CN105957022A - Recovery method of low-rank matrix reconstruction with random value impulse noise deletion image - Google Patents
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Abstract
The present invention belongs to the field of the computer vision, and the objective of the invention is to realize the recovery of the image with random value impulse noise structural deficiency. The method comprises: combining the matrix reconstruction theory and the sparse expression theory, leading into a dictionary learning model on the basis of the traditional matrix reconstruction model, and therefore solving the problems which cannot be solved in the prior art. The method comprises the following steps: 1) taking the image as a matrix, expressing an original image by using a matrix A, and solving the deletion image recovery problem with random value pulse noise to solve the following optimization equation; 2) training a dictionary [Phi]; 3) converting a sequence to another sequence for solution by using an alternative direction method (ADM), wherein a contraction operator is included; and performing iteration solution to obtain a final result according to the steps. The recovery method of low-rank matrix reconstruction with a random value impulse noise deletion image is mainly applied to the computer image processing.
Description
Technical field
The invention belongs to computer vision field.Particularly to rebuild based on low-rank matrix with Random Valued Impulse Noise
Missing image restoration methods.
Background technology
Matrix Problems of Reconstruction is broadly divided into matrix fill-in and matrix recovers, and has received much attention since proposing the most always, and tool
There is the strongest vitality.Under the historical background of big data, especially increasingly constituted the study hotspot of mathematics and computer realm.
In recent years, the algorithm about solving matrix Problems of Reconstruction has had a lot of achievement in research.Owing to rank of matrix minimization problem is
One non-convex optimization problem, therefore current algorithm approaches original model mainly by the method for the singular value decomposition of iteration
Solve.Such as SVT (singular value threshold value) algorithm, APG (accelerating neighbour's gradient) algorithm, ALM (augmentation Lagrange multiplier) algorithm etc..?
In existing algorithm, when solving matrix fills problem, the internal memory that SVT algorithm needs during programming realization is the least, so ratio
The situation being relatively suitable for extensive matrix uses, but SVT is only applicable to the low-down situation of rank of matrix.APG is (fast from FISTA
Speed iterative shrinkage threshold value) algorithm change obtains, and its convergence rate is more a lot of than SVT scheduling algorithm.Problem is recovered at solving matrix
Time, SVT and APG algorithm still can have good performance, but from the point of view of convergence rate, these algorithms are all sublinears
's.ALM algorithm has convergence rate faster by contrast.
Image completion and image denoising are all problems critically important in computer vision field, digital picture acquisition and
In transmitting procedure, sensor is the most not normal or transmission channel noise all can make image face the damages such as disappearance and noise,
In numerous noises, Random Valued Impulse Noise is common a kind of noise.Therefore, digital picture suffers disappearance with random the most simultaneously
The damage of value impulsive noise, it is also desirable to be simultaneously filled with recovering with denoising.For image completion problem, existing at present
Algorithm such as SVT algorithm, APG algorithm and ALM algorithm etc. be all utilize image low-rank characteristic by disappearance pixel filling good.But
When in image, pixel is structural disappearance such as wire disappearance, and even during full line disappearance, existing algorithm can not solve this figure
As filling problem.Because the matrix that they are acted on has a common feature, i.e. the element vacant locations of observing matrix is dilute
Dredge and random.This has certain effect for the image completion of missing at random, but for the completeest during vacancy full line element
The most inoperative.Because it is to solve that the matrix of a large amount of full line element vacancies recovers only to apply low-rank condition to carry out restriction
's.And also polluted by Random Valued Impulse Noise while image suffers structural disappearance, then problem the most more can solve.Along with
The arrival of big data age, quantity of information sharp increase, image is likely to meet with structural disappearance with random in transmitting procedure
The pollution of value impulsive noise.Therefore a kind of algorithm that can solve the problem that the missing image with Random Valued Impulse Noise recovers is designed
It is necessary.
Summary of the invention
The invention is intended to make up the deficiencies in the prior art, i.e. realize the structural disappearance with Random Valued Impulse Noise
Picture recovers.The present invention adopts the technical scheme that, low-rank matrix is rebuild extensive with Random Valued Impulse Noise missing image
Compound recipe method, combines matrix reconstruction theory with sparse representation theory, introduces dictionary on the basis of traditional matrix reconstruction model
Learning model, thus solve the problem that prior art cannot process.The present invention comprises the following steps:
1) regard image as matrix, then original image matrix A represents, solves the missing image with Random Valued Impulse Noise
Recovery problem is for solving following optimization method:
Wherein | | A | |*The nuclear norm of representing matrix A, | | | |1One norm of representing matrix, Ω is observation space, pΩ
() is projection operator, represents that variable projects the value in spatial domain Ω.λ and γ is weight coefficient, and the Φ in constraints is
The dictionary trained, B is the coefficient that dictionary is corresponding, D be known to degrade observing matrix i.e. damaged image, E represents damaged image
The pixel of middle disappearance and noise；
When solving this equation, using augmented vector approach, equation is as follows:
L(A,B,E,Y1,Y2) it is Augmented Lagrangian Functions, wherein μ1And μ2It it is penalty factor；Y1、Y2It is glug bright
Day multiplier matrix；<,>represents the inner product of two matrixes；||·||FNot Luo Beini this Frobenius model of crow of representing matrix
Number；
(2) the iterative normal equation of formula is as follows:
In above formulaRepresent variables A when making object function take minima, the value of B, E, ρ1And ρ2For multiple
The factor, k is iterations；
2) training dictionary Φ: use on-line learning algorithm to train dictionary Φ on high-quality image data set；
3) use alternating direction method ADM that sequence (3) is converted into following sequence to solve:
Then according to step 4), 5), 6) method be iterated solving and obtain final result；
4) B is solvedk+1: use acceleration neighbour's gradient algorithm to try to achieve Bk+1；
Remove in formula (4) and solve item unrelated with B in the object function of B, obtain equation below:
The method using Taylor expansion, constructs a second order function to approach above formula, then for the letter of this second order
Number solves full scale equation, orderMay finally solve:
Soft therein (x, α) is contraction operator, Representative function f pair
Zj+1Frechet gradient, f is f (B) and Z herej+1Replace B, LfBeing a constant, value isVariable ZjRenewal
Rule is as follows:
tjBeing one group of sequence, j is iteration of variables number of times；
5) A is solvedk+1: use SVT Algorithm for Solving Ak+1；
Remove in formula (4) and solve item unrelated with A in the object function of A, and obtained by formula:
Wherein:
Singular value threshold method is used to solve for formula (9):
Wherein Uk, VkIt is W respectivelykLeft singular matrix and right singular matrix；
6) E is solvedk+1: by Ek+1It is divided into two parts to solve, inside spatial domain Ω, is solved by contraction operator, right
In the part beyond spatial domain Ω i.e.In, use first derivation to solve, two parts be altogether the last solution of E:
7) repeat the above steps 4), 5), 6) until algorithmic statement, at this moment result A of iterationk+1、Bk+1And Ek+1It is exactly former asking
Last solution A, B and E of topic.
The technical characterstic of the present invention and effect:
The inventive method recovers problem for the missing image with Random Valued Impulse Noise, by introducing dictionary by image
Battle array rarefaction representation, it is achieved that the structural missing image with Random Valued Impulse Noise is recovered solving of problem.The present invention has
There is a following characteristics:
1, while image lack part being filled with by recovery process, the Random Valued Impulse Noise in image is carried out
Remove, it is not necessary to the two is separately processed.
2, ALM, APG, SVT scheduling algorithm has been used to solve subproblem, the advantage incorporating existing algorithm.
3, row dictionary is employed, more efficient compared with traditional block dictionary.
4, matrix reconstruction theory and sparse representation theory are combined, traditional matrix reconstruction model introduces dictionary
Practising, the matrix so making full line element lack also is able to rebuilt.
Accompanying drawing explanation
Fig. 1 is algorithm flow chart；
Fig. 2 is original Lena figure；
Fig. 3 is impaired figure；Black is missing pixel, and graininess scatterplot is that Random Valued Impulse Noise is (left: 10% pixel disappearance
Random Valued Impulse Noise is had with 20% pixel；Right: 20% pixel disappearance and 30% pixel have Random Valued Impulse Noise).
Fig. 4 is the result figure after recovering；(left: 10% pixel disappearance and 20% pixel have Random Valued Impulse Noise to recover knot
Really, PSNR=34.83；Right: 20% pixel disappearance and 30% pixel have Random Valued Impulse Noise restoration result, PSNR=
32.25)。
Detailed description of the invention
It is introducing dictionary learning model on the basis of classical matrix reconstruction model, enabling reconstruct with random value
The low-rank matrix of the structural disappearance of impulsive noise thus image after being restored, i.e. based on low-rank matrix rebuild with
The missing image restoration methods of machine value impulsive noise, thus solve the problem that prior art cannot process.Below in conjunction with the accompanying drawings and
The present invention is elaborated by embodiment.
1) regard image as matrix, then original image can represent by matrix A, solves the disappearance with Random Valued Impulse Noise
Image recovers problem for solving following optimization method:
||A||*The nuclear norm of representing matrix A.||·||1One norm of representing matrix.Ω is observation space, pΩ() is
Projection operator, represents that variable projects the value in spatial domain Ω.λ and γ is weight coefficient.Φ in constraints is for training
Dictionary, B is the coefficient matrix that dictionary is corresponding.D degrades observing matrix i.e. damaged image known to being, E represents in damaged image
The pixel of disappearance and noise.Experiment use Lena image as test data.
11) when solving this equation, the present invention uses augmented vector approach, and equation is as follows:
L(A,B,E,Y1,Y2) it is Augmented Lagrangian Functions.μ therein1And μ2It it is penalty factor；Y1、Y2It is glug
Bright day multiplier matrix；<,>represents the inner product of two matrixes, such as<M, N>=trace (MTN), MTThe transposition of representing matrix M,
Matrix trace is sought in trace () expression；||·||FThe Frobenius of representing matrix (not Luo Beini crow this) norm.
12) solving (2) formula by iterative method, the equation of iteration is as follows:
In above formulaRepresent variables A when making object function take minima, the value of B, E.ρ1And ρ2For multiple
The factor.K is iterations.
2) Online Learning dictionary learning method is used to train dictionary Φ.
21) structure dictionary Φ makes matrix A can i.e. be met A=Φ B by dictionary rarefaction representation, and wherein B is coefficient matrix
And be sparse.The present invention uses Online Learning algorithm to train dictionary Φ on Kodak image set.
22) when training dictionary, it is as follows that we set dictionary relevant parameter: in the line number of matrix A to be reconstructed and dictionary Φ
The dimension m of element is equal, i.e. the line number of A and the line number of Φ are m.The dictionary Φ trained is a dictionary the most complete, i.e.
The columns of dictionary have to be larger than its line number.
3) present invention uses alternating direction method to solve sequence (3), sequence (3) will be converted into following sequence and ask
Solve:
Set each initial parameter values, then according to step 4), 5), 6) method be iterated solving final result.
4) acceleration neighbour's gradient algorithm is used to solve first unknown Bk+1。
41) remove and after formula (4) solves item unrelated with B in the object function of B, obtain equation below:
The method using Taylor expansion, constructs a second order function to approach above formula, then for the letter of this second order
Number solves full scale equation.
Order
It is re-introduced into variable Z, is defined as follows function:
Wherein,The representative function f Frechet gradient to Z, f is here
F (B) and Zj+1Replace B, LfBeing a constant, value isIt is used for ensureing for all of Z, has F (Z)≤Q (B, Z).
42) converting through upper step, formula (5) is converted into and solves Q (B, Zj) minima solve, then obtained by formula
To following form:
WhereinVariable ZjMore new regulation as follows:
tjBeing one group of sequence, j is iteration of variables number of times.Use contraction operator solves:
Wherein soft (x, α) is contraction operator.
5) A is solvedk+1, remove and formula (4) solves item unrelated with A in the object function of A obtain:
Method of completing the square is used to be rewritten into by above formula:
Wherein:
Singular value threshold method is used to solve for formula (11):
Wherein Uk, VkIt is W respectivelykLeft singular matrix and right singular matrix.
6) E is solvedk+1, about Ek+1Equation be shown below:
By Ek+1It is divided into two parts to solve, for the part in spatial domain Ω, uses contraction operator to solve.For space
Place beyond Ω is i.e.In, use first derivation to solve, two parts are altogether the last solution of E.
61) in the place within spatial domain Ω.The solution that contraction operator can be used to try to achieve formula (14) is:
62) in the place outside spatial domain Ω.Can try to achieve the solution of formula (14) with first derivation is:
63) solution inside and outside for spatial domain Ω being joined together is exactly the last solution of E again:
7) repeat the above steps 4), 5), 6) until algorithmic statement.At this moment result A of iterationk+1、Bk+1And Ek+1It is exactly former asking
Last solution A, B and E of topic.
Matrix reconstruction theory is combined by the inventive method with sparse representation theory, on traditional matrix reconstruction model basis
Upper introducing dictionary learning model, thus solve the problem that prior art cannot process, i.e. realize with Random Valued Impulse Noise
Disappearance picture carry out recovering (experiment flow figure is as shown in Figure 1).Detailed description in conjunction with the accompanying drawings and embodiments is as follows:
1) experiment uses the Lena picture (as shown in Figure 2) of 512 × 512 pixels as original graph, construct 2 thereon
Plant the damaged image polluting in various degree and lacking to carry out testing (as shown in Figure 3).The present invention uses the dictionary of fixed size, institute
By first by figure to be restored according to sliding window from top to bottom in the way of be divided into the image block of several m × 512.M represents code word in dictionary
Dimension, in experiment, m value is 30, and the step-length of sliding window is 5 pixels.The image block of these several m × 512 is recovered successively,
Final recombinant gets up i.e. to can get the recovery figure of original size 512 × 512.When processing first image block, then used
Matrix A represents, recovers currently processed image block problem for solving following optimization method:
||A||*The nuclear norm of representing matrix A.||·||1One norm of representing matrix.Ω is observation space, pΩ() is
Projection operator, represents that variable projects the value in spatial domain Ω.λ and γ is weight coefficient, is 0.02 by λ value in an experiment,
γ value is 0.3.Φ in constraints is the dictionary trained, and B is the coefficient matrix that dictionary is corresponding.D degrades known to being
The damaged image that observing matrix i.e. lacks.E represents the pixel of disappearance in damaged image.
11) when solving this equation, using augmented vector approach, equation is as follows:
L(A,B,E,Y1,Y2) it is Augmented Lagrangian Functions.μ therein1And μ2It it is penalty factor；Y1、Y2It is glug
Bright day multiplier matrix；The inner product of two matrixes of<,>expression, the transposition of<M, N>=trace (MTN), MT representing matrix M,
Matrix trace is sought in trace () expression；||·||FThe Frobenius of representing matrix (not Luo Beini crow this) norm.
12) solving (2) formula by iterative method, the equation of iteration is as follows:
In above formulaRepresent variables A when making object function take minima, the value of B, E.ρ1And ρ2For multiple
The factor.K is iterations.
2) Online Learning dictionary learning method is used to train dictionary Φ.
21) structure dictionary Φ makes matrix A can i.e. be met A=Φ B by dictionary rarefaction representation, and wherein B is coefficient matrix
And be sparse.Experiment use Online Learning algorithm train dictionary Φ on Kodak image set.Scheme at Kodak
Randomly selecting 50000 sizes in image set on all images altogether is that the pixel column of 30 × 1 is as training data.
22) when training dictionary, it is as follows that we set dictionary relevant parameter: in the line number of matrix A to be reconstructed and dictionary Φ
The dimension m of element is equal, i.e. the line number of A and the line number of Φ are m.The dictionary Φ trained is a dictionary the most complete, i.e.
The columns of dictionary have to be larger than its line number.Taking dictionary columns in experiment is 300, then the specification of dictionary Φ is 30 × 300.
3) when solving sequence (3), the present invention uses alternating direction method that sequence (3) is converted into following sequence and asks
Solve:
Set each initial parameter values, then according to step 4), 5), 6) method be iterated solving.At the beginning of experiment sets
Value is: K=1；ρ1=ρ2=1.1；A1=E1=B1=0.
4) APG method is used to solve first unknown Bk+1。
41) remove and after formula (4) solves item unrelated with B in the object function of B, obtain equation below:
The method using Taylor expansion, constructs a second order function to approach above formula, then for the letter of this second order
Number solves full scale equation.
Order
It is re-introduced into variable Z, is defined as follows function:
Wherein,The representative function f Frechet gradient to Z；Here f is
F (B) and Zj+1Replace B, LfBeing a constant, value isIt is used for ensureing for all of Z, has F (Z)≤Q (B, Z).
42) converting through upper step, formula (5) is converted into and solves Q (B, Zj) minima solve, then obtained by formula
To following form:
WhereinVariable ZjMore new regulation as follows:
tjBeing one group of sequence, j is iteration of variables number of times.Through above-mentioned conversion, set each initial parameter value as follows: j=1；t1
=1；Z1=0.Can solve during convergence:
Wherein soft (x, α) is contraction operator.
5) A is solvedk+1, remove and formula (4) solves item unrelated with A in the object function of A obtain:
Method of completing the square is used to be rewritten into by above formula:
Wherein:
Singular value threshold method is used to solve for formula (11):
Wherein Uk, VkIt is W respectivelykLeft singular matrix and right singular matrix.
6) E is solvedk+1, about Ek+1Equation be shown below:
By Ek+1It is divided into two parts to solve, for the part in spatial domain Ω, uses contraction operator to solve.For space
Place beyond Ω is i.e.In, use first derivation to solve, two parts are altogether the last solution of E.
61) in the place within spatial domain Ω.The solution that contraction operator can be used to try to achieve formula (14) is:
62) in the place outside spatial domain Ω.Can try to achieve the solution of formula (14) with first derivation is:
63) solution inside and outside for spatial domain Ω being joined together is exactly the last solution of E again:
7) repeat the above steps 4), 5), 6) until convergence.At this moment result A of iterationk+1、Bk+1And Ek+1It it is exactly former problem
Last solution A, B and E.
8) process step 1 successively) in remaining several image block of obtaining until full recovery is good, then by these image blocks
It is combined into final recovery figure (as shown in Figure 4).During combination, the pixel repeatedly recovered takes the average repeatedly recovered as
Final value.
Experimental result: the present invention uses PSNR (Y-PSNR) as the calculated measure of image restoration result:
Wherein I0Representing and do not have impaired true picture, I is the image after recovering, and h is the height of image, and w is image
Width, (x, y) be xth row y row pixel value, Σ represents summation operation, | | for absolute value.This experiment takes n=8, experiment
In the result recovered for 2 pictures the most impaired of test see that Fig. 4 marks.
Claims (1)
1. low-rank matrix is rebuild with a Random Valued Impulse Noise missing image restoration methods, it is characterized in that, including walking as follows
Rapid:
1) regard image as matrix, then original image matrix A represents, solves to recover with the missing image of Random Valued Impulse Noise
Problem is for solving following optimization method:
Wherein | | A | |*The nuclear norm of representing matrix A, | | | |1One norm of representing matrix, Ω is observation space, pΩ() is
Projection operator, represents that variable projects the value in spatial domain Ω, λ and γ is weight coefficient, and the Φ in constraints is for training
Dictionary, B is the coefficient that dictionary is corresponding, D be known to degrade observing matrix i.e. damaged image, E represents in damaged image and lacks
Pixel and noise；
When solving this equation, the present invention uses augmented vector approach, and equation is as follows:
L(A,B,E,Y1,Y2) it is Augmented Lagrangian Functions, wherein μ1And μ2It it is penalty factor；Y1、Y2It is Lagrange to take advantage of
Submatrix；<,>represents the inner product of two matrixes；||·||FNot Luo Beini this Frobenius norm of crow of representing matrix；
(2) the iterative normal equation of formula is as follows:
In above formulaRepresent variables A when making object function take minima, the value of B, E, ρ1And ρ2For multiplying factor,
K is iterations；
2) training dictionary Φ: use on-line learning algorithm to train dictionary Φ on high-quality image data set；
3) use alternating direction method ADM that sequence (3) is converted into following sequence to solve:
Then according to step 4), 5), 6) method be iterated solving and obtain final result；
4) B is solvedk+1: use acceleration neighbour's gradient algorithm to try to achieve Bk+1；
Remove in formula (4) and solve item unrelated with B in the object function of B, obtain equation below:
The method using Taylor expansion, constructs a second order function to approach above formula, and then the function for this second order comes
Solve full scale equation, orderLast solution obtains:
Soft therein (x, α) is contraction operator, Representative function f is to Zj+1's
Frechet gradient, f is f (B) and Z herej+1Replace B, LfBeing a constant, value isVariable ZjMore new regulation such as
Under:
tjBeing one group of sequence, j is iteration of variables number of times；
5) A is solvedk+1: use SVT Algorithm for Solving Ak+1；
Remove in formula (4) and solve item unrelated with A in the object function of A, and obtained by formula:
Wherein:
Singular value threshold method is used to solve for formula (9):
Wherein Uk, VkIt is W respectivelykLeft singular matrix and right singular matrix；
6) E is solvedk+1: by Ek+1Be divided into two parts to solve, inside spatial domain Ω, solved by contraction operator, for
Part beyond spatial domain Ω is i.e.In, use first derivation to solve, two parts be altogether the last solution of E:
7) repeat the above steps 4), 5), 6) until algorithmic statement, at this moment result A of iterationk+1、Bk+1And Ek+1It it is exactly former problem
Last solution A, B and E.
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| CN106780369B (en) * | 2016-11-28 | 2019-12-03 | 江苏宇迪光学股份有限公司 | A kind of visible near-infrared broad wavelength band image noise suppressing method |
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| CN106683178A (en) * | 2016-12-30 | 2017-05-17 | 天津大学 | Method for recovering three-dimensional framework by low-rank matrix on basis of graph theory |
| CN106683178B (en) * | 2016-12-30 | 2020-04-28 | 天津大学 | Graph theory-based low-rank matrix three-dimensional framework recovery method |
| CN111025385A (en) * | 2019-11-26 | 2020-04-17 | 中国地质大学（武汉） | Seismic data reconstruction method based on low rank and sparse constraint |
| CN111314000A (en) * | 2020-02-18 | 2020-06-19 | 五邑大学 | Radio frequency tomography method based on low-rank data-driven weight model |
| CN112561842A (en) * | 2020-12-07 | 2021-03-26 | 昆明理工大学 | Multi-source damaged image fusion and recovery combined implementation method based on dictionary learning |
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