CN109064412A - A kind of denoising method of low-rank image - Google Patents
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Abstract
The invention discloses a kind of denoising methods of low-rank image, this method is applied to the reconstruct containing noise image, pass through the analysis to low-rank matrix correlation properties, the Denoising Problems of image are modeled as a nonlinear restriction problem, then specific iterative step is derived using alternating direction multipliers method, to obtain the image after removal noise.The present invention more accurately approaches rank of matrix using truncation nuclear norm, avoids result error caused by big singular value；In addition Frobenius norm is added in a model is used as regular terms, and truncation nuclear norm to constitute elastic network(s) about singular value, so that the result finally found out while having sparsity and stability, so that reaching better denoises effect.
Description
Technical field
The invention belongs to Image Denoising Technology fields, and in particular to a kind of denoising method of low-rank image.
Background technique
Image is a kind of important information source, and the intension of people's understanding information can be helped by image procossing.But scheme
As usually making image deterioration due to interference and influence by various noises during generation and transmission, this is to subsequent image
Processing (such as segmentation, compression and image understanding) will have an adverse effect.There are many noise type, such as: electrical noise, mechanical noise,
Interchannel noise and other noises.In order to inhibit noise, improving image quality, convenient for higher level processing, it is necessary to image into
Row noise suppression preprocessing.
It is studied by the several years, there are many Image denoising algorithms at present, but can't fully meet growing application
Demand.Image sparse was indicated increasingly by the favor of experts and scholars in recent years, by way of linear combination, extracts figure
As important and crucial characteristic, by the most simple most complete expression of image.Lot of documents proves, rarefaction representation relative to
Other algorithm storage capacities are strong, and the speed of service also relatively rapidly, can obtain more extensive development in field of image processing.
But in the current low-rank image de-noising method based on rarefaction representation, the most commonly used is the nuclear norms for using matrix
Rank of matrix is replaced, in the case where matrix low-rank, is easy to cause solution excessively sparse, so that unstable solution is generated, so that
Influence the precision that image restores.Therefore the stability for improving low-rank matrix restoration result becomes more have practical significance and application
Prospect.
Summary of the invention
Goal of the invention: in order to overcome the deficiencies in the prior art, provide it is a kind of based on truncation nuclear norm and
The low-rank image de-noising method of Frobenius norm, by converting a nonlinear restriction problem for image denoising problem, so
Model is solved using alternating direction multipliers method algorithm afterwards, to obtain the image after removal noise.
Technical solution: to achieve the above object, the present invention provides a kind of denoising methods of low-rank image, including step is such as
Under:
1) a noise-containing image D is inputted；
2) by matrix D ∈ R to be restoredm×nCarry out singular value decompositionIt is unusual to obtain its left side
Vector U, right singular vector V and singular value matrix Σr；
3) according to the definition of truncation nuclear normAcquire the left and right singular value vector F of truncation nuclear norm
And G；
4) according to objective function and constraint condition, the low-rank image based on truncation nuclear norm and Frobenius norm is established
Denoising model；
5) model is solved to obtain specific iterative algorithm step using alternating direction multipliers method and separation of variables
Suddenly；
6) it will be truncated in the corresponding left and right singular vector F and G of nuclear norm and matrix D to be restored input algorithm, and be arranged
Threshold condition ε and the number of iterations, are iterated operation.When algorithm iteration is to less than threshold epsilon0When, end loop, to obtain
Image array A after recovery.
Further, matrix D is subjected to singular value decomposition in the step 2Obtain matrix
The left singular vector U and right singular vector V and singular value Σ of Dr.Wherein U=(u1,u2,...,um)∈Rm×m, V=(v1,
v2,...,vm)∈Rn×nAnd Σr=diag (σ1,σ2,...,σr)。
Further, the left and right singular vector of truncation nuclear norm is acquired in the step 3 method particularly includes: by singular value
It arranges from small to large, removes maximum r-t singular value, take the corresponding left singular vector F=(u of the smallest t singular value1,
u2,...,ut)TWith right singular vector G=(v1,v2,...,vt)T。
Further, according to the objective function and its constraint condition of construction in the step 4, building is based on truncation nuclear norm
Image denoising model with Frobenius norm is
Wherein A represents original image；E indicates noise, | | E | |1Indicate the l of E1Norm；||A||FRepresenting matrix A's
Frobenius norm；||A||t=| | A | |*What-Tr (FAG') was represented is truncation nuclear norm；λ is indicated | | E | |1Weight, λ gets over
The big sparsity for indicating E is stronger；γ is indicated | | A | |FWeight parameter, that is, the extent of stability of the solution found out.λ and γ should be reasonable
Choosing value, so that the solution finally found out is i.e. sparse and stablizes.
Further, as follows using specific derivation of the alternating direction multipliers method method to model in the step 5:
The Augmented Lagrangian Functions expression formula of above-mentioned model is write out first:
Then H and B is used to substitute respectively second and third A in above formula according to separation of variables,
S.t.H=A, B=A
Wherein B, H are the intermediate variable of separation of variables.
The constrained expression formula of above formula is removed bound term using Augmented Lagrangian Functions again:
Wherein uiFor penalty term parameter.
A) intermediate variable B is calculated firstk+1, fixed Yik,Ek,Hk, minimize L (A, E, Yi,ui), Bk+1Abbreviation are as follows:
Its minimum value can be acquired by infinitesimal method
It b) similarly, can be in the hope of
C) E is then calculatedk+1, fixed Yik,Ak,Bk,Hk, Ek+1Expression formula is
Using soft-threshold solving methodAcquire Ek+1Expression formula:
Wherein soft-threshold Su(X)=sgn (X) max (| X |-u, 0)
D) it calculates and restores matrix A, ignore constant term, fixed Yik,Ek+1,Bk+1,Hk+1, pass through singular value contraction operatorAcquire the solution for restoring matrix A
Wherein Dτ(X)=UDτ(Σ)VT,Dτ(Σ)=diag (max { σi-τ,0}),σiFor the singular value of X.
E) Lagrange multiplier Y is finally solvedi k+1
WhereinFor the parameter of penalty term,ρ is indicatedIncreased times, umaxIndicate that u's is upper
Limit.
Further, ε is set by threshold epsilon in the step 60, wait algorithmWhen, end loop,
Image array A after acquiring final removal noise, and the image after being denoised.
The present invention is in low-rank image denoising, by establishing a kind of mathematical modulo that nuclear norm and Frobenius norm is truncated
Type avoids the excessively sparse of image restoration result to establish the elastic network(s) about singular value, enhances image and restores knot
The stability of fruit, to improve the precision of image recovery.
The utility model has the advantages that compared with prior art, the present invention having the advantage that
1, the present invention replaces traditional nuclear norm to approach rank of matrix using truncation nuclear norm, avoids big singular value
Influence so that solve effect it is more preferable.
2, the present invention avoids the solution acquired excessively sparse, enhances understanding using Frobenius norm as regular terms
Stability improves image and restores precision.
3, Frobenius norm and truncation nuclear norm joint are constituted the elastic network(s) in relation to singular value by the present invention, so that most
Solution afterwards had not only had sparsity but also had had stability.
4, the present invention solves model using alternating direction multipliers method algorithm, solves accurate algorithm iteration step
Suddenly, the complexity for reducing algorithm, enhances algorithm performance.
Detailed description of the invention
Fig. 1 is overview flow chart of the invention；
Fig. 2 is different denoising methods to the denoising effect contrast figure of image, and wherein measurement index is using peak value
Signal-to-noise ratio (PSNR)；
Fig. 3 is whole algorithm flow chart of the invention.
Specific embodiment
In the following with reference to the drawings and specific embodiments, the present invention is furture elucidated, it should be understood that these embodiments are merely to illustrate
It the present invention rather than limits the scope of the invention, after the present invention has been read, those skilled in the art are to of the invention each
The modification of kind equivalent form falls within the application range as defined in the appended claims.
Embodiment 1:
As shown in figures 1 and 3, the present invention provides a kind of denoising method of low-rank image, includes the following steps:
1) a noise-containing image D is inputted；
2) first by matrix D ∈ R to be restoredm×nCarry out singular value decompositionObtain his left side
Singular vector U=(u1,u2,...,um)∈Rm×m, right singular vector V=(v1,v2,...,vm)∈Rn×nAnd singular value Σr=
diag(σ1,σ2,...,σr)；
3) according to the definition of truncation nuclear normSingular value is arranged from small to large, is removed maximum
R-t singular value takes the corresponding left singular vector F=(u of the smallest t singular value1,u2,...,ut)TWith right singular vector G=
(v1,v2,...,vt)T；
4) according to objective function and constraint condition, the image denoising based on truncation nuclear norm and Frobenius norm is established
Model,
Wherein A represents original image；E indicates noise, | | E | |1Indicate the l of E1Norm；||A||FRepresenting matrix A's
Frobenius norm；||A||t=| | A | |*What-Tr (FAG') was represented is truncation nuclear norm；λ is indicated | | E | |1Weight, λ gets over
The big sparsity for indicating E is stronger；γ is indicated | | A | |FWeight parameter, that is, the extent of stability of the solution found out.λ and γ should be reasonable
Choosing value, so that the solution finally found out is i.e. sparse and stablizes.
5) model is carried out pushing over solution using alternating direction multipliers method method and separation of variables and is specifically changed
For algorithm.
The Augmented Lagrangian Functions expression formula of above-mentioned model is write out first:
Then H and B is used to substitute respectively second and third A in above formula according to separation of variables,
S.t.H=A, B=A
Wherein B, H are the intermediate variable of separation of variables.
The constrained expression formula of above formula is removed bound term using Augmented Lagrangian Functions again:
Wherein uiFor the parameter of penalty term.
The specific steps of which are as follows:
A) intermediate variable B is calculated firstk+1, fixed Yik,Ek,Hk, minimize L (A, E, Yi,ui), Bk+1Abbreviation are as follows:
Its minimum value can be acquired by infinitesimal method
It b) similarly, can be in the hope of
C) E is then calculatedk+1, fixed Yik,Ak,Bk,Hk, Ek+1Expression formula is
Using soft-threshold solving methodAcquire Ek+1Expression formula:
Wherein soft-threshold Su(X)=sgn (X) max (| X |-u, 0)
D) it calculates and restores matrix A, ignore constant term, fixed Yik,Ek+1,Bk+1,Hk+1, pass through singular value contraction operatorAcquire the solution for restoring matrix A
Wherein Dτ(X)=UDτ(Σ)VT,Dτ(Σ)=diag (max { σi-τ,0})
E) Lagrange multiplier is finally solved
WhereinFor the parameter of penalty term,ρ is indicatedIncreased times, umaxIndicate that u's is upper
Limit.
6) the corresponding left and right singular vector F and G of nuclear norm will be truncated and matrix D to be restored is sent into iterative algorithm, it will
Threshold epsilon is set as ε0, wait algorithmWhen, end loop.Image moment after acquiring final removal noise
Battle array A.
Embodiment 2:
As shown in Fig. 2, (a) is original image, (b) it is the noisy image of original image, is utilized respectively NNR, NNF, TNNR
Denoising carried out to noisy image (b) with the method for embodiment 1, the PSNR of four kinds of methods is respectively 25.3,26.1,27.5,
28.4, image (c), (d), (e) and (f) after denoising are respectively obtained, according to comparison it is found that obtaining by the method for embodiment 1
Image (f) closest to original image (a), it is best to denoise effect.
Claims (6)
1. a kind of denoising method of low-rank image, which comprises the steps of:
1) a noise-containing image D is inputted；
2) by matrix D ∈ R to be restoredm×nCarry out singular value decompositionObtain its left singular vector
U, right singular vector V and singular value matrix Σr；
3) according to the definition of truncation nuclear normAcquire left and right the singular value vector F and G of truncation nuclear norm；
4) according to objective function and constraint condition, the low-rank image denoising based on truncation nuclear norm and Frobenius norm is established
Model；
5) model is solved to obtain specific iterative algorithm step using alternating direction multipliers method and separation of variables；
6) it will be truncated in the corresponding left and right singular vector F and G of nuclear norm and matrix D to be restored input algorithm, and threshold value is set
Condition ε and the number of iterations, are iterated operation.When algorithm iteration is to less than threshold epsilon0When, end loop, to be restored
Image array A afterwards.
2. a kind of denoising method of low-rank image according to claim 1, it is characterised in that: by matrix D in the step 2
Carry out singular value decompositionObtain the left singular vector U and right singular vector V of matrix D and unusual
Value matrix Σr, wherein U=(u1,u2,...,um)∈Rm×m, V=(v1,v2,...,vm)∈Rn×nAnd Σr=diag (σ1,
σ2,...,σr)。
3. a kind of denoising method of low-rank image according to claim 1, it is characterised in that: solve and cut in the step 3
The left and right singular vector of disconnected nuclear norm method particularly includes: singular value is arranged from small to large, it is a unusual to remove maximum r-t
Value, takes the corresponding left singular vector F=(u of the smallest t singular value1,u2,...,ut)TWith right singular vector G=(v1,
v2,...,vt)T。
4. a kind of denoising method of low-rank image according to claim 1, it is characterised in that: according to structure in the step 4
The objective function and its constraint condition made, the low-rank image denoising mould based on truncation nuclear norm and Frobenius norm of building
Type are as follows:
Wherein A represents original image, and E indicates noise, | | E | |1Indicate the l of E1Norm；||A||tThe truncation core model of representing matrix A
Number, | | A | |FThe Frobenius norm of representing matrix A；λ is indicated | | E | |1Weight, λ it is bigger indicate E sparsity it is stronger；γ
Indicate | | A | |FWeight parameter, that is, the extent of stability of the solution found out.
5. a kind of denoising method of low-rank image according to claim 1, it is characterised in that: utilize friendship in the step 5
Model is solved for direction multiplier method algorithm, obtains specific algorithm iteration step are as follows:
A) Lagrange multiplier Y is initializedik, penalty term parameter ui, left and right the singular vector F and G of input truncation nuclear norm, and
Matrix D to be restored；
B) intermediate variable B and H are solved,
C) matrix A after noise matrix E and denoising is updated according to intermediate variable, specific as follows:
Wherein k is the number of iterations, SτFor soft-threshold contraction operator, Su(X)=sgn (X) max (| X |-u, 0), DτFor singular value receipts
Contracting operator, Dτ(X)=UDτ(Σ)VT,Dτ(Σ)=diag (max { σi- τ, 0 }), σiFor the singular value of X；
D) final updating Lagrange multiplier
Wherein uiFor the parameter of penalty term,ρ is indicatedIncreased times, umaxIndicate the upper limit of u.
6. a kind of denoising method of low-rank image according to claim 1, it is characterised in that: by threshold epsilon in the step 6
It is set as ε0, wait algorithmWhen, end loop, the image array A after acquiring final removal noise, and
Image after being denoised.
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| CN110310234A (en) * | 2019-05-17 | 2019-10-08 | 浙江工业大学 | The facial image denoising method and device with Shannon fully differential are approached based on low-rank |
| CN110675331A (en) * | 2019-08-13 | 2020-01-10 | 南京人工智能高等研究院有限公司 | Image denoising method and device, computer readable storage medium and electronic device |
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| CN116738764A (en) * | 2023-08-08 | 2023-09-12 | 中国海洋大学 | Ocean platform cabin comfort level assessment method based on singular value threshold algorithm |
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| CN109919878A (en) * | 2019-03-14 | 2019-06-21 | 上海市第一人民医院 | Noise-reduction method and its system based on optical coherence tomography |
| CN109919878B (en) * | 2019-03-14 | 2023-04-18 | 上海市第一人民医院 | Noise reduction method and system based on optical coherence tomography |
| CN110060219A (en) * | 2019-04-24 | 2019-07-26 | 北京理工大学 | One kind being based on low-rank approximately true figure noise-reduction method |
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| CN110728641A (en) * | 2019-10-12 | 2020-01-24 | 浙江工业大学 | Remote sensing image impulse noise removing method and device |
| CN111028162B (en) * | 2019-11-26 | 2023-03-31 | 广东石油化工学院 | Image missing recovery method based on truncated Schattenp-norm |
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| CN110992292B (en) * | 2019-12-09 | 2023-04-18 | 河北工业大学 | Enhanced low-rank sparse decomposition model medical CT image denoising method |
| CN114820387A (en) * | 2022-05-27 | 2022-07-29 | 山东财经大学 | Image recovery method and terminal based on probability induced kernel norm minimization |
| CN116738764A (en) * | 2023-08-08 | 2023-09-12 | 中国海洋大学 | Ocean platform cabin comfort level assessment method based on singular value threshold algorithm |
| CN116738764B (en) * | 2023-08-08 | 2023-10-20 | 中国海洋大学 | Ocean platform cabin comfort level assessment method based on singular value threshold algorithm |
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