figure FDA0002285001760000064
a function h (-) is established, mathematically defined as:
Figure FDA0002285001760000071
a function Λ (·) is established, mathematically defined as:
Λ(x)＝(p,q) (22)
formula (21) is as in (p)k-1,qk-1,ak-1,bk-1) A first derivative value h' (p) ofk-1,qk-1,ak-1,bk-1) The formula is as follows:
Figure FDA0002285001760000072
a function P (-) is established, mathematically defined as:
Figure FDA0002285001760000073
combining the equations (23) and (24), and calculating the matrix pair (p) after the k iteration by using a gradient projection algorithmk,qk) The calculation is as follows:
Figure FDA0002285001760000074
in the formula (25), the reaction mixture,
Figure FDA0002285001760000075
is the gradient of the gradient projection algorithm,
Figure FDA0002285001760000076
is the projection step size of the gradient projection algorithm.
8. The method of non-blind restoration of a noise-blurred image as claimed in claim 7, wherein the image x is restored after the k-th iterationkThe values are calculated as follows:
Figure FDA0002285001760000077
9. a noise-blurred image non-blind restoration system is characterized by comprising a processor and a storage medium;
the storage medium is used for storing instructions;
the processor is configured to operate in accordance with the instructions to perform the steps of the method according to any one of claims 1 to 8.
10. Computer-readable storage medium, on which a computer program is stored which, when being executed by a processor, carries out the steps of the method according to any one of claims 1 to 8.
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