CN110930331B - 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 PDFInfo
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Abstract
The invention discloses a method, a system and a storage medium for non-blind restoration of a noise blurred image, 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
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 system and a storage medium, which can be used for denoising and deblurring a noise blurred image and simultaneously improving the restoration effect of a texture part and a detail part in the 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 x k-1 Generating a pair of matrices (p, q) from a pair of pre-constructed matrices (p, q) k-1 ,q k-1 )；x k-1 Representing the value of the digital image x after the k-1 iteration; p represents the difference in gray value between a pixel in the digital image x and its adjacent pixel below, q represents the difference in gray value between a pixel in the digital image x and its adjacent pixel to the right; p is a radical of k-1 Representing the digital image x after the k-1 iteration k-1 Difference of gray value between middle pixel and its lower adjacent pixel, q k-1 Representing the digital image x after the k-1 iteration k-1 K 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 ,q k-1 ) Generating a longitudinal adaptive weighting parameter a k-1 And a transverse adaptive weighting parameter b k-1 ；
According to matrix pair (p) k-1 ,q k-1 )、a k-1 And b k-1 And a pre-constructed matrix set (p, q, a, b) resulting in a matrix set (p) k -1 ,q k-1 ,a k-1 ,b k-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 is k-1 Representing a digital image x k-1 Longitudinal adaptive weighting parameter of middle pixel, b k-1 Representing a digital image x k-1 A horizontal adaptive weighting parameter of the middle pixel;
according to a matrix set (p) k-1 ,q k-1 ,a k-1 ,b k-1 ) Obtaining matrix pairs (p) by using gradient projection algorithm k ,q k )；p k Representing the value of p after the kth iteration, q k Represents the value of q after the kth iteration;
according to matrix pair (p) k ,q k ) Generating a longitudinal adaptive weighting parameter a k And a laterally adaptive weighting parameter b k Obtaining a matrix set (p) k ,q k ,a k ,b k )；a k Representing the digital image x after the kth iteration k Longitudinal adaptive weighting parameter of middle pixel, b k Representing the digital image x after the kth iteration k A horizontal adaptive weighting parameter of the middle pixel;
according to matrix set (p) k ,q k ,a k ,b k ) Obtaining a digital image x after the kth iteration k ；
If the iteration times are less than the set iteration times N, adding 1 to the iteration times to execute the iteration operation again, otherwise, ending the iteration to obtain the final restored image x N ；
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＝Ax original +n additive (1)
in the formula (1), f is ∈ R m×n For blurred images containing noise, R m×n Representing a matrix of m rows and n columns, x original ∈R m×n For sharp images, A is a blurring operator, n additive ∈R m×n To be added to the blurred image Ax original Additive noise in (2);
establishing a self-adaptive weighted total variation noise blurred image restoration model as follows:
in the formula (2), the reaction mixture is,
is a fidelity term, 2 λ AWTV (x) is adaptive weightingA total variation regularization term, AWTV (x) represents the adaptive weighted total variation of the digital image x, | · | | non-calculation 2 Representing a vector 2 norm, λ > 0 being a regularization parameter;
establishing the following self-adaptive weighted total variation regular term model:
in formula (3), i and j represent the row and column coordinates of the elements in the matrix, x i,j Represents the pixel value at image pixel coordinate (i, j) in the digital image x; x is a radical of a fluorine atom i+1,j Represents the pixel value at image pixel coordinate (i +1, j) in the digital image x; x is the number of i,j+1 Represents the pixel value at image pixel coordinate (i, j + 1) in the digital image x; x is the number of i,n Represents the pixel value at image pixel coordinates (i, n) in the digital image x; x is the number of i+1,n Represents the pixel value at image pixel coordinate (i +1, n) in the digital image x; x is the number of m,j Represents the pixel value at image pixel coordinates (m, j) in the digital image x; x is the number of m,j+1 Represents the pixel value at image pixel coordinate (m, j + 1) in the digital image x; a is i,j Represents the value of the longitudinal adaptive weighting parameter at image pixel coordinates (i, j) in the digital image x; b is a mixture of i,j Represents the value of the laterally adaptive weighting parameter at image pixel coordinates (i, j) in the digital image x; a is i,n Represents the value of the longitudinal adaptive weighting parameter at image pixel coordinates (i, n) in the digital image x; b m,j Represents the value of the laterally adaptive weighting parameter at the image pixel coordinates (m, j) in the digital image x; i =0,1, \8230;, m-1,j =0,1, \8230;, 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 a i,j Not more than 1, and the value range of the transverse self-adaptive weighting parameter is not less than 0 and not more than b i,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)×n Constructing a number matrix of size mx (n + 1) as q, q ∈ R m×(n+1) ；
Let p be i,j Representing digital momentsElement value, q, at element coordinate (i, j) in array p i,j Representing the value of an element at the element coordinate (i, j) in the number matrix q, i and j respectively representing the row and column coordinates of the element in the matrix, i =0,1, \8230;, m, j =0,1, \8230;, n, then:
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) ，a i,j Representing the value of an element at element coordinate (i, j) in the number matrix a, then:
in the formula: i and j represent the row and column coordinates of the elements in the matrix, i =0,1, \8230;, m-1, j =0,1, \8230;, 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) ，b i,j Representing the value of an element at element coordinate (i, j) in the number matrix b, then:
and constructing a matrix group (p, q, a, b) according to the constructed number matrix p, q, a, b.
Further, x is k-1 The initial assignment formula of (a) is as follows:
x 0 ＝0 m×n (7)
in the formula (7), x 0 Represents the start x of the 1 st iteration k-1 Value of (1), 0 m×n Representing a zero matrix of size mxn。
Further, a longitudinal adaptive weighting parameter a k-1 The generation formula of (c) is as follows:
transverse adaptive weighting parameter b k-1 The generation formula of (c) is as follows:
in the formula: a is a k-1 ∈R (m-1)×(n-1) Is a numerical matrix of size (m-1) × (n-1), ω is a constant greater than 0; b k-1 ∈R (m-1)×(n-1) Is a matrix of numbers of size (m-1) × (n-1).
Further, the matrix pair (p) k ,q k ) The calculation method of (2) is as follows:
a function g (-) is established, mathematically defined as:
the first derivative g' (x) of g (x) with respect to x is found by the formula:
g'(x)＝2A T (Ax-f) (11)
in formula (11), A T Transpose of fuzzy operator A;
the second derivative g "(x) of g (x) with respect to x is found as follows:
g″(x)＝2A T A (12)
the mathematical definition of the Lipschitz constant L of equation (10) is:
the Tylor formula is used for expanding the formula (10), and the formula (10) is in x k-1 The Tylor expansion of (A) is expressed in the following mathematical form:
in the formula (14), 0! Represents a hierarchy of 0, 1! Represents the hierarchy level 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 x k-1 The value of the first derivative, g ″ (x) k-1 ) Represents g (x) in x k-1 Value of the second derivative of (g) (t) (x k-1 ) Represents g (x) in x k-1 T derivative value of (g) (t+1) (x k-1 +θ(x-x k-1 ) Is g (x) in x k-1 +θ(x-x k-1 ) T +1 order derivative value, t is more than or equal to 0, and theta is more than 0 and less than 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:
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:
a function Γ (·) is established, mathematically defined as:
Γ(p i,j ,q i,j ,a i,j ,b i,j )＝a i,j (p i,j -p i-1,j )+b i,j (q i,j -q i,j-1 ) (17)
combining equations (2), (3), (16), and (17), equation (2) after the k-1 iteration is expressed in mathematical form as follows:
a function Φ (·) is established, mathematically defined as:
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), equation (18) is expressed in mathematical form as follows:
a function h (-) is established, mathematically defined as:
a function Λ (·) is established, mathematically defined as:
Λ(x)＝(p,q) (22)
formula (21) is as in (p) k-1 ,q k-1 ,a k-1 ,b k-1 ) A first derivative value h' (p) of (d) k-1 ,q k-1 ,a k-1 ,b k-1 ) The formula is as follows:
the function p (·) is established, mathematically defined as:
combining the equations (23) and (24), and calculating the matrix pair (p) after the k iteration by using a gradient projection algorithm k ,q k ) The calculation is as follows:
in the formula (25), the reaction mixture,
is the gradient of the gradient projection algorithm,
is the projection step size of the gradient projection algorithm.
Further, the image x is restored after the k-th iteration k The values are calculated as follows:
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 a total variation method for 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 the method of the present invention for the noise average blurred image of "Peppers";
FIG. 3a is a "Milkdrop" sharp image;
FIG. 3b is a "Milkdrop" noise motion blurred image;
FIG. 3c is a restored image obtained by applying the total variation method to the "Milldrop" noise motion blurred image;
FIG. 3d is a restored image obtained by applying the method of the present invention to a "Milkdrop" noise motion blurred image;
FIG. 3e is a "Milkdrop" noise Gaussian blur image;
FIG. 3f is a restored image obtained by applying a total variation method to a "Milkdrop" noise Gaussian blur image;
FIG. 3g is a restored image obtained by applying the method of the present invention to a "Milkdrop" noise Gaussian blur image;
FIG. 3h is a "Milkdrop" noise averaged blurred image;
FIG. 3i is a restored image obtained by applying a total variation method to the "Milkdrop" noise-averaged blurred image;
fig. 3j 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＝Ax original +n additive (1)
in the formula (1), f is epsilon of R m×n For blurred images containing noise, R m×n Representing a matrix of m rows and n columns, x original ∈R m×n For sharp images, A is the blur operator, n additive ∈R m×n To be added to the blurred image Ax original Additive noise in (1);
establishing a self-adaptive weighted total variation noise blurred image restoration model as follows:
in the formula (2), the reaction mixture is,
is a fidelity term, 2 lambda AWTV (x) is an adaptive weighted total variation regular term, AWTV (x) represents the adaptive weighted total variation of the digital image x, | | · | (| white space) 2 Representing a vector 2 norm, λ > 0 being a regularization parameter;
establishing the following self-adaptive weighted total variation regular term model:
in formula (3), i and j represent the row and column coordinates of the elements in the matrix, respectively (the digital image x is also a matrix), x i,j Representing the pixel value at the image pixel coordinate (i, j) in the digital image x, x i+1,j Representing the pixel value at the image pixel coordinate (i +1, j) in the digital image x, x i,j+1 Representing the pixel value at the image pixel coordinate (i, j + 1) in the digital image x, x i,n Representing the pixel value at image pixel coordinate (i, n) in image x, x i+1,n Representing the pixel value at the image pixel coordinate (i +1, n) in the digital image x, x m,j Representing the pixel value at the image pixel coordinate (m, j) in the digital image x, x m,j+1 Representing the pixel value, a, at the image pixel coordinate (m, j + 1) in the digital image x i,j Representing the value of a longitudinal adaptive weighting parameter at the image pixel coordinates (i, j) in the digital image x, b i,j Representing the value of a laterally adaptive weighting parameter at the image pixel coordinates (i, j) in the digital image x, a i,n Representing the value of a longitudinal adaptive weighting parameter at the image pixel coordinates (i, n) in the digital image x, b m,j Representing the value of the horizontal self-adaptive weighting parameter at the image pixel coordinate (m, j) in the digital image x, i =0,1, \ 8230, m-1, j =0,1, \ 8230, n-1, the value range of the vertical self-adaptive weighting parameter is 0 ≦ a i,j B is not less than 0 and not more than 1, and the value range of the transverse self-adaptive weighting parameter is not less than 0 i,j ≤1。
Step 2: constructing a matrix pair (p, q), a matrix group (p, q, a, b), and pairing x k-1 Performing an initial assignment, wherein: x is a radical of a fluorine atom k-1 Representing the digital image after the (k-1) th 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 of k-1 Representing the digital image x after the k-1 iteration k-1 Difference of gray value between middle pixel and its lower adjacent pixel, q k-1 Representing the digital image x after the k-1 iteration k-1 The 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:
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)×n Is a number matrix of size (m + 1) x n, p i,j Representing the element value at the element coordinate (i, j) in the digital matrix p, representing the difference in 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 ∈ R m×(n+1) Is a number matrix of size mx (n + 1), q i,j Representing the value of an element at the element coordinate (i, j) in the digital matrix q, representing the difference in gray value between the pixel at the coordinate (i, j) and the coordinate (i, j + 1) in the digital image x (due to the limitation of the image x boundary, columns 0 and n in the digital matrix q cannot be computed, the values are all assigned 0), i =0,1, \ 8230;, m, j =0,1, \8230;, 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 a horizontal adaptive weighting parameter of a pixel in the digital image x for characterizing a degree of comparison between the difference of the gray values of the pixel in the digital image x and its right neighboring pixel and the difference of the gray values of its lower neighboring pixel, and a vertical adaptive weighting parameter a i,j The values of (c) were calculated as follows:
in the formula (5), a is R (m-1)×(n-1) Is a number matrix of size (m-1) x (n-1), i and j respectively representing the row and column coordinates of the elements of the matrix, a i,j Representing 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 value between the pixel at coordinate (i, j) and coordinate (i +1, j) in the digital image x and the difference in gray value between the pixel at coordinate (i, j) and coordinate (i, j + 1), i =0,1, \ 8230;, m-1, j =0,1, \ 8230;, n-1, ω is a constant slightly larger than 0 (avoiding denominator of 0);
step203, transverse adaptive weighting parameter b i,j The values of (c) were calculated as follows:
in the formula (6), b ∈ R (m-1)×(n-1) Is a number matrix of size (m-1) x (n-1), i and j respectively representing the row and column coordinates of the elements of the matrix, b i,j Representing 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 value between the pixel at coordinate (i, j) and coordinate (i, j + 1) in the digital image x and the difference in gray value between the pixel at coordinate (i, j) and coordinate (i +1, j), i =0,1, \ 8230;, m-1, j =0,1, \ 8230;, n-1, ω is a constant slightly larger than 0 (avoiding denominator of 0);
for x k-1 The initial assignment is made, as shown below:
x 0 ＝0 m×n (7)
in the formula (7), x 0 Represents the start x of the 1 st iteration k-1 Value of (1), 0 m×n Representing a zero matrix of size m x n.
And 3, step 3: starting the k-th iteration according to x k-1 Value of (b) to generate a matrix pair (p) k-1 ,q k-1 ) The value of (c).
The k-1 th iteration is followed by the matrix pair (p) according to equation (4) k-1 ,q k-1 ) The values of (c) were calculated as follows:
in the formula (8), p k-1 Representing the digital image x after the k-1 iteration k-1 Difference of gray value between middle pixel and its lower adjacent pixel, q k-1 Representing the digital image x after the k-1 iteration k-1 The difference in gray value between the middle pixel and its right adjacent pixel, k =1, \8230, N, i and j represent the row and column coordinates of the matrix element, p k-1 ∈R (m+1)×n Is a digital matrix of size (m + 1) x n,
representing a digital matrix p k-1 The element value at the middle element coordinate (i, j) represents the digital image x after the k-1 iteration k-1 Difference in gray value between the pixel at the middle coordinate (i, j) and the coordinate (i +1, j) (due to image x) k-1 Limitation of boundaries, number matrix p k-1 Row 0 and row m cannot be calculated, all values are assigned 0), q k-1 ∈R m×(n+1) Is a matrix of numbers of size m x (n + 1),
representing a matrix of numbers q k-1 The value of the element at the medium element coordinate (i, j), representing the digital image x k-1 Difference in gray value between the pixel at coordinate (i, j) and coordinate (i, j + 1) (due to image x) k-1 Limitation of boundaries, number matrix q k-1 The 0 th and nth columns cannot be calculated, and all values are assigned 0), i =0,1, \ 8230;, m, j =0,1, \8230;, n.
And 4, step 4: according to matrix pair (p) k-1 ,q k-1 ) Adaptively generating values a of longitudinal and lateral weighting parameters k-1 And b k -1 Obtaining a matrix set (p) k-1 ,q k-1 ,a k-1 ,b k-1 ) A value of (d);
a k-1 representing a digital image x k-1 Longitudinal adaptive weighting parameter of middle pixel for characterizing digital image x k-1 A 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; b k-1 Representing a digital image x k-1 Transversal adaptive weighting parameter of medium pixel for characterizing digital image x k-1 A 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;
step401: combining equations (5) and (8), longitudinal adaptive weighting parameter a after k-1 iteration k-1 The values of (c) were calculated as follows:
in the formula (9), a k-1 Represents the basis matrix pair (p) after the k-1 iteration k-1 ,q k-1 ) Is used to adaptively generate the value of the longitudinal weighting parameter, k =1, \8230;, N, a k-1 ∈R (m-1)×(n-1) Is a number matrix of size (m-1) x (n-1), i and j respectively representing the row and column coordinates of the elements in the matrix,
representing a digital matrix a k-1 The value of the element at the medium element coordinate (i, j), i =0,1, \8230;, m-1, j =0,1, \8230;, n-1, ω is a constant slightly greater than 0 (avoiding the denominator being 0);
step402: combining equations (6) and (8), transversal adaptive weighting parameter b after k-1 iteration k-1 The values of (c) were calculated as follows:
in formula (10), b k-1 Represents the basis matrix pair (p) after the k-1 iteration k-1 ,q k-1 ) Is adaptively generated to add transverselyThe value of the weight parameter, k =1, \8230, N, b k-1 ∈R (m-1)×(n-1) Is a matrix of numbers of size (m-1) × (n-1), i and j representing the row and column coordinates respectively in which the elements of the matrix are located,
representing a digital matrix b k-1 The value of the element at the medium element coordinate (i, j), i =0,1, \8230;, m-1, j =0,1, \8230;, n-1, ω is a constant slightly greater than 0 (avoiding a denominator of 0);
step403: by combining equations (8), (9) and (10), the k-1 th iteration matrix set (p) can be obtained k-1 ,q k-1 ,a k-1 ,b k-1 ) The value of (c).
And 5: according to a matrix set (p) k-1 ,q k-1 ,a k-1 ,b k-1 ) Using a gradient projection algorithm to obtain a matrix pair (p) k ,q k ) A value of (d);
step501: a function g (-) is established, mathematically defined as:
the first derivative g' (x) of equation (11) for x is shown below:
g'(x)＝2A T (Ax-f) (12)
in formula (12), A T Transpose of fuzzy operator A;
the second derivative g "(x) of equation (11) for x is given by:
g″(x)＝2A T A (13)
the mathematical definition of the Lipschitz constant L of equation (11) is:
step502: the Tylor formula is used for developing the formula (11), and the formula (11) is in x k-1 The Tylor expansion of (A) is expressed in the following mathematical form:
in the formula (15), 0! Represents a hierarchy of 0, 1! Represents the hierarchy level 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 x k-1 The value of the first derivative, g ″ (x) k-1 ) Represents g (x) in x k-1 Value of the second derivative of (g) (t) (x k-1 ) Represents g (x) in x k-1 T derivative value of (g) (t+1) (x k-1 +θ(x-x k-1 ) Is g (x) in x k-1 +θ(x-x k-1 ) T +1 order derivative value, t is more than or equal to 0, and theta is more than 0 and less than 1;
step503: combining equations (12), (13), (14), and (15), taking the first three terms of equation (15), equation (15) is expressed in mathematical form as follows:
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:
step504: a function Γ () is established, mathematically defined as:
Γ(p i,j ,q i,j ,a i,j ,b i,j )＝a i,j (p i,j -p i-1,j )+b i,j (q i,j -q i,j-1 ) (18)
combining equations (2), (3), (17), and (18) can express equation (2) after the k-1 th iteration in the following mathematical form:
step505: a function Φ (·) is built, mathematically defined as:
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:
step506: a function h (-) is established, mathematically defined as:
step507: a function Λ (·) is established, mathematically defined as:
Λ(x)＝(p,q) (23)
formula (22) is as in (p) k-1 ,q k-1 ,a k-1 ,b k-1 ) A first derivative value h' (p) of k-1 ,q k-1 ,a k-1 ,b k-1 ) The formula is as follows:
step508: a function p (·) is established, mathematically defined as:
combinations (24) and (25) of which the gradient projection algorithm can be usedComputing a k-th iteration matrix pair (p) k ,q k ) The calculation is as follows:
in the formula (26), the reaction mixture is,
is the gradient of the gradient projection algorithm,
is the projection step size, p, of the gradient projection algorithm k And q is k Representing the values of p and q after the kth iteration, respectively, k =1, \8230, N.
Step 6: according to matrix pair (p) k ,q k ) Adaptively generating values a of longitudinal and lateral weighting parameters k And b k Obtaining a matrix set (p) k ,q k ,a k ,b k ) A value of (d);
step601: combining equations (5) and (26), longitudinal adaptive weighting parameter a after kth iteration k The values of (c) were calculated as follows:
in the formula (27), a k Represents the basis matrix pair (p) after the kth iteration k ,q k ) Is adaptively generated, k =1, \ 8230;, N, a, the value of the longitudinal weighting parameter k ∈R (m-1)×(n-1) Is a number matrix of size (m-1) x (n-1), i and j respectively representing the row and column coordinates of the elements in the matrix,
representing a digital matrix a k The value of the element at the medium element coordinate (i, j), representing the digital image x k The difference in gray value between the pixel at coordinate (i, j) and the pixel at coordinate (i +1, j) and the gray value between the pixel at coordinate (i, j) and the pixel at coordinate (i, j + 1)A degree of comparison between the differences in values, i =0,1, \8230;, m-1, j =0,1, \8230;, n-1, ω is a constant slightly greater than 0 (avoiding a denominator of 0);
step602: combining equations (6) and (26), the transverse adaptive weighting parameter b after the kth iteration k The value of (c) is calculated as follows:
in the formula (28), b k Represents the basis matrix pair (p) after the k-th iteration k ,q k ) Is adaptively generated, k =1, \8230;, N, b, the value of the lateral weighting parameter k ∈R (m-1)×(n-1) Is a number matrix of size (m-1) x (n-1), i and j respectively representing the row and column coordinates of the elements in the matrix,
representing a matrix of numbers b k The value of the element at the medium element coordinate (i, j), representing the digital image x k A degree of comparison between the difference in gray value between the pixel at the middle coordinate (i, j) and the coordinate (i, j + 1) and the difference in gray value between the pixel at the coordinate (i, j) and the coordinate (i +1, j), i =0,1, \ 8230;, m-1, j =0,1, \ 8230;, n-1, ω is a constant slightly greater than 0 (avoiding a denominator of 0);
step603: combining equations (26), (27) and (28) to obtain the k-th iteration matrix set (p) k ,q k ,a k ,b k ) The value of (c).
And 7: according to matrix set (p) k ,q k ,a k ,b k ) The value of (a) is obtained as a restored image x after the kth iteration k 。
Combinations of formulae (19) and (p) k ,q k ,a k ,b k ) Value of (c), restoring image x after the kth iteration k The values are calculated as follows:
and step 8: judging whether the iteration number k is smaller than the set iteration number N, if k is smaller than N, enabling k = k +1, and re-entering the step 3; if k = N, the iteration is finished, and the final restored image x is obtained N 。
According to the embodiment of the invention, the pixel transverse and longitudinal regularization weighting parameters 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 of 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:
in the formula (30), M ≧ 1 represents the number of gray-scale bits of the digital image pixel, 2 M -1 is the maximum value of the gray value of the M-bit gray image, the size of the image being M × n, X i,j Representing the pixel value at the image pixel coordinate (i, j) in a sharp image, Y i,j Representing 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" and "Milkdrop" are respectively transformed into a "Peppers" motion blurred image, a "Peppers" gaussian blurred image, a "Peppers" average blurred image, a "Milkdrop" motion blurred image, a "Milkdrop" gaussian blurred image, and a "Milkdrop" average blurred image by a blurring operator a.
Adding normally distributed Gaussian white noise with the average value of 0.255 into the motion blurred image of Peppers, the Gaussian blurred image of Peppers, the average blurred image of Peppers, the motion blurred image of Millkdrop, the Gaussian blurred image of Millkdrop and the average blurred image of Millkdrop to respectively obtain the noise motion blurred image of Peppers, the noise Gaussian blurred image of Peppers, the noise average blurred image of Millkdrop, the noise motion blurred image of Millkdrop and the noise average blurred image of Millkdrop.
Taking the regularization parameter lambda =0.0001, the Lipschitz constant L =2, the constant omega =0.00001 and the iteration number N =3000, and respectively restoring the Pepers noise motion blurred image, the Pepers noise Gaussian blurred image, the Pepers noise average blurred image, the Milkdrop noise motion blurred image, the Milkdrop noise Gaussian blurred image and the Milkdrop noise average blurred image by using the total variation method and the method.
FIG. 2a is a "Peppers" sharp image; FIG. 2b is a "Peppers" noise motion blurred image; FIG. 2c is a 'Peppers' noise motion blurred image, which is restored by a total variation method; FIG. 2d is a 'Peppers' noise motion blurred image, and the image is restored by adopting the method of the 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, and the image is restored by adopting 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 "Milkdrop" sharp image; FIG. 3b is a "Milkdrop" noise motion blurred image; FIG. 3c is a diagram of a "Milkdrop" noise motion blur image restored by a total variation method; FIG. 3d is a "Milldrop" noise motion blurred image which is restored by the method of the present invention; FIG. 3e is a "Milkdrop" noise Gaussian blur image; FIG. 3f is a diagram of a "Milkdrop" noise Gaussian blur image restored by a total variation method; FIG. 3g is a "Milkdrop" noise Gaussian blur image which is restored by the method of the present invention; FIG. 3h is a "Milkdrop" noise averaged blurred image; FIG. 3i is a diagram of an image restored by a total variation method from a "Milkdrop" noise-averaged blurred image; FIG. 3j 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:
| image of a person | Noise blurred images | Total variation method | The method of the invention |
| "Peppers" noise motion blur | 26.1545 | 33.6495 | 35.7877 |
| "Peppers" noise Gaussian blur | 26.1073 | 32.8509 | 33.8906 |
| "Peppers" noise mean blur | 27.1975 | 33.497 | 35.2097 |
| "Milkdrop" noise motion blur | 27.8874 | 34.8188 | 39.4625 |
| "Milkdrop" noise Gaussian blur | 28.4388 | 35.7334 | 38.3575 |
| "Milkdrop" noise averaging blur | 29.6016 | 35.7101 | 39.5546 |
Table 1 shows PSNR values of restored images after restoring "Peppers" noise motion blurred images, "Peppers" noise gaussian blurred images, "Peppers" noise average blurred images, "Milkdrop" noise motion blurred images, "Milkdrop" noise gaussian blurred images, "and" Milkdrop "noise average 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 to store 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 further provide a computer-readable storage medium, on which a computer program is stored, where the computer program is used to implement the steps of any one of the foregoing methods when executed by a processor.
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 so forth) 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 (8)
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 x k-1 Generating a pair of matrices (p, q) from a pair of pre-constructed matrices (p, q) k-1 ,q k-1 )；x k-1 Represents 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 of k-1 Representing the digital image x after the k-1 iteration k-1 Difference in gray value between the middle pixel and its lower neighboring pixel, q k-1 Representing the digital image x after the k-1 iteration k-1 K 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 ,q k-1 ) Generating a longitudinal adaptive weighting parameter a k-1 And a laterally adaptive weighting parameter b k-1 ；
According to matrix pair (p) k-1 ,q k-1 )、a k-1 And b k-1 And a set of pre-constructed matrices (p,q, a, b) to obtain a matrix set (p) k-1 ,q k -1 ,a k-1 ,b k-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 is k-1 Representing a digital image x k-1 Longitudinal adaptive weighting parameter of middle pixel, b k-1 Representing a digital image x k-1 A horizontal adaptive weighting parameter of the middle pixel;
according to matrix set (p) k-1 ,q k-1 ,a k-1 ,b k-1 ) Obtaining matrix pairs (p) by using gradient projection algorithm k ,q k )；p k Representing the value of p after the kth iteration, q k Represents the value of q after the kth iteration;
according to matrix pair (p) k ,q k ) Generating a longitudinal adaptive weighting parameter a k And a transverse adaptive weighting parameter b k Obtaining a matrix set (p) k ,q k ,a k ,b k )；a k Representing the digital image x after the kth iteration k Longitudinal adaptive weighting parameter of middle pixel, b k Representing the digital image x after the kth iteration k A horizontal adaptive weighting parameter of the middle pixel;
according to a matrix set (p) k ,q k ,a k ,b k ) Obtaining a digital image x after the kth iteration k ；
If the iteration times are less than the set iteration times N, adding 1 to the iteration times to execute the iteration operation again, otherwise, ending the iteration to obtain the final restored image x N ；
The noise blurred image restoration model comprises a fidelity term and an adaptive weighted total variation regular term;
the method for establishing the noise blurred image restoration model comprises the following steps of:
establishing a mathematical model of the noise blurred image f:
f＝Ax original +n additive (1)
in the formula (1), f is ∈ R m×n For blurred images containing noise, R m×n RepresentMatrix of size m rows and n columns, x original ∈R m×n For sharp images, A is a blurring operator, n additive ∈R m×n To be added to the blurred image Ax original Additive noise in (1);
establishing a self-adaptive weighted total variation noise blurred image restoration model as follows:
in the formula (2), the reaction mixture is,
is a fidelity term, 2 lambda AWTV (x) is an adaptive weighted total variation regular term, AWTV (x) represents the adaptive weighted total variation of the digital image x, | | · | (| white space) 2 Representing a vector 2 norm, λ > 0 being a regularization parameter;
establishing the following self-adaptive weighted total variation regular term model:
in formula (3), i and j represent the row and column coordinates of the elements in the matrix, x i,j Represents the pixel value at image pixel coordinate (i, j) in the digital image x; x is a radical of a fluorine atom i+1,j Represents the pixel value at image pixel coordinate (i +1, j) in the digital image x; x is a radical of a fluorine atom i,j+1 Represents the pixel value at image pixel coordinate (i, j + 1) in the digital image x; x is the number of i,n Represents the pixel value at image pixel coordinates (i, n) in the digital image x; x is the number of i+1,n Represents the pixel value at image pixel coordinate (i +1, n) in the digital image x; x is the number of m,j Represents the pixel value at image pixel coordinates (m, j) in the digital image x; x is the number of m,j+1 Represents the pixel value at image pixel coordinate (m, j + 1) in the digital image x; a is i,j Represents the value of the longitudinal adaptive weighting parameter at image pixel coordinates (i, j) in the digital image x; b is a mixture of i,j Representing the laterally adaptive weighting parameters at image pixel coordinates (i, j) in a digital image xThe value of the number; a is a i,n A value representing a longitudinal adaptive weighting parameter at image pixel coordinates (i, n) in the digital image x; b m,j Represents the value of the laterally adaptive weighting parameter at the image pixel coordinates (m, j) in the digital image x; i =0,1, \8230;, m-1,j =0,1, \8230;, 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 a i,j B is not less than 0 and not more than 1, and the value range of the transverse self-adaptive weighting parameter is not less than 0 i,j ≤1；
Wherein the matrix pair (p) k ,q k ) The calculation method of (2) is as follows:
a function g (-) is established, mathematically defined as:
the first derivative g' (x) of g (x) with respect to x is found by the formula:
g'(x)＝2A T (Ax-f) (11)
in formula (11), A T Transpose of fuzzy operator A;
the second derivative g "(x) of g (x) with respect to x is found by the formula:
g”(x)＝2A T A (12)
the mathematical definition of the Lipschitz constant L of equation (10) is:
the Tylor formula is used for expanding the formula (10), and the formula (10) is in x k-1 The Tylor expansion of (A) is expressed in the following mathematical form:
in the formula (14), 0! Represents a factorial of 0, 1! Represents a factorial of 1, 2! Represents a factorial of 2, t! Represents the factorial of t, (t + 1)! Represents a factorial, g' (x) of t +1 k-1 ) Represents g (x) in x k-1 The value of the first derivative, g ″ (x) k-1 ) Represents g (x) in x k-1 Value of the second derivative of (g) (t) (x k-1 ) Represents g (x) in x k-1 T derivative value of g (t+1) (x k-1 +θ(x-x k-1 ) Is g (x) in x k-1 +θ(x-x k -1 ) T +1 order derivative value, t is more than or equal to 0, and theta is more than 0 and less than 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:
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:
a function Γ (·) is established, mathematically defined as:
Γ(p i,j ,q i,j ,a i,j ,b i,j )＝a i,j (p i,j -p i-1,j )+b i,j (q i,j -q i,j-1 ) (17)
in the formula, p i,j Representing the value of an element at the element coordinate (i, j) in the number matrix p, q i,j Representing the value of an element at the element coordinate (i, j) in the number matrix q, where i =1, \ 8230;, m, j =1, \ 8230;, n, then:
combining equations (2), (3), (16), and (17), equation (2) after the k-1 th iteration is expressed in mathematical form as follows:
a function Φ (·) is established, mathematically defined as:
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:
a function h (-) is established, mathematically defined as:
a function Λ (·) is established, mathematically defined as:
Λ(x)＝(p,q) (22)
formula (21) is as in (p) k-1 ,q k-1 ,a k-1 ,b k-1 ) A first derivative value h' (p) of k-1 ,q k-1 ,a k-1 ,b k-1 ) The formula is as follows:
a function P (-) is established, mathematically defined as:
combining the equations (23) and (24), and calculating the matrix pair (p) after the k iteration by using a gradient projection algorithm k ,q k ) The calculation is as follows:
in the formula (25), the reaction mixture,
is the gradient of the gradient projection algorithm,
is the projection step size of the gradient projection algorithm.
2. The method for non-blind restoration of a noise-blurred image, according to claim 1, wherein 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 m x n, wherein p belongs to R (m+1)×n Constructing a number matrix of size mx (n + 1) as q, q ∈ R m×(n+1) ；
And constructing a matrix pair (p, q) according to the constructed number matrixes p and q.
3. The method for non-blind restoration of a noise-blurred image according to claim 2, 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) ，a i,j Representing the value of an element at element coordinate (i, j) in the number matrix a, then:
in the formula: i and j represent the row and column coordinates of the elements in the matrix, i =0,1, \8230;, m-1, j =0,1, \8230;, 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) ，b i,j Representing the value of an element at element coordinate (i, j) in the number matrix b, then:
and constructing a matrix group (p, q, a, b) according to the constructed number matrixes p, q, a, b.
4. The method of non-blind restoration of a noise-blurred image according to claim 1, wherein x is k-1 The initial assignment formula of (a) is as follows:
x 0 ＝0 m×n (7)
in formula (7), x 0 Represents the start x of the 1 st iteration k-1 Value of (1), 0 m×n Representing a zero matrix of size m x n.
5. The method of non-blind restoration of a noise-blurred image as claimed in claim 1, wherein the longitudinal adaptive weighting parameter a is k-1 The generation formula of (c) is as follows:
the transverse adaptive weighting parameter b k-1 The generation formula of (c) is as follows:
in the formula: a is k-1 ∈R (m-1)×(n-1) Is a numerical matrix of size (m-1) x (n-1), ω is a constant greater than 0; b k-1 ∈R (m -1)×(n-1) Is a matrix of numbers of size (m-1) × (n-1).
6. The method of non-blind restoration of a noise-blurred image according to claim 1, wherein the method is characterized in thatIn that the image x is restored after the k-th iteration k The values are calculated as follows:
7. a noise-blurred image non-blind restoration system is characterized by comprising a processor and a storage medium;
the storage medium is to store 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 6.
8. 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 6.
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