CN107945121A - A kind of image recovery method and system based on full variation - Google Patents
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Abstract
The invention discloses a kind of image recovery method and system based on full variation, including：Step S100 initializes the 0th generation restored image f when obtaining blurred picture0, the 0th generation parameter information；Step S200 setting iterationses k=0；Step S300 calculates the generation restored image of kth+1 f according to the parameter information in kth generationk+1；Step S400 is according to the generation of kth+1 restored image fk+1With the kth for restored image fk, judge whether to meet iteration stopping condition；Step S500 is if so, the then generation of kth+1 restored image fk+1For reconstructed image；Step S600 is if it is not, then calculate the parameter information in the generation of kth+1；Step S700 updates iterations k=k+1, and jumps to step S300.Regular parameter λ can be adaptively adjusted in the case of known fuzzy core in the present invention, reduce artificial blindness, improve calculating speed, while recover the image of high quality.
Description
Technical field
The present invention relates to image processing field, espespecially a kind of image recovery method and system based on full variation.
Background technology
In actual image acquisition process, digital imagery by motion blur, optical dimming, random noise etc. due to being moved back
The influence of change factor, what is often obtained is the image of a width blur degradation.The purpose of image restoration, exactly from the degeneration obtained
Image recovers original image with maximum fidelity.
The linear regression model of image restoration is：G=hf+n；Wherein, g is the blurred picture obtained, and f is original image,
H is fuzzy core, and n is random noise.
Since image restoration problem is an ill inversion process, it will cause the nonuniqueness without solution or solution.For this reason,
More technology uses full Variational Restoration method, i.e., is constrained using regularization method, image restoration problem is changed into good state and asks
Topic.
The energy minimization function of variational regularization is entirely：
Wherein, Section 1 is full variation (Total Variation, abbreviation TV) regular terms；Latter is fidelity item, is used
To weigh the error of reconstructed image, the fitting degree of reconstructed image is represented；λ is regular parameter, for balancing the proportion of items.
To numerous researchs based on Total Variation derived version, find regular parameter λ, play very important effect.
The principle of selection λ be should ensure restoration result and initial data have it is preferable coincide, while to be reduced as far as noise and
Ringing effect.Assuming that using the flatness of image as prior-constrained condition, λ chooses excessive, and regular terms accounts for the power of object function
Weight is larger, and smooth phenomenon occurred in reflection on the image；λ chooses too small, and regularization will not have prior-constrained work
With, it is little to eliminating ill-posed problem effect, so as to cause image to have certain ring and noise.Therefore, how certainly
Adaptively, it is one up for solving the problems, such as to restore clearly image while rapidly estimating λ.
The content of the invention
The object of the present invention is to provide a kind of image recovery method and system based on full variation, in known fuzzy core
In the case of, by introducing two auxiliary variables, using separating variables technology, overcome the non-differentiability of TV, in combination with
Morozov ' s deviation criterions, can be adaptively adjusted regular parameter λ, reduce artificial blindness, improve calculating speed, at the same time
Recover the image of high quality.
Technical solution provided by the invention is as follows：
A kind of image recovery method based on full variation, including：Step S100 is when obtaining blurred picture, initialization the 0th
For restored image f0, the 0th generation parameter information；The parameter information in the 0th generation includes：First auxiliary variable x0, second auxiliary
Variable y0, the first Lagrange multiplier u0, the second Lagrange multiplier ξ0；Step S200 setting iterationses k=0；Step
S300 calculates the generation restored image of kth+1 f according to the parameter information in kth generationk+1；Step S400 is restored according to the generation of kth+1
Image fk+1With the kth for restored image fk, judge whether to meet iteration stopping condition；Step S500 is if so, the then kth
+ 1 generation restored image fk+1For reconstructed image；Step S600 is if it is not, then calculate the parameter information in the generation of kth+1；The generation of kth+1
Parameter information includes：First auxiliary variable xk+1, the second auxiliary variable yk+1, regular parameter λk+1, the first Lagrange multiplier uk +1, the second Lagrange multiplier ξk+1；Step S700 updates iterations k=k+1, and jumps to step S300.
In the above-mentioned technical solutions, two auxiliary variables x and y are introduced, using separating variables technology, overcome TV can not
Micro- property, with reference to Lagrangian theory, by iteration, is adaptively adjusted regular parameter λ, reduces artificial blindness, improve and calculate
Speed, while recover the image of high quality.
Further, the step S300 includes：The generation of kth+1 restored image is calculated according to formula (1)：
fk+1=(β1hT-β2Δ)-1[hT(β1xk-uk)-div(β2yk-ξk)]………(1)
Wherein, hTFor the transposed matrix of fuzzy core h, β1For the default first positive parameter, β2For the default second positive parameter,
(β1hT-β2Δ)-1For (β1hT-β2Δ) invert, ukIt is first Lagrange multiplier in kth generation, ξkIt is second drawing in kth generation
Ge Lang multipliers, xkFor first auxiliary variable in kth generation, ykFor second auxiliary variable in kth generation, Δ is laplacian,
Div is divergence operator.
In the above-mentioned technical solutions, introduce and calculate fk+1Method；By introducing the auxiliary of the first auxiliary variable x and second
Variable y, only the first auxiliary variable x is related with λ, and fk+1It is unrelated with λ, so as to allow fk+1Easily update.
Further, the step S400 includes：According to the generation of kth+1 restored image fk+1With the kth for restored map
As fk, according to formula (2), calculate iteration error A：
A=| | fk+1-fk||2/||fk||2……………………(2)
Judge whether the iteration error is less than default iteration error threshold value；If, then it is assumed that meet iteration stopping bar
Part；If not, then it is assumed that be unsatisfactory for iteration stopping condition.
In the above-mentioned technical solutions, iteration stopping condition is given, when iteration stopping condition is met, the generation of kth+1
Restored image is reconstructed image.
Further, the step S600 includes：When being unsatisfactory for iteration stopping condition, calculate the auxiliary of the generation of kth+1 second and become
AmountWherein,For yk+1The i-th row, jth row element, i=1,2 ... M, j=1,
2,...N；
According to formula (3), calculate
Wherein,It is gradient operator,The i-th row, the element of jth row for the gradient of the generation original image of kth+1,Be second Lagrange multiplier in kth generation the i-th row, jth row element, β2For the default second positive parameter；
According to formula (4), the generation intermediate variable of kth+1 a is calculatedk+1；
ak+1=hfk+1+(uk/β1)……………(4)
Wherein, fk+1For the generation restored image of kth+1, ukIt is first Lagrange multiplier in kth generation, β1For default first just
Parameter；
Whether judgment formula (5) is set up；
Wherein, g is the blurred picture obtained, and c is default 3rd parameter；
When formula (5) is invalid, the generation regular parameter of kth+1 λ is calculated according to formula (6)k+1, and according to formula (7) meter
Calculate the generation of kth+1 first auxiliary variable xk+1；
xk+1=(λk+1g+β1ak+1)/(λk+1+β1)……………(7)
Wherein, g is the blurred picture obtained, and c is default 3rd parameter, β1For the default first positive parameter, ak+1For
The generation intermediate variable of kth+1；
The first Lagrange multiplier u in the generation of kth+1 is calculated according to formula (8)k+1；
uk+1=uk-β1(xk+1-hfk+1)……………(8)
Wherein, ukFor first Lagrange multiplier in kth generation, β1For the default first positive parameter, h is fuzzy core, fk+1For
The generation restored image of kth+1；
The second Lagrange multiplier ξ in the generation of kth+1 is calculated according to formula (9)k+1；
Wherein, ξkFor second Lagrange multiplier in kth generation, β2For the default second positive parameter,It is former for the generation of kth+1
The gradient of beginning image, yk+1For second auxiliary variable of the generation of kth+1.
Further, further include：When formula (5) is set up, then by the generation intermediate variable of kth+1 ak+1It is assigned to the generation of kth+1
One auxiliary variable xk+1, and by the generation regular parameter of kth+1 λk+1It is set to 0.
In the above-mentioned technical solutions, give the parameter information update method in the generation of kth+1, ensure that restored image into
Single-step iteration updates.
Further, when the image of shooting is coloured image, the coloured image is obtained respectively by tri- passages of RGB
The blurred picture of each passage, and obtain tri- passages of RBG according to step S100 to step S700 processing respectively and correspond to
Reconstructed image, then three respective reconstructed images of passage are combined to the coloured image to form reconstruct.
In the above-mentioned technical solutions, the restored method based on gray scale blurred picture, gives the recovery side of coloured image
Method.
The present invention also provides a kind of image restoration system based on full variation, it is characterised in that including：Initialization module,
For when obtaining blurred picture, initializing the 0th generation restored image f0, the 0th generation parameter information；The parameter letter in the 0th generation
Breath includes：First auxiliary variable x0, the second auxiliary variable y0, the first Lagrange multiplier u0, the second Lagrange multiplier ξ0；With
And setting iterations k=0；Computing module, is electrically connected with the initialization module, for the parameter information according to kth generation,
Calculate the generation restored image of kth+1；Judgment module, is electrically connected with the computing module, for according to the generation of kth+1 restored map
As fk+1With the kth for restored image fk, judge whether to meet iteration stopping condition；The computing module, is further used for
If so, the then generation of kth+1 restored image fk+1For reconstructed image；And if it is not, then calculate the parameter information in the generation of kth+1；Institute
Stating the parameter information in the generation of kth+1 includes：First auxiliary variable xk+1, the second auxiliary variable yk+1, regular parameter λk+1, first draw
Ge Lang multipliers uk+1, the second Lagrange multiplier ξk+1；And renewal iterations k=k+1, and recalculate the generation of kth+1 and answer
Original image.
In the above-mentioned technical solutions, two auxiliary variables x and y are introduced, using separating variables technology, overcome TV can not
Micro- property, with reference to Lagrangian theory, by iteration, is adaptively adjusted regular parameter λ, reduces artificial blindness, improve and calculate
Speed, while recover the image of high quality.
Further, the computing module is further used for calculating the generation of kth+1 restored image according to formula (1)：
fk+1=(β1hT-β2Δ)-1[hT(β1xk-uk)-div(β2yk-ξk)]………(1)
Wherein, hTFor the transposed matrix of fuzzy core h, β1For the default first positive parameter, β2For the default second positive parameter,
(β1hT-β2Δ)-1For (β1hT-β2Δ) invert, ukIt is first Lagrange multiplier in kth generation, ξkIt is second drawing in kth generation
Ge Lang multipliers, xkFor first auxiliary variable in kth generation, ykFor second auxiliary variable in kth generation, Δ is laplacian,
Div is divergence operator.
In the above-mentioned technical solutions, the method for introducing calculating；Become by introducing the auxiliary of the first auxiliary variable x and second
Y is measured, only the first auxiliary variable x is with related, and nothing to do with, is easily updated so as to allow.
Further, the judgment module is further used for according to the generation of kth+1 restored image fk+1It is multiple with kth generation
Original image fk, according to formula (2), calculate iteration error A；
A=| | fk+1-fk||2/||fk||2……………………(2)
And judge whether the iteration error is less than default iteration error threshold value；If, then it is assumed that meet that iteration is stopped
Only condition；If not, then it is assumed that be unsatisfactory for iteration stopping condition.If not, then it is assumed that be unsatisfactory for iteration stopping condition.
In the above-mentioned technical solutions, iteration stopping condition is given, when iteration stopping condition is met, the generation of kth+1
Restored image is reconstructed image.
Further, the computing module is further used for when being unsatisfactory for iteration stopping condition, calculates the generation of kth+1 second
Auxiliary variableWherein,For yk+1The i-th row, jth row element, i=1,2 ... M, j
=1,2 ... N；
And according to formula (3), calculate
Wherein,It is gradient operator,The i-th row, the element of jth row for the gradient of the generation original image of kth+1,Be second Lagrange multiplier in kth generation the i-th row, jth row element, β2For the default second positive parameter；
And according to formula (4), calculate the generation intermediate variable of kth+1 ak+1；
ak+1=hfk+1+(uk/β1)……………(4)
Wherein, fk+1For the generation restored image of kth+1, ukIt is first Lagrange multiplier in kth generation, β1For default first just
Parameter；
And whether judgment formula (5) is set up；
Wherein, g is the blurred picture obtained, and c is default 3rd parameter；
And when formula (5) is invalid, the generation regular parameter of kth+1 λ is calculated according to formula (6)k+1, and according to formula
(7) generation of kth+1 first auxiliary variable x is calculatedk+1；
xk+1=(λk+1g+β1ak+1)/(λk+1+β1)……………(7)
Wherein, g is the blurred picture obtained, and c is default 3rd parameter, β1For the default first positive parameter, ak+1For
The generation intermediate variable of kth+1；
And the first Lagrange multiplier u in the generation of kth+1 is calculated according to formula (8)k+1；
uk+1=uk-β1(xk+1-hfk+1)……………(8)
Wherein, ukFor first Lagrange multiplier in kth generation, β1For the default first positive parameter, h is fuzzy core, fk+1For
The generation restored image of kth+1；
And the second Lagrange multiplier ξ in the generation of kth+1 is calculated according to formula (9)k+1；
Wherein, ξkFor second Lagrange multiplier in kth generation, β2For the default second positive parameter,It is former for the generation of kth+1
The gradient of beginning image, yk+1For second auxiliary variable of the generation of kth+1.
In the above-mentioned technical solutions, give the parameter information update method in the generation of kth+1, ensure that restored image into
Single-step iteration updates.
By a kind of image recovery method and system based on full variation provided by the invention, can bring it is following at least
A kind of beneficial effect：
1st, the present invention is by introducing two auxiliary variables x and y, using separating variables technology, overcomes the non-differentiability of TV,
With reference to Lagrangian theory, by iteration, regular parameter λ is adaptively adjusted, reduces artificial blindness, improves calculating speed,
The image of high quality is recovered at the same time.
2nd, the restored method of the invention based on gray scale blurred picture, is equally applicable to multichannel image, such as coloured image
Restore.
Brief description of the drawings
Below by a manner of clearly understandable, preferred embodiment is described with reference to the drawings, to a kind of figure based on full variation
As above-mentioned characteristic, technical characteristic, advantage and its implementation of restored method and system are further described.
Fig. 1 is the flow chart of image recovery method one embodiment of the invention based on full variation；
Fig. 2 is the flow chart of the invention based on another embodiment of the image recovery method of full variation；
Fig. 3 is the flow chart of the invention based on another embodiment of the image recovery method of full variation；
Fig. 4 is the structure diagram of image restoration system one embodiment of the invention based on full variation；
Fig. 5 is the APE-ADMM methods and Wen- of image recovery method one embodiment of the invention based on full variation
The restoration result figure of Chan methods；
Fig. 6 is the APE-ADMM methods and Wen- of image recovery method one embodiment of the invention based on full variation
The PSNR- time diagrams of Chan methods；
Fig. 7 is the APE-ADMM methods and Wen- of image recovery method one embodiment of the invention based on full variation
Regular parameter-time diagram of Chan methods.
Drawing reference numeral explanation：
10. initialization module, 20. computing modules, 30. judgment modules.
Embodiment
In order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, control attached drawing is said below
Bright embodiment of the invention.It should be evident that drawings in the following description are only some embodiments of the present invention,
For those of ordinary skill in the art, without creative efforts, can also be obtained according to these attached drawings
Other attached drawings, and obtain other embodiments.
To make simplified form, part related to the present invention is only schematically show in each figure, they are not represented
Its practical structures as product.In addition, so that simplified form readily appreciates, there is identical structure or function in some figures
Component, only symbolically depict one of those, or only marked one of those.Herein, "one" not only table
Show " only this ", can also represent the situation of " more than one ".
In one embodiment of the invention, as shown in Figure 1, a kind of image recovery method based on full variation, including：
Step S100 initializes the 0th generation restored image f when obtaining blurred picture0, the 0th generation parameter information；
The parameter information in the 0th generation includes：First auxiliary variable x0, the second auxiliary variable y0, the first Lagrange multiplies
Sub- u0, the second Lagrange multiplier ξ0；
Step S200 setting iterationses k=0；
Step S300 calculates the generation restored image of kth+1 f according to the parameter information in kth generationk+1；
Step S400 is according to the generation of kth+1 restored image fk+1With the kth for restored image fk, judge whether to meet
Iteration stopping condition；
Step S500 is if so, the then generation of kth+1 restored image fk+1For reconstructed image；
Step S600 is if it is not, then calculate the parameter information in the generation of kth+1；
The parameter information in the generation of kth+1 includes：First auxiliary variable xk+1, the second auxiliary variable yk+1, regular parameter
λk+1, the first Lagrange multiplier uk+1, the second Lagrange multiplier ξk+1；
Step S700 updates iterations k=k+1, and jumps to step S300.
Specifically, blurred picture is gray level image.Assuming that g is the blurred picture obtained, h is fuzzy nuclear matrix；Initially
Change the 0th generation restored image f0=g, initializes the parameter information in the 0th generation：Such as the first auxiliary variable x0According to h*f0Obtain, the
Two auxiliary variable y0According toIt is calculated, the first Lagrange multiplier u0Set by unit matrix, the second Lagrange multiplies
Sub- ξ0Set by unit matrix.
Set iterations k=0；f0、x0、y0、u0、ξ0, be the 0th generation parameter information；Believed according to the parameter in the 0th generation
Breath, calculates 1st generation restored image f1, according to f1And f0, judge whether to meet iteration stopping condition；If so, then f1For reconstruct image
Picture, whole iteration terminate；If it is not, then calculate the parameter information of 1st generation：First auxiliary variable x1, the second auxiliary variable y1, canonical
Parameter lambda1, the first Lagrange multiplier u1, the second Lagrange multiplier ξ1；Iterations k=1 is updated, 2nd generation is calculated and restores
Image f2, according to f2And f1, judge whether to meet iteration stopping condition；If so, then f2For reconstructed image, whole iteration terminates；
If it is not, then calculate the parameter information of 2nd generation；Such iterative cycles, until meeting iteration stopping condition, the obtained generation of kth+1
Restored image fk+1As reconstructed image.
In another embodiment of the present invention, as shown in Fig. 2, a kind of image recovery method based on full variation, bag
Include：
Step S100 initializes the 0th generation restored image f when obtaining blurred picture0, the 0th generation parameter information；
The parameter information in the 0th generation includes：First auxiliary variable x0, the second auxiliary variable y0, the first Lagrange multiplies
Sub- u0, the second Lagrange multiplier ξ0；
Step S200 setting iterationses k=0；
Step S310 calculates the generation of kth+1 restored image f according to formula (1)k+1；
fk+1=(β1hT-β2Δ)-1[hT(β1xk-uk)-div(β2yk-ξk)]………(1)
Wherein, hTFor the transposed matrix of fuzzy core h, β1For the default first positive parameter, β2For the default second positive parameter,
(β1hT-β2Δ)-1For (β1hT-β2Δ) invert, ukIt is first Lagrange multiplier in kth generation, ξkIt is second drawing in kth generation
Ge Lang multipliers, xkFor first auxiliary variable in kth generation, ykFor second auxiliary variable in kth generation, Δ is laplacian,
Div is divergence operator；
Step S410 is according to the generation of kth+1 restored image fk+1With the kth for restored image fk, according to formula (2),
Calculate iteration error A；
A=| | fk+1-fk||2/||fk||2……………………(2)
Step S420 judges whether the iteration error is less than default iteration error threshold value；
Step S500 is if so, the then generation of kth+1 restored image fk+1For reconstructed image；
Step S610 is if it is not, then calculate second auxiliary variable of the generation of kth+1Wherein,For
yk+1The i-th row, jth row element, i=1,2 ... M, j=1,2 ... N；
According to formula (3), calculate
Wherein,It is gradient operator,The i-th row, the element of jth row for the gradient of the generation original image of kth+1,Be second Lagrange multiplier in kth generation the i-th row, jth row element, β2For the default second positive parameter；
Step S620 calculates the generation intermediate variable of kth+1 a according to formula (4)k+1；
ak+1=hfk+1+(uk/β1)……………(4)
Wherein, fk+1For the generation restored image of kth+1, ukIt is first Lagrange multiplier in kth generation, β1For default first just
Parameter；
Whether step S630 judgment formulas (5) are set up；
Wherein, g is the blurred picture obtained, and c is default 3rd parameter；
Step S640 calculates the generation regular parameter of kth+1 λ when formula (5) is invalid, according to formula (6)k+1, and according to public affairs
Formula (7) calculates the generation of kth+1 first auxiliary variable xk+1；
xk+1=(λk+1g+β1ak+1)/(λk+1+β1)……………(7)
Wherein, g is the blurred picture obtained, and c is default 3rd parameter, β1For the default first positive parameter, ak+1For
The generation intermediate variable of kth+1；
Step S650 is when formula (5) is set up, then by the generation intermediate variable of kth+1 ak+1Being assigned to the generation of kth+1 first aids in
Variable xk+1, and by the generation regular parameter of kth+1 λk+1It is set to 0；
Step S660 calculates the first Lagrange multiplier u in the generation of kth+1 according to formula (8)k+1；
uk+1=uk-β1(xk+1-hfk+1)……………(8)
Wherein, ukFor first Lagrange multiplier in kth generation, β1For the default first positive parameter, h is fuzzy core, fk+1For
The generation restored image of kth+1；
Step S670 calculates the second Lagrange multiplier ξ in the generation of kth+1 according to formula (9)k+1；
Wherein, ξkFor second Lagrange multiplier in kth generation, β2For the default second positive parameter,It is former for the generation of kth+1
The gradient of beginning image, yk+1For second auxiliary variable of the generation of kth+1；
Step S710 updates iterations k=k+1, and jumps to step S310.
Specifically, full Variational Image Restoration problem is to find suitable f, meet following formula：
Wherein, Section 1 is TV regular terms, and latter is fidelity item, and λ is regular parameter.
The full Variational Calculation formula of image f is：
Wherein, Ω is the support region of f.
According to Morozov ' s deviation criterions, the above problem is equivalent to following constrained optimum problem：
Wherein, c is default 3rd parameter, according to c=τ mn σ2Calculate, σ2For noise variance, τ is one dependent on making an uproar
The parameter of sound, such as, it is set to 1；M, n is the dimension of original image.
Using variable separation, two auxiliary variables are introduced, is the second auxiliary variable y, y ∈ Q respectively, is equal toWith
First auxiliary variable x, x ∈ V, equal to hf, wherein, V represents theorem in Euclid space Rmm, Q=V × V, so as to obtain equivalent constraint form
For：
Lagrangian (Augmented Lagrangian, the abbreviation AL) functional of the augmentation of formula (10) is defined as：
Wherein, u is the first Lagrange multiplier, uTIt is the transposed matrix of u, ξ is the second Lagrange multiplier, ξTIt is ξ
Transposed matrix, | | y | | it is the L1 norms of y, β1For the default first positive parameter, β2For the default second positive parameter；
The minimization problem of formula (10) is switched to the saddle of solution formula (11) augmentation Lagranian functional
Point, specific iteration frame are as follows：
Due to formula (12), iteration is directed to inner iteration every time, so solution (fk+1, xk+1, yk+1) it is comparatively laborious.
Therefore this is solved the problems, such as using improved alternative manner, i.e., only needs a step computing can be in the hope of solution in each iteration
(fk+1, xk+1, yk+1)。
Improved alternative manner is as follows：
Formula (13) subproblem, is the minimization problem on f, is a quadratic equation, form is such as shown in (16)；Root
Formula (1) is obtained according to formula (16)：
fk+1=(β1hT-β2Δ)-1[hT(β1xk-uk)-div(β2yk-ξk)]………………(1)
Wherein,It is laplacian；For the image of m × n size, by FFT twice and once
Inverse FFT, you can try to achieve the solution of formula (1), its algorithm complex is O (mnlog (mn)).
Formula (14) subproblem, is the minimization problem on y：
This formula is that near-end minimizes (Proximal Minimization, PM) problem, its solution is tried to achieve using two-dimensional contraction
For：
Formula (15) subproblem, is the minimization problem on x, wherein, first calculate the generation intermediate variable of kth+1 ak+1：
ak+1=hfk+1+(uk/β1)…………………(4)
Formula (17) is obtained by solution formula (15)：
Formula (17) is a quadratic minimization problem on x, and it has the solution of a closed form：
xk+1=(λk+1g+β1ak+1)/(λk+1+β1) ... ... ... ... (7) are according to ak+1Scope, λ in each iteration
Solution there are two kinds of situations：A kind of situation is
λ at this timek+1=0 while xk+1=ak+1, it is evident that xk+1Meet Morozov ' s deviation criterions；Another situation isAccording to Morozov ' s deviation criterions, following equation should be solved：
By the x in formula (7)k+1Formula (18) is substituted into, is obtained：
In this way, releasing out from stringent Morozov ' s deviation criterions by introducing auxiliary variable x, hf, benefit
In this, the λ that can obtain a closed form without additional conditions in each iteration is solved.
The generation restored image of kth+1 f is being calculated according to formula (1)k+1Afterwards, iteration error can be calculated according to formula (2)
A；
Judge whether to need to continue iteration；
When iteration error is less than default iteration error threshold value, then the generation restored image of kth+1 fk+1For reconstructed image, entirely
Iteration terminates；Default iteration error threshold value, is rule of thumb set, such as 10-5。
When iteration error is not less than default iteration error threshold value, it can determine whether iterations is more than
In default total iterations；When iterations reaches default total iterations, then whole iteration, f are terminatedk+1For reconstruct image
Picture.
When iteration error is not less than default iteration error threshold value, and iterations not up to presets total iterations, need
Update iterations, further iteration.
Above is for the processing in the case of known fuzzy core, for the blurred picture that truly shoots, it is necessary to using current
Existing method such as recognition status, first carries out the estimation of fuzzy core.
To this method, (the TV models based on original-dual problem, solve saddle point, Ran Houli first with Wen-Chan methods
Optimal solution is solved with original-antithesis proximal point algorithm) it is tested and compares：
(a) of Fig. 5 is 300 × 300 original image；(b) of Fig. 5 is blurred picture, by the motion blur in the lower left corner
The result of (fuzzy core size is 53 × 53) and gaussian random noise that variance is 1；(c) of Fig. 5 is to utilize Wen-Chan methods
The result of recovery；(d) of Fig. 5 is the result that context of methods (referred to as APE-ADMM) restores；Both convergent conditions are 10-5,
Wherein, APE-ADMM iteration 50 times, Wen-Chan method iteration 69 times.
From the details enlarged drawing in the lower left corner it is apparent that (d) figure details of Fig. 5 becomes apparent from, relatively connect with artwork
Closely, and (c) figure of Fig. 5 shows slightly smooth.
Fig. 6 is the Y-PSNR PSNR (Peak Signal to Noise Ratio) of two methods with the change of time
Change relation, it is seen that APE-ADMM methods can reach higher PSNR with faster speed；
(f) of Fig. 7 is that the regular parameter of APE-ADMM methods changes with time relation, and (g) is Wen-Chan methods
Regular parameter changes with time relation, is understood from figure, and the regular parameter that APE-ADMM methods are estimated about becomes near 1.8s
In stabilization, and Wen-Chan methods are almost in 4.4s or so, it is seen that this method is better than Wen-Chan methods in terms of speed.
Table 1 is the contrast of the objective evaluation parameter of two methods, and objective evaluation parameter includes Y-PSNR PSNR, knot
Structure similitude SSIM (structural similarity index measurement)；As it can be seen from table 1 APE-ADMM
Algorithm is better than Wen-Chan methods.
Table 1
In another embodiment of the present invention, as shown in figure 3, a kind of image recovery method based on full variation, except with
It is above-mentioned it is identical outside, further include：
Step S800 obtains the coloured image respectively when the image of shooting is coloured image, by tri- passages of RGB
The blurred picture of each passage；
Step S810 by the blurred picture of three passages respectively according to step S100 to step S700 processing,
Obtain the corresponding reconstructed image of tri- passages of RBG；
The respective reconstructed image of three passages is combined the coloured image to form reconstruct by step S820.
Specifically, when the image of shooting is coloured image, the coloured image is obtained respectively by tri- passages of RGB
The blurred picture of each passage；According to previous embodiment, restoration disposal is carried out to the blurred picture of three passages respectively,
Obtain corresponding reconstructed image；Three respective reconstructed images of passage are formed to the cromogram of reconstruct according to RGB combination again
Picture.
In another embodiment of the present invention, as shown in figure 4, a kind of image restoration system based on full variation, bag
Include：
Initialization module 10, for when obtaining blurred picture, initializing the 0th generation restored image f0, the 0th generation parameter
Information；The parameter information in the 0th generation includes：First auxiliary variable x0, the second auxiliary variable y0, the first Lagrange multiplier
u0, the second Lagrange multiplier ξ0；And setting iterations k=0；
Computing module 20, is electrically connected with the initialization module 10, for the parameter information according to kth generation, calculate kth+
1 generation restored image；
Judgment module 30, is electrically connected with the computing module 20, for according to the generation of kth+1 restored image fk+1And institute
Kth is stated for restored image fk, judge whether to meet iteration stopping condition；
The computing module 20, is further used for if so, the then generation of kth+1 restored image fk+1For reconstructed image；With
And if it is not, then calculate the parameter information in the generation of kth+1；The parameter information in the generation of kth+1 includes：First auxiliary variable xk+1,
Two auxiliary variable yk+1, regular parameter λk+1, the first Lagrange multiplier uk+1, the second Lagrange multiplier ξk+1；And renewal changes
Generation number k=k+1, and recalculate the generation restored image of kth+1.
Specifically, blurred picture is gray level image.Assuming that g is the blurred picture obtained, h is fuzzy nuclear matrix；Initially
Change the 0th generation restored image f0=g, initializes the parameter information in the 0th generation：Wherein the first auxiliary variable x0According to h*f0Obtain, the
Two auxiliary variable y0According toIt is calculated, the first Lagrange multiplier u0Set by unit matrix, the second Lagrange multiplies
Sub- ξ0Set by unit matrix.
Set iterations k=0；f0、x0、y0、u0、ξ0, be the 0th generation parameter information；Believed according to the parameter in the 0th generation
Breath, calculates 1st generation restored image f1, according to f1And f0, judge whether to meet iteration stopping condition；If so, then f1For reconstruct image
Picture, whole iteration terminate；If it is not, then calculate the parameter information of 1st generation：First auxiliary variable x1, the second auxiliary variable y1, canonical
Parameter lambda1, the first Lagrange multiplier u1, the second Lagrange multiplier ξ1；Iterations k=1 is updated, 2nd generation is calculated and restores
Image f2, according to f2And f1, judge whether to meet iteration stopping condition；If so, then f2For reconstructed image, whole iteration terminates；
If it is not, then calculate the parameter information of 2nd generation；Such iterative cycles, until meeting iteration stopping condition, the obtained generation of kth+1
Restored image fk+1As reconstructed image.
In another embodiment of the present invention, as shown in figure 4, a kind of image restoration system based on full variation, bag
Include：
Initialization module 10, for when obtaining blurred picture, initializing the 0th generation restored image f0, the 0th generation parameter
Information；The parameter information in the 0th generation includes：First auxiliary variable x0, the second auxiliary variable y0, the first Lagrange multiplier
u0, the second Lagrange multiplier ξ0；And setting iterations k=0；And setting iterations k=0；
Computing module 20, is electrically connected with the initialization module 10, for being answered according to formula (1) the calculating generation of kth+1
Original image fk+1：
fk+1=(β1hT-β2Δ)-1[hT(β1xk-uk)-div(β2yk-ξk)]………(1)
Wherein, hTFor the transposed matrix of fuzzy core h, β1For the default first positive parameter, β2For the default second positive parameter,
(β1hT-β2Δ)-1For (β1hT-β2Δ) invert, ukIt is first Lagrange multiplier in kth generation, ξkIt is second drawing in kth generation
Ge Lang multipliers, xkFor first auxiliary variable in kth generation, ykFor second auxiliary variable in kth generation, Δ is laplacian,
Div is divergence operator；
Judgment module 30, is electrically connected with the computing module 20, for according to the generation of kth+1 restored image fk+1And institute
Kth is stated for restored image fk, according to formula (2), calculate iteration error A；
A=| | fk+1-fk||2/||fk||2……………………(2)
And judge whether the iteration error is less than default iteration error threshold value；If, then it is assumed that meet that iteration is stopped
Only condition；If not, then it is assumed that be unsatisfactory for iteration stopping condition；
The computing module, is further used for when meeting iteration stopping condition, then the generation of kth+1 restored image fk+1
For reconstructed image；And when being unsatisfactory for iteration stopping condition, then calculate second auxiliary variable of the generation of kth+1Wherein,For yk+1The i-th row, jth
According to formula (3), calculate
Wherein,It is gradient operator,The i-th row, the element of jth row for the gradient of the generation original image of kth+1,Be second Lagrange multiplier in kth generation the i-th row, jth row element, β2For the default second positive parameter；
And according to formula (4), calculate the generation intermediate variable of kth+1 ak+1；
ak+1=hfk+1+(uk/β1)……………(4)
Wherein, fk+1For the generation restored image of kth+1, ukIt is first Lagrange multiplier in kth generation, β1For default first
Positive parameter；
And whether judgment formula (5) is set up；
Wherein, g is the blurred picture obtained, and c is default 3rd parameter；
And when formula (5) is invalid, the generation regular parameter of kth+1 λ is calculated according to formula (6)k+1, and according to formula
(7) generation of kth+1 first auxiliary variable x is calculatedk+1；
xk+1=(λk+1g+β1ak+1)/(λk+1+β1)……………(7)
Wherein, g is the blurred picture obtained, and c is default 3rd parameter, β1For the default first positive parameter, ak+1For
The generation intermediate variable of kth+1；
And the first Lagrange multiplier u in the generation of kth+1 is calculated according to formula (8)k+1；
uk+1=uk-β1(xk+1-hfk+1)……………(8)
Wherein, ukFor first Lagrange multiplier in kth generation, β1For the default first positive parameter, h is fuzzy core, fk+1For
The generation restored image of kth+1；
And the second Lagrange multiplier ξ in the generation of kth+1 is calculated according to formula (9)k+1；
Wherein, ξkFor second Lagrange multiplier in kth generation, β2For the default second positive parameter,It is former for the generation of kth+1
The gradient of beginning image, yk+1For second auxiliary variable of the generation of kth+1；
And renewal iterations k=k+1, and recalculate the generation restored image of kth+1；
The computing module, is further used for when formula (5) is set up, then by the generation intermediate variable of kth+1 ak+1It is assigned to
The generation of kth+1 first auxiliary variable xk+1, and by the regular parameter λ in the generation of kth+1k+1It is set to 0.
Specifically, full Variational Image Restoration problem is to find suitable f, meet following formula：
Wherein, Section 1 is TV regular terms, and latter is fidelity item, and λ is regular parameter.
The full Variational Calculation formula of image f is：
Wherein, Ω is the support region of f.
According to Morozov ' s deviation criterions, the above problem is equivalent to following constrained optimum problem：
Wherein, c is default 3rd parameter, according to c=τ mn σ2Calculate, σ2For noise variance, τ is one dependent on making an uproar
The parameter of sound, such as, it is set to 1；M, n is the dimension of original image.
Using variable separation, two auxiliary variables are introduced, is the second auxiliary variable y, y ∈ Q respectively, is equal toWith
First auxiliary variable x, x ∈ V, equal to hf, wherein, V represents theorem in Euclid space Rmm, Q=V × V, so as to obtain equivalent constraint form
For：
Lagrangian (Augmented Lagrangian, the abbreviation AL) functional of the augmentation of formula (10) is defined as：
Wherein, u is the first Lagrange multiplier, uTIt is the transposed matrix of u, ξ is the second Lagrange multiplier, ξTIt is ξ
Transposed matrix, | | y | | it is the L1 norms of y, β1For the default first positive parameter, β2For the default second positive parameter；
The minimization problem of formula (10) is switched to the saddle point of solution formula (11) augmentation Lagranian functional, it is specific to change
It is as follows for frame：
Due to formula (12), iteration is directed to inner iteration every time, so solution (fk+1,xk+1,yk+1) it is comparatively laborious.
Therefore this is solved the problems, such as using improved alternative manner, i.e., only needs a step computing can be in the hope of solution in each iteration
(fk+1,xk+1,yk+1)。
Improved alternative manner is as follows：
Formula (13) subproblem, is the minimization problem on f, is a quadratic equation, form is such as shown in (16)；Root
Formula (1) is obtained according to formula (16)：
fk+1=(β1hT-β2Δfk)-1[hT(β1xk-uk)-div(β2yk-ξk)]………………(1)
Wherein,It is laplacian；For the image of m × n size, by FFT twice and once
Inverse FFT, you can try to achieve the solution of formula (1), its algorithm complex is O (mn log (mn)).
Formula (14) subproblem, is the minimization problem on y：
This formula is that near-end minimizes (Proximal Minimization, PM) problem, its solution is tried to achieve using two-dimensional contraction
For：
Formula (15) subproblem, is the minimization problem on x, wherein, first calculate the generation intermediate variable of kth+1 ak+1：
ak+1=hfk+1+(uk/β1)…………………(4)
Formula (17) is obtained by solution formula (15)：
Formula (17) is a quadratic minimization problem on x, and it has the solution of a closed form：
xk+1=(λk+1g+β1ak+1)/(λk+1+β1) ... ... ... ... (7) are according to ak+1Scope, λ in each iteration
Solution there are two kinds of situations：A kind of situation is
λ at this timek+1=0 while xk+1=ak+1, it is evident that xk+1Meet Morozov ' s deviation criterions；Another situation isAccording to Morozov ' s deviation criterions, following equation should be solved：
By the x in formula (7)k+1Formula (18) is substituted into, is obtained：
In this way, releasing out from stringent Morozov ' s deviation criterions by introducing auxiliary variable x, hf, benefit
In this, the λ that can obtain a closed form without additional conditions in each iteration is solved.
The generation restored image of kth+1 f is being calculated according to formula (1)k+1Afterwards, iteration error can be calculated according to formula (2)
A；
A=| | fk+1-fk||2/||fk||2……………………(2)
Judge whether to need to continue iteration；
When iteration error is less than default iteration error threshold value, then the generation restored image of kth+1 fk+1For reconstructed image, entirely
Iteration terminates；Default iteration error threshold value, is rule of thumb set, such as 10-5。
When iteration error is not less than default iteration error threshold value, it can determine whether iterations is more than
In default total iterations；When iterations reaches default total iterations, then whole iteration, f are terminatedk+1For reconstruct image
Picture.
When iteration error is not less than default iteration error threshold value, and iterations not up to presets total iterations, need
Update iterations, further iteration.
Above is for the processing in the case of known fuzzy core, for the blurred picture that truly shoots, it is necessary to using current
Existing method such as recognition status, first carries out the estimation of fuzzy core.
It should be noted that above-described embodiment can be freely combined as needed.The above is only the preferred of the present invention
Embodiment, it is noted that for those skilled in the art, do not departing from the premise of the principle of the invention
Under, some improvements and modifications can also be made, these improvements and modifications also should be regarded as protection scope of the present invention.
Claims (10)
- A kind of 1. image recovery method based on full variation, it is characterised in that including：Step S100 initializes the 0th generation restored image f when obtaining blurred picture0, the 0th generation parameter information；The parameter information in the 0th generation includes：First auxiliary variable x0, the second auxiliary variable y0, the first Lagrange multiplier u0、 Second Lagrange multiplier ξ0；Step S200 setting iterationses k=0；Step S300 calculates the generation restored image of kth+1 f according to the parameter information in kth generationk+1；Step S400 is according to the generation of kth+1 restored image fk+1With the kth for restored image fk, judge whether to meet that iteration is stopped Only condition；Step S500 is if so, the then generation of kth+1 restored image fk+1For reconstructed image；Step S600 is if it is not, then calculate the parameter information in the generation of kth+1；The parameter information in the generation of kth+1 includes：First auxiliary variable xk+1, the second auxiliary variable yk+1, regular parameter λk+1, One Lagrange multiplier uk+1, the second Lagrange multiplier ξk+1；Step S700 updates iterations k=k+1, and jumps to step S300.
- 2. the image recovery method according to claim 1 based on full variation, it is characterised in that the step S300 bags Include：The generation of kth+1 restored image f is calculated according to formula (1)k+1：fk+1=(β1hT-β2Δ)-1[hT(β1xk-uk)-div(β2yk-ξk)]………(1)Wherein, hTFor the transposed matrix of fuzzy core h, β1For the default first positive parameter, β2For the default second positive parameter, (β1hT- β2Δ)-1For (β1hT-β2Δ) invert, ukIt is first Lagrange multiplier in kth generation, ξkIt is that second Lagrange in kth generation multiplies Son, xkFor first auxiliary variable in kth generation, ykFor second auxiliary variable in kth generation, Δ is laplacian, and div is divergence Operator.
- 3. the image recovery method according to claim 1 based on full variation, it is characterised in that the step S400 bags Include：According to the generation of kth+1 restored image fk+1With the kth for restored image fk, according to formula (2), calculate iteration error A：A=| | fk+1-fk||2/||fk||2……………………(2)Judge whether the iteration error is less than default iteration error threshold value；If, then it is assumed that meet iteration stopping condition；If not, then it is assumed that be unsatisfactory for iteration stopping condition.
- 4. the image recovery method according to claim 1 based on full variation, it is characterised in that the step S600 bags Include：When being unsatisfactory for iteration stopping condition, second auxiliary variable of the generation of kth+1 is calculatedWherein,For yk +1The i-th row, the element of jth row, i=1,2 ..., M, j=1,2 ..., N；According to formula (3), calculateWherein, ▽ is gradient operator, (▽ fk+1)i,jThe i-th row, the element of jth row for the gradient of the generation original image of kth+1, Be second Lagrange multiplier in kth generation the i-th row, jth row element, β2For the default second positive parameter；According to formula (4), the generation intermediate variable of kth+1 a is calculatedk+1；ak+1=hfk+1+(uk/β1)……………(4)Wherein, fk+1For the generation restored image of kth+1, ukIt is first Lagrange multiplier in kth generation, β1For the default first positive ginseng Number；Whether judgment formula (5) is set up；Wherein, g is the blurred picture obtained, and c is default 3rd parameter；When formula (5) is invalid, the generation regular parameter of kth+1 λ is calculated according to formula (6)k+1, and according to formula (7) calculate kth+ 1 generation the first auxiliary variable xk+1；xk+1=(λk+1g+β1ak+1)/(λk+1+β1)……………(7)Wherein, g is the blurred picture obtained, and c is default 3rd parameter, β1For the default first positive parameter, ak+1For the generation of kth+1 Intermediate variable；The first Lagrange multiplier u in the generation of kth+1 is calculated according to formula (8)k+1；uk+1=uk-β1(xk+1-hfk+1)……………(8)Wherein, ukFor first Lagrange multiplier in kth generation, β1For the default first positive parameter, h is fuzzy core, fk+1For kth+1 For restored image；The second Lagrange multiplier ξ in the generation of kth+1 is calculated according to formula (9)k+1；ξk+1=ξk-β2(yk+1-▽fk+1)……………(9)Wherein, ξkFor second Lagrange multiplier in kth generation, β2For the default second positive parameter, ▽ fk+1For the generation original graph of kth+1 The gradient of picture, yk+1For second auxiliary variable of the generation of kth+1.
- 5. the image recovery method according to claim 4 based on full variation, it is characterised in that：When formula (5) is set up, then by the generation intermediate variable of kth+1 ak+1It is assigned to the generation of kth+1 first auxiliary variable xk+1, and by K+1 is for regular parameter λk+1It is set to 0.
- 6. the image recovery method according to claim 1 based on full variation, it is characterised in that：When the image of shooting is coloured image, the mould of each passage of the coloured image is obtained respectively by tri- passages of RGB Image is pasted, and obtains the corresponding reconstructed image of tri- passages of RBG according to step S100 to step S700 processing respectively, then Three respective reconstructed images of passage are combined to the coloured image to form reconstruct.
- A kind of 7. image restoration system based on full variation, it is characterised in that including：Initialization module, for when obtaining blurred picture, initializing the 0th generation restored image f0, the 0th generation parameter information；It is described The parameter information in the 0th generation includes：First auxiliary variable x0, the second auxiliary variable y0, the first Lagrange multiplier u0, the second glug Bright day multiplier ξ0；And setting iterations k=0；Computing module, is electrically connected with the initialization module, for the parameter information according to kth generation, calculates the generation restored map of kth+1 Picture；Judgment module, is electrically connected with the computing module, for according to the generation of kth+1 restored image fk+1It is multiple with kth generation Original image fk, judge whether to meet iteration stopping condition；The computing module, is further used for if so, the then generation of kth+1 restored image fk+1For reconstructed image；And if it is not, Then calculate the parameter information in the generation of kth+1；The parameter information in the generation of kth+1 includes：First auxiliary variable xk+1, second auxiliary become Measure yk+1, regular parameter λk+1, the first Lagrange multiplier uk+1, the second Lagrange multiplier ξk+1；And renewal iterations k= K+1, and recalculate the generation restored image of kth+1.
- 8. the image restoration system according to claim 7 based on full variation, it is characterised in that：The computing module is further used for calculating the generation of kth+1 restored image f according to formula (1)k+1：fk+1=(β1hT-β2Δ)-1[hT(β1xk-uk)-div(β2yk-ξk)]………(1)Wherein, hTFor the transposed matrix of fuzzy core h, β1For the default first positive parameter, β2For the default second positive parameter, (β1hT- β2Δ)-1For (β1hT-β2Δ) invert, ukIt is first Lagrange multiplier in kth generation, ξkIt is that second Lagrange in kth generation multiplies Son, xkFor first auxiliary variable in kth generation, ykFor second auxiliary variable in kth generation, Δ is laplacian, and div is divergence Operator.
- 9. the image restoration system according to claim 7 based on full variation, it is characterised in that：The judgment module is further used for according to the generation of kth+1 restored image fk+1With the kth for restored image fk, according to Formula (2), calculates iteration error A；And judge whether the iteration error is less than default iteration error threshold value；If, then it is assumed that meet iteration stopping bar Part；If not, then it is assumed that be unsatisfactory for iteration stopping condition.
- 10. the image restoration system according to claim 7 based on full variation, it is characterised in that：The computing module is further used for when being unsatisfactory for iteration stopping condition, calculates second auxiliary variable of the generation of kth+1Wherein,For yk+1The i-th row, jth row element, i=1,2 ... M, j=1,2, ...N；And according to formula (3), calculateWherein, ▽ is gradient operator, (▽ fk+1)i,jThe i-th row, the element of jth row for the gradient of the generation original image of kth+1, Be second Lagrange multiplier in kth generation the i-th row, jth row element, β2For the default second positive parameter；And according to formula (4), calculate the generation intermediate variable of kth+1 ak+1；ak+1=hfk+1+(uk/β1)……………(4)Wherein, fk+1For the generation restored image of kth+1, ukIt is first Lagrange multiplier in kth generation, β1For the default first positive ginseng Number；And whether judgment formula (5) is set up；Wherein, g is the blurred picture obtained, and c is default 3rd parameter；And when formula (5) is invalid, the generation regular parameter of kth+1 λ is calculated according to formula (6)k+1, and according to formula (7) meter Calculate the generation of kth+1 first auxiliary variable xk+1；xk+1=(λk+1g+β1ak+1)/(λk+1+β1)……………(7)Wherein, g is the blurred picture obtained, and c is default 3rd parameter, β1For the default first positive parameter, ak+1For the generation of kth+1 Intermediate variable；And the first Lagrange multiplier u in the generation of kth+1 is calculated according to formula (8)k+1；uk+1=uk-β1(xk+1-hfk+1)……………(8)Wherein, ukFor first Lagrange multiplier in kth generation, β1For the default first positive parameter, h is fuzzy core, fk+1For kth+1 For restored image；And the second Lagrange multiplier ξ in the generation of kth+1 is calculated according to formula (9)k+1；ξk+1=ξk-β2(yk+1-▽fk+1)……………(9)Wherein, ξkFor second Lagrange multiplier in kth generation, β2For the default second positive parameter, ▽ fk+1For the generation original graph of kth+1 The gradient of picture, yk+1For second auxiliary variable of the generation of kth+1.
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| CN111986123A (en) * | 2020-09-24 | 2020-11-24 | 南京航空航天大学 | Based on Kullback-Leibler divergence and l0Norm-constrained blurred image restoration method |
| CN112215779A (en) * | 2020-10-28 | 2021-01-12 | 青岛大学 | Image processing method, device, equipment and computer readable storage medium |
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| CN113570522B (en) * | 2021-07-28 | 2024-04-16 | 南京航空航天大学 | Haze image restoration method and terminal based on mixed constraint condition |
| CN114022387A (en) * | 2021-11-11 | 2022-02-08 | 上海联影医疗科技股份有限公司 | Image optimization method and device based on total variation minimization and computer equipment |
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| JP4177128B2 (en) * | 2003-01-29 | 2008-11-05 | 三菱電機株式会社 | Radar image processing device |
| RU2629432C2 (en) * | 2011-11-23 | 2017-08-29 | Конинклейке Филипс Н.В. | Removal of noise in image area |
| CN104751420B (en) * | 2015-03-06 | 2017-12-26 | 湖南大学 | A kind of blind restoration method based on rarefaction representation and multiple-objection optimization |
| CN106780645B (en) * | 2016-11-28 | 2020-02-14 | 四川大学 | Dynamic MRI image reconstruction method and device |
| CN106934778B (en) * | 2017-03-10 | 2019-11-29 | 北京工业大学 | A kind of MR image rebuilding method based on small echo domain structure and non local grouping sparsity |
| CN107945121A (en) * | 2017-11-06 | 2018-04-20 | 上海斐讯数据通信技术有限公司 | A kind of image recovery method and system based on full variation |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| WO2019085433A1 (en) * | 2017-11-06 | 2019-05-09 | 上海斐讯数据通信技术有限公司 | Total variation-based image restoration method and system |
| CN111986123A (en) * | 2020-09-24 | 2020-11-24 | 南京航空航天大学 | Based on Kullback-Leibler divergence and l0Norm-constrained blurred image restoration method |
| CN111986123B (en) * | 2020-09-24 | 2024-03-12 | 南京航空航天大学 | Blurred image restoration method based on KL divergence and L0 norm constraint |
| CN112215779A (en) * | 2020-10-28 | 2021-01-12 | 青岛大学 | Image processing method, device, equipment and computer readable storage medium |
| CN112215779B (en) * | 2020-10-28 | 2023-10-03 | 青岛大学 | Image processing method, device, equipment and computer readable storage medium |
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