CN105761223A - Iterative noise reduction method based on image low-rank performance - Google Patents

Abstract

The invention provides an iterative noise reduction method based on image low-rank performance. The method concretely relates to the field of image processing and computer vision. In the image representation, Y refers to an image having noise, X refers to an ideal image, N refers to Gaussian white noise of which the standard deviation is tau, and the image is described as Y=X+N and denoising means to reduce the value of N. In order to better utilize the image low-rank performance to perform denoising, firstly an observation image is divided into multiple small blocks which are identical in size. As for any image block P<i>, image blocks similar to the image block P<i> are searched according to a certain similarity criterion. The similar image blocks are vectorized so that a similar block matrix P<i> is formed, based on which the P<i> is estimated by utilizing the low-rank performance of the similar block matrix and the theory of minimum variance estimation so that the objective of suppressing image block noise can be achieved. Finally the denoised image is obtained through reconstruction of the image block estimation value. As for the problem of a small amount of residual noise, the denoised image is processed by adopting a synthetic iterative method.

Description

A kind of iteration noise-reduction method based on image low-rank

Technical field

The method relates to computer information technology field, more particularly to image procossing, computer vision field.

Background technology

Impact by image-forming condition and external interference factor, image can produce noise in various degree in acquisition, compression, transmitting procedure, that reduce the visual quality of image and affect the subsequent treatment of image, so the noise reduction of image obtains bigger concern in the field such as image procossing and computer vision.

In research, scholar is had to adopt evaluation method and wavelet coefficient to propose the denoising method based on wavelet transformation, also scholar is had to adopt generalized Gaussian distribution that wavelet coefficient is modeled, then Bayesian Estimation deduced image constriction coefficient is utilized to carry out noise reduction, also scholar is had to utilize the dependency between wavelet coefficient yardstick to set up bivariate denoising model, researcher is also had according to the directional dependency between shearing wave coefficient, coefficient to be modeled, propose the ternary denoising method based on shearing wave conversion, the method can keep edge and the texture of image preferably while effectively suppressing noise. but, although the line that the above-mentioned denoising method based on multiple dimensioned geometric transformation can process in image preferably is unusual, but the self adaptation of image is represented that ability still has much room for improvement and the learning process of dictionary is more complicated.

At present, the self-similarity of image is included into sight line gradually, and its main manifestations is there is substantial amounts of similar pixel in image, and the position of these pixels is not necessarily adjacent, but its gray scale and architectural feature are usually closely similar.Utilizing this similarity between pixel by being weighted on average suppressing noise to the similar pixel in noisy image, test result indicate that, the utilization of self-similarity can improve the visual quality of image after denoising.In order to more efficiently utilize image self-similarity, can being many little image blocks by picture breakdown, and be grouped according to the similarity between image block, often group similar block can pass through block vectorization process generation similar block matrix；Adopt nonlinear IEM model and regularization method that similar block matrix is estimated, to reach to suppress the purpose of noise.Owing to there is stronger dependency between similar block matrix column vector so that when this matrix is carried out wavelet transformation, it represents that coefficient has well openness.How to make full use of the key that the prior information that image self comprises is structure denoising method, the self-similarity of image means have higher dependency between the similar block matrix column vector being made up of similar image block, and therefore similar block rank of matrix is usually relatively low.This low-rank utilizing image can extract the core information of view data easily, thus contributing to realizing separating of image and noise.In consideration of it, this method utilizes the low-rank that image self-similarity is implied, derive a kind of image de-noising method by minimum variance estimate theory.

Summary of the invention

For above-mentioned weak point, the present invention proposes a kind of iteration noise-reduction method based on image low-rank, the method utilizes image self similarity to construct the similar block matrix with low-rank, thus image denoising problem is converted into low-rank matrix estimation problem, it is derived a kind of effective image de-noising method based on minimum variance estimate theory, meanwhile, for a small amount of noise residue problem, alternative manner is adopted to improve the denoising performance of method further.Making full use of the redundancy contained by image to reach the purpose of denoising, similar image block is had certain self adaptation and represents ability by the singular value decomposition of employing.

It is an object of the invention to: improve the noise reduction capability of image so that the visual quality of image is better.

What the present invention adopted for achieving the above object technical scheme is: a kind of iteration noise-reduction method based on image low-rank.The enforcement step of the method is as follows:

Step one: input noisy image Y；

Step 2: utilize absolute deviation median method estimate image Y (or) noise level τ；

Step 3: to image Y (or) carry out piecemeal process, and each image block is found out similar image block according to algorithm, thus forming C similar block block matrix

Step 4: to each similar block matrix P_j, calculate its noisy similar block matrix P_jEstimated value

Step 5: calculateThe weight w of contained image block_j；

Step 6: utilize the estimated value reconstruct image of image block

Step 7: utilize formula 12 to generate new noisy image

Step 8: repeat step 2 to step 7, until image reaches desirable level；

Step 9: the image after output denoising

The invention has the beneficial effects as follows: 1, there is higher PSNR value；2 have higher FSIM (structural similarity)；3, visual effect is improved；4 keep image detail.

Detailed description of the invention

One, problem describes

In graphical representation, making Y represent containing noisy image, X represents that ideal image, N represent the white Gaussian noise that standard deviation is τ, then image can be described as: Y=X+N.

It can thus be seen that denoising seeks to reduce the value of N, difficulty is in that how to retain in noise reduction process the minutias such as edge and the texture of image.

Image denoising it is crucial that how to make full use of the priori rule of image self and adopt which kind of data evaluating method that image is estimated.In order to better utilize image low-rank to carry out denoising, first observed image is divided into the fritter that many sizes are identical by the present invention, to arbitrary image block P_iSearch out the image block similar to it by a certain similarity criterion, these similar image blocks are carried out vectorization and forms similar block matrix P_i, on this basis, utilize the low-rank of similar block matrix and minimum variance estimate theoretical to P_iEstimate, thus arriving the purpose suppressing image block noise；Finally obtained the image after denoising by the reconstruct of image block estimated value.For a small amount of noise residue problem, adopt back projection method that denoising image is processed, promote the denoising effect of image further.

Two, the judgement of the packet similarity of image block

We divide the image into the image subblock of 7*7, use P_jRepresent, P_j∈ P}, j, i, r ∈ 1,2 ..., N}, owing to consider computation complexity and the speed of image procossing, so adopting the Euclidean distance based on gray feature to measure the similarity between image block；To any two image block P_jAnd P_iIts similarity can represent:

$\mathrm{\&Delta;}(P_j,P_i)=\left|\right|P_i-P_j|{|}_2^2$

Δ(P_j, P_i) value more little, then P_jAnd P_iMore similar, for arbitrary image block P_r, adopt above formula to carry out the similarity between coupling image block, find out K the image block similar to it, be recorded asNamely can be expressed as:

$P_r=\&lsqb;\begin{array}{cccc}{P}_r,& P_r^1,& ...,& P_r^K\end{array}\&rsqb;$

Then image block can be expressed in matrix as:

P_r=Q_r+N_rFormula 1

Wherein Q_rRepresent the ideal image block matrix of not Noise, N_rFor noise matrix, denoising target is conversion P just_rAs much as possible accurately estimate Q_r.By noisy similar block matrix P_rConstitute it can be seen that have higher dependency between its column vector, it means that matrix P_rIt is low-rank, it is possible to use similar block matrix Q_rThis low-rank it is estimated, thus reach remove picture noise purpose.

Three, the estimation of similar block matrix

Make Q ∈ R^m*nOrder be k, m > n is then according to singular value decomposition theorem, the singular value decomposition of noisy similar block matrix P can be expressed as:

$P=U_P{E}_P{V}_P^T=\left(\begin{array}{cc}{U}_{P1}& U_{P2}\end{array}\right)\left(\begin{array}{cc}{E}_{P1}& 0\\ 0& E_{P2}\end{array}\right)\left(\begin{array}{c}{V}_{P1}^T\\ V_{P2}^T\end{array}\right)$ Formula 2

Wherein, U_P1∈R^m*k, U_P2∈R^m*(n-k), V_P1∈R^k*n, V_P2∈R^n*(n-k),

E_P1=diag (δ₁, δ₂..., δ_k)∈R^k*k,

E_P2=diag (δ_1+k, δ_2+k..., δ_n)∈R^(n-k)*(n-k)

Owing to Q is low-rank, k < n, so $Q=U_Q{E}_Q{V}_Q^T=\left(\begin{array}{cc}{U}_{Q1}& U_{Q2}\end{array}\right)\left(\begin{array}{cc}{E}_{Q1}& 0\\ 0& E_{Q2}\end{array}\right)\left(\begin{array}{c}{V}_{Q1}^T\\ V_{Q2}^T\end{array}\right),$ Adopting minimum variance that Q is estimated, targeted transformation is the matrix M solving and meeting minimization problem:

$\arg min\left|\right|PM-Q|{|}_2^2,M\&Element;R^{m*n}$

Object function in above formula is Strict Convex and can be micro-, by asking first derivative to be zero:

M=(P^TP)^-1P^TQ

Thus, the minimum variance of QCan be expressed as:

$\hat{Q}=PM=P{\left(P^{T}P\right)}^{-1}{P}^{T}Q=U_P{U}_P^{T}Q$ Formula 3

Owing to noise is white Gaussian noise, therefore noise matrix N_rIt is full rank, so that noisy matrix P_rNon-singular matrix.Known P^TP is reversible.Owing to Q is unknown, formula (3) only has theory significance.Q is actually Q rectangular projection on P column space, can release the estimated value expression formula based on matrix P further with the relation of noisy matrix P and ideal matrix Q.

From formula (1) and Singular Value Decomposition Using character:

$\begin{array}{c}P=Q+N=\left(\begin{array}{cc}{U}_{Q1}& U_{Q2}\end{array}\right)\left(\begin{array}{cc}{E}_{Q1}& 0\\ 0& 0\end{array}\right)\left(\begin{array}{c}{V}_{Q1}^T\\ V_{Q2}^T\end{array}\right)+N\left(\begin{array}{cc}{V}_{Q1}& V_{Q2}\end{array}\right)\left(\begin{array}{c}{V}_{Q1}^T\\ V_{Q2}^T\end{array}\right)\\ =U_{Q1}{E}_{Q1}{V}_{Q1}^T+{\mathrm{NV}}_{Q1}{V}_{Q1}^T+{\mathrm{NV}}_{Q2}{V}_{Q2}^T\\ =\left\{\left(U_{Q1}{E}_{Q1}+{\mathrm{NV}}_{Q1}\right){\left(E_{Q1}^2+{\mathrm{\&tau;}}^{2}I\right)}^{-\frac{1}{2}}{\mathrm{\&tau;}}^{-1}{\mathrm{NV}}_{Q2}\right\}*\left\{\left(\begin{array}{cc}{\left(E_{Q1}^2+{\mathrm{\&tau;}}^{2}I\right)}^{-\frac{1}{2}}& 0\\ 0& \mathrm{\&tau;}I\end{array}\right)\left(\begin{array}{c}{V}_{Q1}^T\\ V_{Q2}^T\end{array}\right)\right\}\end{array}$

Formula 4

From formula 4 and formula 2:

$U_P=\left(\begin{array}{cc}{U}_{P1}& U_{P2}\end{array}\right)=\left\{(U_{Q1}{E}_{Q1}+{\mathrm{NV}}_{Q1}){(E_{Q1}^2+{\mathrm{\&tau;}}^{2}I)}^{-\frac{1}{2}}{\mathrm{\&tau;}}^{-1}{\mathrm{NV}}_{Q2}\right\}$ Formula 5

V_P=(V_P1V_P2)=(V_Q1V_Q2),

$E_{P1}={(E_{Q1}^2+{\mathrm{\&tau;}}^{2}I)}^{\frac{1}{2}}$ Formula 6

E_P2=τ I

Can be obtained by formula 5 and formula 6:

$U_P=\left(\begin{array}{cc}{U}_{P1}& U_{P2}\end{array}\right)=\left\{\begin{array}{cc}(U_{Q1}{E}_{Q1}+{\mathrm{NV}}_{Q1})E_{P1}^{-1}& U_{Q1}{E}_{Q1}{V}_{Q1}^T{\mathrm{NV}}_{Q2}\end{array}\right\}$ Formula 7

Formula 7 substitutes into formula 3 and obtains:

$\hat{Q}=U_P{U}_P^{T}Q=\left(\begin{array}{cc}{U}_{P1}& U_{P2}\end{array}\right)\left(\begin{array}{c}{U}_{Q1}^T\\ U_{Q2}^T\end{array}\right)U_{Q1}{E}_{Q1}{V}_{Q1}^T=U_{P1}{U}_{P1}^T{U}_{Q1}{E}_{Q1}{V}_{Q1}^T+$

$U_{P2}{U}_{P2}^T{U}_{Q1}{E}_{Q1}{V}_{Q1}^T$

$=U_{P1}{E}_{P1}^{-1}(E_{Q1}{V}_{Q1}^T+V_{Q1}^T{N}^T)U_{Q1}{E}_{Q1}{V}_{Q1}^T+{\mathrm{\&tau;}}^{-1}{U}_{P2}{V}_{P2}^T{N}^T{U}_{Q1}{E}_{Q1}{V}_{Q1}^T$ Formula 8

If noise is additive white Gaussian noise, then have:Formula 9

And $E_{Q1}^2=E_{P1}^2-{\mathrm{\&tau;}}^{2}I$ Formula 10

Formula 9 formula 10 substitutes into formula 8 and obtainsFinal estimated value be:

$\hat{Q}=U_{P1}(E_{P1}-{\mathrm{\&tau;}}^2{\mathrm{IE}}_{P1}^{-1})V_{P1}^T$ Formula 11

Due to E_P1For diagonal matrix, i.e. E_P1=diag (δ₁, δ₂..., δ_k)

Then: $E_{P1}-{\mathrm{\&tau;}}^2{\mathrm{IE}}_{P1}^{-1}=diag\left(\begin{array}{cccc}{\mathrm{\&delta;}}_1-\frac{{\mathrm{\&tau;}}^2}{{\mathrm{\&delta;}}_1},& {\mathrm{\&delta;}}_2-\frac{{\mathrm{\&tau;}}^2}{{\mathrm{\&delta;}}_2},& ...,& {\mathrm{\&delta;}}_k-\frac{{\mathrm{\&tau;}}^2}{{\mathrm{\&delta;}}_k}\end{array}\right)$

Thus it is known that the core that matrix Q carries out minimum variance estimate is the singular value to noisy matrix P

Carry out shrink process:

$Shrink({\mathrm{\&delta;}}_1=\widehat{{\mathrm{\&delta;}}_1}=)\left\{\begin{array}{cc}{\mathrm{\&delta;}}_i-\frac{{\mathrm{\&tau;}}^2}{{\mathrm{\&delta;}}_i},& i<k\\ 0,& i\&GreaterEqual;k\end{array}\right.$

Singular value after processing is reconstructed and obtains similar block matrix after denoisingRightAll column vectors carry out the image block after rearranging and can obtain denoising.

Four, image reconstruction

Overlap is had time due to piecemeal, namely a pixel would be likely to occur in multiple image block, that is after noise reduction process, each pixel or have multiple estimated value, need to carry out fusion treatment, adopt the weighted average of estimated value to obtain the end value of this pixel, in order to reduce complexity, this method adopts fixing weights, the namely similar block matrix to jthThe all pixels comprised carry out weights definition:

$w_j=\left\{\begin{array}{cc}1-\frac{k}{k+1},& k<K+1\\ \frac{1}{k+1},& k=K+1\end{array}\right.$

Define based on these weights, pixel x_iEstimated value be represented by:j∈S(x_i)

$W={\mathrm{\&Sigma;}}_j^L{w}_j,$

S(x_i)={ j | x_i∈Q_j, j=1 ..., L}

Once after the pixel estimated value of image determines, the estimated value of original image XAlso determine that.Namely complete the reconstruction of image.

Five, denoising quality is promoted

For noisy image, the existence of noise can reduce the accuracy of image block packet, thus affecting the estimated accuracy of successive image block matrix, ultimately resulting in and there is a small amount of noise residual in the image after denoising.Generally, the signal to noise ratio of noisy image is more low, and noise residual phenomena is more obvious.In order to solve this problem, adopting iterative processing, it may be assumed that by image after noisy image Y and denoisingResidual imageBy a certain percentage with denoising after image blend generate the noisy image that a width is newThen rightAgain carry out denoising.Generation process be represented by:

$\hat{Y}=\hat{X}+\mathrm{\&gamma;}(Y-\hat{X}),0<\mathrm{\&gamma;}<1$ Formula 12

γ is relaxation factor, in an experiment, is obtained in that better result when γ=0.2.

Above-described embodiment is intended merely to better explains the present invention, is not limit the present invention, and protection scope of the present invention is determined according to the content of claims.

Claims (6)

1. based on an iteration noise-reduction method for image low-rank, the method relates to computer information technology field, more particularly to image procossing, computer vision field, it is characterized in that: the enforcement step of the method is as follows:

Step one: input noisy image Y；

Step 2: utilize absolute deviation median method estimate image Y(or) noise level；

Step 3: to image Y(or) carry out piecemeal process, and each image block is found out similar image block according to algorithm, thus forming C similar block block matrix；

Step 4: to each similar block matrix, calculate its noisy similar block matrixEstimated value；

Step 5: calculateThe weights of contained image block；

Step 6: utilize the estimated value reconstruct image of image block；

Step 7: by noisy image Y(or) with denoising after imageResidual imageBy a certain percentage with denoising after image blend generate new noisy image；

Step 8: repeat step 2 to step 7 it is known that image reaches desirable level；

Step 9: the image after output denoising。

2. a kind of iteration noise-reduction method based on image low-rank according to claim 1, is characterized in that: image in step 3

Piecemeal and similarity decision method be:

Divide the image into the image subblock of 7*7, useRepresent,, owing to consider computation complexity and the speed of image procossing, so adopting the Euclidean distance based on gray feature to measure the similarity between image block；To any two image blocksWithIts similarity can represent:

Value more little, thenWithMore similar, for arbitrary image block, adopt above formula to carry out the similarity between coupling image block, find out K the image block similar to it, be recorded asNamely it is represented by:

Then image block can be expressed in matrix as:

。

3. a kind of iteration noise-reduction method based on image low-rank according to claim 1, is characterized in that: calculate in step 4

The estimated value method of its noisy similar block matrix is as follows:

OrderOrder be k,Then according to singular value decomposition theorem, the singular value decomposition of noisy similar block matrix P can be expressed as:

Owing to Q is low-rank,, so:

Adopting minimum variance that Q is estimated, targeted transformation is the matrix M solving and meeting minimization problem:

Object function in above formula is Strict Convex and can be micro-, by asking first derivative to be zero:

Thus, the minimum variance of QCan be expressed as:

Wherein

Thus calculate:

If noise is additive white Gaussian noise, then have:

And

So:。

4. a kind of iteration noise-reduction method based on image low-rank according to claim 1, is characterized in that: in step 5The weights of contained image blockComputational methods are as follows:

。

5. a kind of iteration noise-reduction method based on image low-rank according to claim 1, is characterized in that: utilize the estimated value reconstruct image of image block in step 6Method is as follows:

Define based on these weights, pixelEstimated value can be expressed as:

Once after the pixel estimated value of image determines, the estimated value QUOTE of original image X Also the reconstruction namely completing image has been determined that.

6. a kind of iteration noise-reduction method based on image low-rank according to claim 1, is characterized in that: in step 7, new noisy imageGeneration method as follows:

For relaxation factor, whenTime be obtained in that better result.