CN103279959B - A kind of two-dimension analysis sparse model, its dictionary training method and image de-noising method - Google Patents

Abstract

The present invention disclose a kind of make full use of image spatial coherence, need less training sample, a large amount of two-dimension analysis sparse model of saving the storage space of dictionary, and based on the dictionary training method of this model and image de-noising method.

Description

A kind of two-dimension analysis sparse model, its dictionary training method and image de-noising method

Technical field

The invention belongs to the technical field of signal modeling, relate to a kind of two-dimension analysis sparse model particularly, and based on the dictionary training method of this model and image de-noising method.

Background technology

Rarefaction representation be a kind of comparative maturity modeling pattern, to be widely studied, and in most of signal transacting field widespread use, as image denoising, textures synthesis, Video processing and Images Classification.Utilize rarefaction representation to signal modeling mainly comprises two classes: synthesize sparse modeling and analyze sparse modeling.

Synthetic model is defined as follows: x=Db, s.t.||b|| ₀=k, here be a complete dictionary of mistake, wherein an atom (primitive) is shown in each list. it is a sparse vector.L ₀norm || || ₀for characterizing the degree of rarefication of sparse signal, be defined as k nonzero element in a vector.In synthetic model, signal x can by the k in D atom linear expression ^[2].

Analytical model refers to a signal x, when being analyzed dictionary by one after taking advantage of, its result b meets b=Ω x, s.t.||b|| ₀=p-l, be a sparse signal, l represents the joint sparse degree of signal x, and wherein synthesizing sparse coefficient vector b has l neutral element.Contrast is with synthetic model, and analytical model emphasizes the position of neutral element in sparse coefficient more, because these positions indicate the orthogonal space of signal x, that is understands the orthogonal complement space belonging to signal.

Following three aspects are mainly comprised for the research analyzing sparse model: tracing algorithm (PursuitAlgorithm), dictionary training problem, and theoretical analysis.Past all analysis sparse model is when processing two dimensional image, all by arranging or being converted into one dimension height dimensional signal by row by it, result in following problem: first, the two-dimensional space of image is destructurized, local correlations between image is not used effectively, secondly, in order to obtain the estimation of effective robustness, the training sample in a large amount of higher dimensional spaces is needed.

Summary of the invention

Technology of the present invention is dealt with problems and is: overcome the deficiencies in the prior art, provides a kind of and makes full use of the spatial coherence of image, needs the two-dimension analysis sparse model of the storage space of less training sample, a large amount of saving dictionary.

Technical solution of the present invention is: this two-dimension analysis sparse model, and this model is formula (2)

$\{{\widehat{\mathrm{\Ω}}}_1,{\widehat{\mathrm{\Ω}}}_2,{\left\{{\hat{X}}_j\right\}}_{j=1}^M\}=\underset{{\mathrm{\Ω}}_1,{\mathrm{\Ω}}_2,{\left\{X_j\right\}}_{j=1}^M}{\arg \min }\underset{j=1}{\overset{M}{\mathrm{\Σ}}}{\left|\right|X_j-Y_j\left|\right|}_F^2$

$s.t.{\left|\right|{\mathrm{\Ω}}_1{X}_j{\mathrm{\Ω}}_2^T\left|\right|}_0\≤p-l,\∀1\≤j\≤M,---\left(2\right)$

${\left|\right|w_i^{\left(1\right)}\left|\right|}_2=1,\∀1\≤i\≤p_1,$

${\left|\right|w_k^{\left(2\right)}\left|\right|}_2=1,\∀1\≤k\≤p_2,$

Wherein p=p ₁× p ₂, with be respectively and characterize image block X _jthe horizontal dictionary of horizontal nature and vertical property and vertical dictionary, with be respectively Ω ₁the i-th row and Ω ₂row k vector, deriving the number of null element in sparse signal is l, and M is the number of image block in sample set, Y _jfor the image block element of the jth in sample set, be X _jadd the image block to be made an uproar after making an uproar.

Additionally provide the dictionary training method based on two-dimension analysis sparse model, comprise the following steps:

(1) construct training sample set I: to a secondary noisy image, the some image blocks of stochastic sampling are carried out to this image, and image block is combined to training sample concentrates, obtain training sample set wherein Y _jrepresent the size of the jth image block obtained that image sampled, represent real number field, its dimension is d ₁, M ₀=M × d ₁, M represents image block sample size;

(2) initialization two dictionary Ω ₁, Ω ₂: the pseudoinverse utilizing the discrete cosine transform dictionary of redundancy, initialization dictionary Ω ₁, Ω ₂;

(3) sparse coding: first pass through obtain the dictionary Ω that tensor generates, by training sample set in each block permutatation, obtain new samples collection wherein vec (Y _j) represent image block Y _jcarry out the result by column weight arrangement, wherein d=d ₁× d ₁so, to each column signal vec (Y _j), utilize formula (5) to solve reconstructed results vec (X _j)

$\left\{\mathrm{vec}\left({\hat{X}}_j\right)\right\}=\arg \underset{\mathrm{vec}\left(X_j\right)}{\min }{\left|\right|\mathrm{vec}\left(Y_j\right)-\mathrm{vec}\left(X_j\right)\left|\right|}_2^2,---\left(5\right);$

$s.t.{\mathrm{\Ω}}_{{\mathrm{\Λ}}_j}\mathrm{vec}\left(X_j\right)=0,\mathrm{Rank}\left({\mathrm{\Ω}}_{{\mathrm{\Λ}}_j}\right)=d-r$

(4) dictionary updating: as renewal Ω ₂time, suppose Ω ₁be given, utilize U ₁=[(Ω ₁x ₁) ^t, (Ω ₁x ₂) ^t, Λ, (Ω ₁x _m) ^t] obtain estimate V ₁, objective function is formula (6), when trying to achieve the V of estimation ₁after, by finding V ₁in with current certain row primitive to be estimated orthogonal index line J, then correspondence finds U ₁submatrix under manipulative indexing J, then solution formula (7), realizes dictionary Ω ₂row k upgrade, to Ω ₁renewal identical therewith

$\{{\widehat{\mathrm{\Ω}}}_2,{\left\{{\hat{v}}_j\right\}}_{j=1}^N\}=\underset{{\mathrm{\Ω}}_2,{\left\{v_j\right\}}_{j=1}^N}{\arg \min }\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{\left|\right|u_j-v_j\left|\right|}_2^2$

$s.t.{\mathrm{\Ω}}_{{2\mathrm{\Λ}}_j}{u}_j=0,\∀1\≤j\≤N,$

$\mathrm{Rank}\left({\mathrm{\Ω}}_{{2\mathrm{\Λ}}_j}\right)=d_1-r_2$

${\left|\right|w_k^{\left(2\right)}\left|\right|}_2=1,\∀1\≤i\≤p_2---\left(6\right)$

$\left\{w_k^{\left(2\right)}\right\}=\arg \min \left|\right|w_k^{\left(2\right)}{U}_J\left|\right|,s.t.{\left|\right|w_k^{\left(2\right)}\left|\right|}_2=1---\left(7\right);$

(5) judge whether to reach iteration stopping condition: if meet iteration stopping condition, get back to step (3), otherwise return output dictionary Ω ₁, Ω ₂, complete the training of dictionary.

Additionally provide a kind of based on two-dimension analysis sparse model and the image de-noising method of training dictionary thereof, comprise the following steps:

(1) image block utilizing noisy image configuration to be solved;

(2) two are utilized to train dictionary Ω ₁, Ω ₂solve required dictionary Ω in one dimension sparse coding;

(3) one dimension sparse reconstruction method is utilized to solve the reconstructed value of Y ;

(4) above-mentioned N number of reconstruction image block is utilized to obtain denoising image.

Two dictionary Ω are considered in the two-dimension analysis model proposed in the present invention ₁, Ω ₂respectively from the characteristic of horizontal direction and Vertical Square always picture engraving, the space structure of image and the spatial coherence of image can be made full use of like this, so only need to train the less dictionary of storage space just can reflect picture characteristics adaptively, utilize and train the dictionary obtained to carry out image denoising, original image characteristic can be excavated better from image to be made an uproar, thus realize image denoising better.Embodiments of the invention reflection is as dictionary Ω ₁, Ω ₂storage size much smaller than the size of the storage space of dictionary Ω in traditional one-dimensional analytical model, but with dictionary Ω ₁, Ω ₂the Ω that tensor generates ₀when storage space is equal, the denoising effect of denoising effect of the present invention and traditional sparse model is suitable.And when dictionary in conventional model is little of time close with the size of dictionary of the present invention, denoising effect obviously declines, therefore dictionary storage space is being reduced in the present invention, still can ensure the denoising effect of image, and conventional model dictionary is when being reduced to equal with dictionary storage size of the present invention, denoising effect then obviously declines, and denoising effect of the present invention will be better than conventional model far away, and dictionary training algorithm of the present invention has certain reduction compared with on conventional analytical model dictionary training algorithm algorithm complex.

Accompanying drawing explanation

Fig. 1 shows the process flow diagram according to the dictionary training method based on two-dimension analysis sparse model of the present invention;

Fig. 2 shows and utilizes the process flow diagram of image de-noising method based on two-dimension analysis sparse model and training dictionary thereof of the present invention.

Embodiment

This two-dimension analysis sparse model, this model is formula (2)

$\{{\widehat{\mathrm{\Ω}}}_1,{\widehat{\mathrm{\Ω}}}_2,{\left\{{\hat{X}}_j\right\}}_{j=1}^M\}=\underset{{\mathrm{\Ω}}_1,{\mathrm{\Ω}}_2,{\left\{X_j\right\}}_{j=1}^M}{\arg \min }\underset{j=1}{\overset{M}{\mathrm{\Σ}}}{\left|\right|X_j-Y_j\left|\right|}_F^2$

$s.t.{\left|\right|{\mathrm{\Ω}}_1{X}_j{\mathrm{\Ω}}_2^T\left|\right|}_0\≤p-l,\∀1\≤j\≤M,---\left(2\right)$

${\left|\right|w_i^{\left(1\right)}\left|\right|}_2=1,\∀1\≤i\≤p_1,$

${\left|\right|w_k^{\left(2\right)}\left|\right|}_2=1,\∀1\≤k\≤p_2,$

Wherein p=p ₁× p ₂, with be respectively and characterize image block X _jthe horizontal dictionary of horizontal nature and vertical property and vertical dictionary, with be respectively Ω ₁the i-th row and Ω ₂row k vector, deriving the number of null element in sparse signal is l, and M is the number of image block in sample set, Y _jfor the image block element of the jth in sample set, be X _jadd the image block to be made an uproar after making an uproar.

Additionally provide the dictionary training method based on two-dimension analysis sparse model, comprise the following steps:

(3) construct training sample set I: to a secondary noisy image, the some image blocks of stochastic sampling are carried out to this image, and image block is combined to training sample concentrates, obtain training sample set , wherein Y _jrepresent the size of the jth image block obtained that image sampled, represent real number field, its dimension is d ₁, M ₀=M × d ₁, M represents image block sample size;

(4) initialization two dictionary Ω ₁, Ω ₂: the pseudoinverse utilizing the discrete cosine transform dictionary of redundancy, initialization dictionary Ω ₁, Ω ₂;

(3) sparse coding: first pass through obtain the dictionary Ω that tensor generates, by training sample set in each block permutatation, obtain new samples collection wherein vec (Y _j) represent image block Y _jcarry out the result by column weight arrangement, wherein d=d ₁× d ₁so, to each column signal vec (Y _j), utilize formula (5) to solve reconstructed results vec (X _j)

$\left\{\mathrm{vec}\left({\hat{X}}_j\right)\right\}=\arg \underset{\mathrm{vec}\left(X_j\right)}{\min }{\left|\right|\mathrm{vec}\left(Y_j\right)-\mathrm{vec}\left(X_j\right)\left|\right|}_2^2,---\left(5\right);$

$s.t.{\mathrm{\Ω}}_{{\mathrm{\Λ}}_j}\mathrm{vec}\left(X_j\right)=0,\mathrm{Rank}\left({\mathrm{\Ω}}_{{\mathrm{\Λ}}_j}\right)=d-r$

(4) dictionary updating: as renewal Ω ₂time, suppose Ω ₁be given, utilize

U ₁=[(Ω ₁x ₁) ^t, (Ω ₁x ₂) ^t, Λ, (Ω ₁x _m) ^t] obtain estimate V ₁, objective function is formula (6), when trying to achieve the V of estimation ₁after, by finding V ₁in with current certain row primitive to be estimated orthogonal index line J, then correspondence finds U ₁submatrix under manipulative indexing J, then solution formula (7), realizes dictionary Ω ₂row k upgrade, to Ω ₁renewal identical therewith

$\{{\widehat{\mathrm{\Ω}}}_2,{\left\{{\hat{v}}_j\right\}}_{j=1}^N\}=\underset{{\mathrm{\Ω}}_2,{\left\{v_j\right\}}_{j=1}^N}{\arg \min }\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{\left|\right|u_j-v_j\left|\right|}_2^2$

$s.t.{\mathrm{\Ω}}_{{2\mathrm{\Λ}}_j}{u}_j=0,\∀1\≤j\≤N,$

$\mathrm{Rank}\left({\mathrm{\Ω}}_{{2\mathrm{\Λ}}_j}\right)=d_1-r_2$

${\left|\right|w_k^{\left(2\right)}\left|\right|}_2=1,\∀1\≤i\≤p_2---\left(6\right)$

$\left\{w_k^{\left(2\right)}\right\}=\arg \min \left|\right|w_k^{\left(2\right)}{U}_J\left|\right|,s.t.{\left|\right|w_k^{\left(2\right)}\left|\right|}_2=1---\left(7\right);$

(5) judge whether to reach iteration stopping condition: if meet iteration stopping condition, get back to step (3), otherwise return output dictionary Ω ₁, Ω ₂, complete the training of dictionary.

Additionally provide a kind of based on two-dimension analysis sparse model and the image de-noising method of training dictionary thereof, comprise the following steps:

(1) image block utilizing noisy image configuration to be solved;

(2) two are utilized to train dictionary Ω ₁, Ω ₂solve required dictionary Ω in one dimension sparse coding;

(3) one dimension sparse reconstruction method is utilized to solve the reconstructed value of Y ;

(4) above-mentioned N number of reconstruction image block is utilized to obtain denoising image.

In traditional analysis sparse model by image by row or be scanned into vector form by row, destroy the space structure of image like this, the spatial coherence of image cannot be made full use of, as the element of image block the first row and the element of last column non-conterminous, correlativity is more weak, and have passed through by after column scan, in training dictionary process, think that they are strong correlations, train the result of the dictionary obtained necessarily can not reflect the spatial coherence of image well like this.In tradition sparse representation model, dictionary reflects the characteristic of the vectorization image block orthogonal complement space, namely only considered the characteristic of an image dimension, and considers two dictionary Ω in the two-dimension analysis model proposed in the present invention ₁, Ω ₂respectively from the characteristic of horizontal direction and Vertical Square always picture engraving, the space structure of image and the spatial coherence of image can be made full use of like this, so only need to train the less dictionary of storage space just can reflect picture characteristics adaptively, utilize and train the dictionary obtained to carry out image denoising, original image characteristic can be excavated better from image to be made an uproar, thus realize image denoising better.Embodiments of the invention reflection is as dictionary Ω ₁, Ω ₂storage size much smaller than the size of the storage space of dictionary Ω in traditional one-dimensional analytical model, but with dictionary Ω ₁, Ω ₂the Ω that tensor generates ₀when storage space is equal, the denoising effect of denoising effect of the present invention and traditional sparse model is suitable.And when dictionary in conventional model is little of time close with the size of dictionary of the present invention, denoising effect obviously declines, therefore dictionary storage space is being reduced in the present invention, still can ensure the denoising effect of image, and conventional model dictionary is when being reduced to equal with dictionary storage size of the present invention, denoising effect then obviously declines, and denoising effect of the present invention will be better than conventional model far away, and dictionary training algorithm of the present invention has certain reduction compared with on conventional analytical model dictionary training algorithm algorithm complex.

Preferably, iterated conditional is that whether iterations reaches upper limit L or whether noise error reaches designated value.

Illustrate the specific embodiment of this method below.

For the ease of the understanding of hereinafter formula and symbol, first provide the explanation of a little symbolic formula here.Hereinafter black upper case character representing matrix: as matrix X, and this matrix is designated as vec (X) by the vector form after column weight arrangement.Black lowercase character represents vector, as vector x, and vector norm is defined as here x _jit is the element of the jth in vector x.|| || ₀represent l ₀norm, is used for the number of the vectorial nonzero element of sign one.The F norm of matrix refers to wherein x _ijthe i-th row jth row of representing matrix X.The Kroneckerproduct (tensor product) of matrix U and V is defined as:

$U\⊗V=\left(\begin{array}{ccc}{u}_{\mathrm{1,1}}V& u_{\mathrm{1,2}}V& \mathrm{\Λ}\\ u_{\mathrm{2,1}}V& u_{\mathrm{2,2}}V& \\ M& & O\end{array}\right).$

In order to introduce two-dimension analysis sparse model better, first the present invention introduces One Dimension Analysis sparse model.Matrix X represents that size is d ₁× d ₁image block, definition Y=X+V, wherein V is noise block.When given training sample set is here M ₀=M × d ₁, the training process of one dimension dictionary is defined as:

$\{\widehat{\mathrm{\Ω}},\{\mathrm{vec}\left({\hat{X}}_j\right){\}}_{j=1}^M\}=\underset{\mathrm{\Ω},{\left\{\mathrm{vec}\left(X_j\right)\right\}}_{j=1}^M}{\arg \min }\underset{j=1}{\overset{M}{\mathrm{\Σ}}}{\left|\right|\mathrm{vec}\left(X_j\right)-\mathrm{vec}\left(Y_j\right)\left|\right|}_2^2$

$s.t.{\left|\right|\mathrm{\Ωvec}\left(X_j\right)\left|\right|}_0\≤-p-l,\∀1\≤j\≤M,---\left(1\right)$

${\left|\right|w_i\left|\right|}_2=1,\∀1\≤i\≤p,$

Here vec (X _j) indicate without noise cancellation signal, vectorial w _irepresent redundancy analysis dictionary the i-th row vector.One Dimension Analysis sparse model hypothesis signal vec (X _j) be at least orthogonal to the capable signal of l in Ω, and meet || Ω vec (X _j) || ₀≤ p-l.

On the basis of above model, it is as follows that the present invention introduces two-dimension analysis model:

$\{{\widehat{\mathrm{\Ω}}}_1,{\widehat{\mathrm{\Ω}}}_2,{\left\{{\hat{X}}_j\right\}}_{j=1}^M\}=\underset{{\mathrm{\Ω}}_1,{\mathrm{\Ω}}_2,{\left\{X_j\right\}}_{j=1}^M}{\arg \min }\underset{j=1}{\overset{M}{\mathrm{\Σ}}}{\left|\right|X_j-Y_j\left|\right|}_F^2$

$s.t.{\left|\right|{\mathrm{\Ω}}_1{X}_j{\mathrm{\Ω}}_2^T\left|\right|}_0\≤p-l,\∀1\≤j\≤M,---\left(2\right)$

${\left|\right|w_i^{\left(1\right)}\left|\right|}_2=1,\∀1\≤i\≤p_1,$

${\left|\right|w_k^{\left(2\right)}\left|\right|}_2=1,\∀1\≤k\≤p_2,$

Here p=p ₁× p ₂, with be respectively horizontal dictionary and vertical dictionary, with be respectively Ω ₁the i-th row and Ω ₂row k vector.Two dictionaries pointed out by two-dimension analysis model---horizontal dictionary and vertical dictionary, characterize image block X respectively _jhorizontal nature and vertical property, when acting on image block X _jtime, a sparse signal can be obtained namely same this point it is emphasized that derive the position of null element in sparse signal, its null element number is l.

Ω ₁and Ω ₂two dictionaries play an important role at two-dimension analysis model.If definition A ₁=Ω ₁x, A ₂=Ω ₂x ^t, mean Ω ₁and Ω ₂be respectively X and X ^tdictionary, now A ₁, A ₂for sparse coefficient, and as dictionary Ω ₁time fixing, then Ω ₂can regard as sparse dictionary, otherwise as dictionary Ω ₂time fixing, then Ω ₁can regard as sparse dictionary.And as dictionary Ω ₁, Ω ₂when all fixing, two-dimension analysis model can be converted into One Dimension Analysis model, and meets these characteristics are using the foundation as dictionary training.

Two benches Aries In The Block By Block Relaxation method can be adopted in the present invention to solve optimization problem-formula (2), mainly comprise two key link sparse codings and dictionary updating.The object of sparse coding is at given dictionary Ω ₁, Ω ₂time, estimate reconstruction signal dictionary updating object is then to use above reconstruction signal to dictionary Ω ₁, Ω ₂upgrade.

The sparse coding stage

As simultaneously given Ω ₁, Ω ₂time, rebuild can be converted into and solve each X _jask following optimization problem:

${\hat{X}}_j=\arg \underset{X_j}{\min }{\left|\right|Y_j-X_j\left|\right|}_F^2,s.t.{\left|\right|{\mathrm{\Ω}}_1{X}_j{\mathrm{\Ω}}_2^T\left|\right|}_0\≤p-l---\left(3\right)$

Owing to working as Ω ₁, Ω ₂when all fixing, order then the sparse Solve problems of two-dimension analysis can be converted into the sparse Solve problems of following One Dimension Analysis:

$\left\{\mathrm{vec}\left({\hat{X}}_j\right)\right\}=\arg \underset{\mathrm{vec}\left(X_j\right)}{\min }{\left|\right|\mathrm{vec}\left(Y_j\right)-\mathrm{vec}\left(X_j\right)\left|\right|}_F^2,---\left(4\right)$

$s.t.{\left|\right|\mathrm{\Ωvec}\left(X_j\right)\left|\right|}_0\≤p-l,{\left|\right|{\mathrm{\ω}}_i\left|\right|}_2=\mathrm{1,1}\≤i\≤p$

Here ω _irepresent i-th row of Ω.Above problem can be solved by one dimension sparse coding method, and as greedy algorithm (GAP), reverse greedy algorithm (BGA), optimizes reverse greedy algorithm (xBG) etc. and solve.

The problems referred to above are converted into following problem solving by the present invention:

$\left\{\mathrm{vec}\left({\hat{X}}_j\right)\right\}=\arg \underset{\mathrm{vec}\left(X_j\right)}{\min }{\left|\right|\mathrm{vec}\left(Y_j\right)-\mathrm{vec}\left(X_j\right)\left|\right|}_2^2,---\left(5\right)$

$s.t.{\mathrm{\Ω}}_{{\mathrm{\Λ}}_j}\mathrm{vec}\left(X_j\right)=0,\mathrm{Rank}\left({\mathrm{\Ω}}_{{\mathrm{\Λ}}_j}\right)=d-r$

Here Λ _jbe orthogonal to signal vec (X _j) the set of the capable index of l, expression manipulative indexing is Λ _jthe submatrix of Ω, therefore and r represents signal vec (X _j) belonging to the dimension in space, therefore $\mathrm{Rank}\left({\mathrm{\Ω}}_{{\mathrm{\Λ}}_j}\right)=d-r.$

The dictionary updating stage:

The object of dictionary training process uses upgrade dictionary Ω ₁, Ω ₂although, but obtain estimated value, but to dictionary Ω ₁, Ω ₂renewal process remain non-convex optimization problem, therefore the present invention adopts alternately optimal algorithm to upgrade dictionary.For Ω ₁, Ω ₂optimization procedure be similar, below with Ω ₂renewal process be example, provide corresponding algorithm.

Define in the present invention n=p ₁× M, utilize the characteristic of two-dimension analysis model in above-mentioned analysis, then Ω ₂training set U ₁dictionary.Therefore Ω ₂renewal process be converted into following one dimension dictionary learning problem:

$\{{\widehat{\mathrm{\Ω}}}_2,{\left\{{\hat{v}}_j\right\}}_{j=1}^N\}=\underset{{\mathrm{\Ω}}_2,{\left\{v_j\right\}}_{j=1}^N}{\arg \min }\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{\left|\right|u_j-v_j\left|\right|}_2^2$

$s.t.{\mathrm{\Ω}}_{{2\mathrm{\Λ}}_j}{u}_j=0,\∀1\≤j\≤N,---\left(6\right)$

$\mathrm{Rank}\left({\mathrm{\Ω}}_{{2\mathrm{\Λ}}_j}\right)=d_1-r_2$

${\left|\right|w_k^{\left(2\right)}\left|\right|}_2=1,\∀1\≤i\≤p_2$

Here u _ju ₁jth row, v _jand V ₁u respectively _jand U ₁estimation.R ₂characterize signal u _jthe dimension in affiliated space.

Be similar to sparse solution procedure (5), first we fix Ω ₂, then one arrange renewal V ₁.And when upgrading dictionary Ω ₂in each row time, we are by V ₁in be orthogonal to the set of row when being designated as J, then U _jrepresent U ₁comprise the submatrix of those row of index in J, therefore right renewal can by solving following problem solving:

$\left\{{\hat{w}}_k^{\left(2\right)}\right\}=\underset{w_k^{\left(2\right)}}{\arg \min }\left|\right|w_k^{\left(2\right)}{U}_J\left|\right|,s.t.{\left|\right|w_k^{\left(2\right)}\left|\right|}_2=1---\left(7\right)$

This is a SVD problem: sample set U _jthe proper vector of minimal eigenvalue corresponding to autocorrelation matrix be exactly required solution.

Here the algorithm complex in the two-dimension analysis sparse model proposed in the present invention in dictionary training process and dictionary storage size is provided.The accurate svd algorithm complexity of the matrix of a m × n is O (min{mn ², nm ²).In One Dimension Analysis sparse model, carrying out more new capital to each row of dictionary needs to carry out a SVD, and its algorithm complex is O (d ²m) (d<<M), then the algorithm complex needed for dictionary upgrading whole one dimension sparse model is O (pd ²m).Then, in two-dimentional sparse model, need the matrix size performing SVD operation to be d ₁× N, N=p ₁× M, therefore each row of dictionary carries out upgrading required algorithm complex is O (d ₁ ²n)=O (dp ₁m), dictionary Ω is upgraded ₁, Ω ₂comprise p respectively ₁and p ₂row primitive, therefore, the algorithm complex upgraded needed for all dictionaries is suppose then the algorithm complex of the dictionary updating of the present invention's proposition is O (pdM).For dictionary storage space, storage size required in the present invention is (p ₁+ p ₂) × d ₁individual pixel, then required in one-dimensional model storage size is p × d, p=p ₁× p ₂, obviously, algorithm complex and the dictionary storage space of training dictionary is greatly reduced in the present invention.

In order to the validity of two-dimension analysis sparse model and the dictionary training method proposed in the present invention is described, the above theory of the present invention general and algorithm application are in image denoising.

The test pattern used in the present invention is: ' Lena', ' Barbara', ' Boats', ' House', ' Pepper'.We add the white Gaussian noise of square error σ=5 to above signal, then extract 20 at random, and 000 block size is the image block composition training set of 7 × 7, for training dictionary.Utilize the pseudoinverse of redundant discrete cosine DCT to two dictionary Ω ₁, Ω ₂carry out initialization.The iterations arranged in training process is 20 times, solves in the process of (5) and makes r=6.When training obtains Ω ₁, Ω ₂after, the present invention utilizes the dictionary of training to carry out denoising to the above-mentioned image of making an uproar that adds, and gives the dictionary training result of One Dimension Analysis model in prior art and corresponding denoising result simultaneously, thus the validity of the model and algorithm proposed in the present invention is described.

In order to verify with the denoising effect of epigraph, mainly by Y-PSNR (PeakSignaltoNoiseRatio, PSNR) tolerance, unit is decibel (dB).Its computing formula is as follows:

$\mathrm{PSNR}=10\&CenterDot;{\log }_{10}\left(\frac{{255}^2}{\mathrm{MSE}}\right)---\left(8\right)$

Two width sizes are that the square error MSE of the image of m*n is defined as follows:

$\mathrm{MSE}=\frac{1}{m\&times;n}\underset{x=0}{\overset{m-1}{\mathrm{\&Sigma;}}}\underset{y=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{\left|\right|I(x,y)-J(x,y)\left|\right|}^2---\left(9\right)$

Wherein I, J represents the image of original not Noise respectively and utilizes sparse coding method to rebuild image, and I (x, y), J (x, y) is for corresponding to position (x, y) pixel value at place, then square error is less, then PSNR is higher, then the denoising effect of the method is higher.

Table 1 gives denoising result with PSNR form.First three columns is the experimental result in the present invention, and corresponding dictionary size is 7 × 7,8 × 7 respectively, the situation of 9 × 7.Last row are that in prior art, dictionary size is the result of 64 × 49.Compared with the method for prior art, the two dimensional model proposed in the present invention is when using less dictionary, but can reach the denoising effect suitable compared with big dictionary with one-dimensional model, and have 3 sub-pictures in 5 sub-pictures, even denoising effect can higher than one-dimensional model (experimental data of black matrix).

The present invention compares two-dimension analysis model and One Dimension Analysis model further when the denoising result in the close situation of storage size needed for dictionary.In this test, the dictionary of One Dimension Analysis model is 4 × 49, and training dictionary result when the dictionary size that the present invention provides is 8 × 7 and denoising result, its result is as shown in table 2.Compared with the method for prior art, when the size of prior art dictionary is suitable with dictionary size of the present invention, denoising result then can not show a candle to result of the present invention.Obviously, when dictionary size is same or equivalent, two-dimension analysis model can obtain better denoising effect than One Dimension Analysis model.

Table 1

Table 2

By above result, can find out that the two-dimension analysis sparse model proposed in the present invention can utilize the Two-Dimensional Correlativity in two dimensional image better.The model provided in the present invention simultaneously, when dictionary size is much smaller than One Dimension Analysis model dictionary, still can reach suitable denoising effect, and when in one dimension dictionary size and the present invention during planar dictionary sizableness, then the denoising effect in the present invention is better than the denoising effect of one-dimensional model far away.Obviously, also illustrate that the validity of dictionary storage space in the present invention.

Two-dimension analysis model dictionary training algorithm embodiment

1. first construct training sample set I

To a secondary noisy image, some image blocks of stochastic sampling can be carried out to this image.As sample 7 × 7 image block, and image block be combined to training sample concentrate, obtain training sample set wherein Y _jrepresent and image is sampled the jth d obtained ₁× d ₁the size of the image block M image block of=7 × 7, represent real number field, its dimension is d ₁, M ₀=M × d ₁, M represents image block sample size.As M=20000 in the present embodiment.

2. initialization two dictionary Ω ₁, Ω ₂

Utilize the pseudoinverse of discrete cosine transform (DCT) dictionary of redundancy, initialization dictionary Ω ₁, Ω ₂, two dictionary size are all set to 8 × 7.

3. dictionary Ω ₁, Ω ₂training---sparse coding

Utilize the algorithm in form 1, first pass through obtain the dictionary Ω that tensor generates, by original training sample collection in each block permutatation, obtain new sample set wherein vec (Y _j) represent image block Y _jcarry out the result by column weight arrangement, wherein d=d ₁× d ₁so, to each column signal vec (Y _j), utilize formula (5) or (10) to solve reconstructed results vec (X _j).

$\left\{\mathrm{vec}\left({\hat{X}}_j\right)\right\}=\arg \underset{\mathrm{vec}\left(X_j\right)}{\min }{\left|\right|\mathrm{vec}\left(Y_j\right)-\mathrm{vec}\left(X_j\right)\left|\right|}_2^2,---\left(10\right)$

$s.t.{\mathrm{\&Omega;}}_{{\mathrm{\&Lambda;}}_j}\mathrm{vec}\left(X_j\right)=0,\mathrm{Rank}\left({\mathrm{\&Omega;}}_{{\mathrm{\&Lambda;}}_j}\right)=d-r$

4. dictionary Ω ₁, Ω ₂training---dictionary updating

To upgrade Ω ₂for example, suppose Ω ₁be given, more than utilizing, solve the reconstructed results obtained and original picture block can calculate

U ₁=[(Ω ₁X ₁) ^T,(Ω ₁X ₂) ^T,Λ,(Ω ₁X _M) ^T]，V ₁=[(Ω ₁Y ₁) ^T,(Ω ₁Y ₂) ^T,Λ,(Ω ₁Y _M) ^T]。Then deteriorate to dictionary Ω at present ₂renewal process.But directly upgrade Ω ₂more difficult, due to Ω ₂v ₁dictionary, therefore to Ω ₂the renewal of dictionary, needs to find V ₁in orthogonal with it submatrix.Therefore first need to utilize U ₁=[(Ω ₁x ₁) ^t, (Ω ₁x ₂) ^t, Λ, (Ω ₁x _m) ^t] obtain estimate V ₁.Objective function is (6), and in solution procedure, essence solves the problem being similar to (10).When trying to achieve the V of estimation ₁after, then by finding V ₁in with current certain row primitive to be estimated orthogonal index line J, then correspondence finds U ₁submatrix under manipulative indexing J, then Solve problems (7) or (11), can to the row k of dictionary upgrade.

$\left\{w_k^{\left(2\right)}\right\}=\arg \min \left|\right|w_k^{\left(2\right)}{U}_j\left|\right|,s.t.{\left|\right|w_k^{\left(2\right)}\left|\right|}_2=1---\left(11\right)$

Complete successively Ω ₂the renewal of every a line.Ω ₁computation process the same.

5. judge whether to reach iteration stopping condition: as whether iterations reaches upper limit L, noise error reaches certain restriction etc.If still meet iterated conditional to get back to the 3rd step, carry out sparse coding again and dictionary updating, when not meeting iterated conditional, then return and export dictionary Ω ₁, Ω ₂.Complete the training of dictionary.

Two dimension synthetic model image denoising embodiment

1. utilize the image block that noisy image configuration is to be solved.

Known noisy image is carried out to the block sampling of 7 × 7, and in sampling process, use has overlapping mode to sample, lap is overlap=1.Sampling N block, then and by image block be arranged in the image treating sparse reconstruction of 7 × 7N, obtain set Y to be reconstructed altogether.

2. utilize two to train dictionary Ω ₁, Ω ₂solve required dictionary Ω in one dimension sparse coding

Pass through obtain the dictionary Ω that tensor generates..

3. utilize traditional one dimension sparse reconstruction method to solve the reconstructed value of Y .

The sparse reconstruction algorithm that the present embodiment adopts one dimension traditional---optimum oppositely greedy algorithm (xBG), carries out permutatation to each image block in above Y, obtains to every column signal vec (Y wherein _j) rebuild, finally then its reconstructed results is vec (X _j), then to [vec (X ₁), vec (X ₂), Λ, vec (X _m)] carry out inverse permutatation operation, obtain the reconstruction image block set of the form of image block.

4. utilize above-mentioned N number of reconstruction image block to obtain denoising image.

According to the overlap mode of the sample mode in sampling process and respective image block, the N number of reconstruction image block obtained at present being recovered back original image size again, corresponding to there being overlapping place, then adopting the operation of averaging.If namely certain pixel is had by m=6 block simultaneously, then the value that this block is final is the mean value corresponding to this pixel on its total sampling block.Finally can recover the image obtaining the denoising of rebuilding.

The above; it is only preferred embodiment of the present invention; not any pro forma restriction is done to the present invention, every above embodiment is done according to technical spirit of the present invention any simple modification, equivalent variations and modification, all still belong to the protection domain of technical solution of the present invention.

Claims (4)

1. a two-dimension analysis sparse model, is characterized in that: this model is formula (2)

$\{{\widehat{\mathrm{\&Omega;}}}_1,{\widehat{\mathrm{\&Omega;}}}_2,{\left\{{\hat{X}}_j\right\}}_{j=1}^M\}=\underset{{\mathrm{\&Omega;}}_1,{\mathrm{\&Omega;}}_2,{\left\{X_j\right\}}_{j=1}^M}{\arg \min }\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}\left|\right|X_j-Y_j|{|}_F^2$

$\begin{array}{cc}s.t.& \left|\right|{\mathrm{\&Omega;}}_1{X}_j{\mathrm{\&Omega;}}_2^T|{|}_0\&le;p-l,\&ForAll;1\&le;j\&le;M,\end{array}---\left(2\right)$

$\left|\right|w_i^{\left(1\right)}|{|}_2=1,\&ForAll;1\&le;i\&le;p_1,$

$\left|\right|w_k^{\left(2\right)}|{|}_2=1,\&ForAll;1\&le;k\&le;p_2,$

Wherein p=p ₁× p ₂, with be respectively and characterize image block X _jthe horizontal dictionary of horizontal nature and vertical property and vertical dictionary, p ₁, p ₂represent dictionary Ω respectively ₁, Ω ₂the number of row vector, with be respectively Ω ₁the i-th row and Ω ₂row k vector, deriving the number of null element in sparse signal is l, and M is the number of image block in sample set, Y _jfor the image block element of the jth in sample set, be X _jadd the image block to be made an uproar after making an uproar.

2., based on a dictionary training method for two-dimension analysis sparse model, it is characterized in that: comprise the following steps:

(1) construct training sample set I: to a secondary noisy image, the some image blocks of stochastic sampling are carried out to this image, and image block is combined to training sample concentrates, obtain training sample set wherein Y _jrepresent the size of the jth image block obtained that image sampled, represent real number field, its dimension is d ₁, M ₀=M × d ₁, M represents image block sample size;

(2) initialization two dictionary Ω ₁, Ω ₂: the pseudoinverse utilizing the discrete cosine transform dictionary of redundancy, initialization dictionary Ω ₁, Ω ₂;

(3) sparse coding: first pass through obtain the dictionary Ω that tensor generates, by training sample set in each block permutatation, obtain new samples collection wherein vec (Y _j) represent image block Y _jcarry out the result by column weight arrangement, wherein d=d ₁× d ₁so, to each column signal vec (Y _j), utilize formula (5) to solve reconstructed results vec (X _j)

$\left\{vec\left({\hat{X}}_j\right)\right\}=\arg \underset{vec\left(X_j\right)}{min}\left|\right|vec\left(Y_j\right)-vec\left(X_j\right)|{|}_2^2,---\left(5\right)$

$\begin{array}{cc}s.t.& {\mathrm{\&Omega;}}_{{\mathrm{\&Lambda;}}_j}vec\left(X_j\right)=0,Rank\left({\mathrm{\&Omega;}}_{{\mathrm{\&Lambda;}}_j}\right)=d-r\end{array}$

Wherein r is signal the dimension of belonging subspace;

(4) dictionary updating: as renewal Ω ₂time, suppose Ω ₁be given, utilize U ₁=[(Ω ₁x ₁) ^t, (Ω ₁x ₂) ^t..., (Ω ₁x _m) ^t] obtain estimate V ₁, objective function is formula (6), when trying to achieve the V of estimation ₁after, by finding V ₁in with current certain row primitive to be estimated orthogonal index line J, then correspondence finds U ₁submatrix under manipulative indexing J, then solution formula (7), realizes dictionary Ω ₂row k upgrade, to Ω ₁renewal identical therewith

$\{{\widehat{\mathrm{\&Omega;}}}_2,{\left\{{\hat{v}}_j\right\}}_{j=1}^N\}=\underset{{\mathrm{\&Omega;}}_2,{\left\{v_j\right\}}_{j=1}^N}{\arg \min }\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}\left|\right|u_j-v_j|{|}_2^2$

$\begin{array}{cc}s.t.& {\mathrm{\&Omega;}}_{2{\mathrm{\&Lambda;}}_j}{u}_j=0,\&ForAll;1\&le;j\&le;N\end{array},$

$Rank\left({\mathrm{\&Omega;}}_{2{\mathrm{\&Lambda;}}_j}\right)=d_1-r_2$

$\left|\right|w_k^{\left(2\right)}|{|}_2=1,\&ForAll;1\&le;i\&le;p_2---\left(6\right)$

$\left\{w_k^{\left(2\right)}\right\}=\arg min\left|\right|w_k^{\left(2\right)}{U}_J\left|\right|,s.t.\left|\right|w_k^{\left(2\right)}|{|}_2=1---\left(7\right);$

(5) judge whether to reach iteration stopping condition: if meet iteration stopping condition, get back to step (3), otherwise return output dictionary Ω ₁, Ω ₂, complete the training of dictionary.

3. the dictionary training method based on two-dimension analysis sparse model according to claim 2, is characterized in that: iterated conditional is that whether iterations reaches upper limit L or whether noise error reaches designated value.

4., based on two-dimension analysis sparse model and an image de-noising method of training dictionary thereof, it is characterized in that: comprise the following steps:

(1) image block utilizing noisy image configuration to be solved;

(2) two are utilized to train dictionary Ω ₁, Ω ₂solve required dictionary Ω in one dimension sparse coding, wherein $\mathrm{\&Omega;}={\mathrm{\&Omega;}}_1\&CircleTimes;{\mathrm{\&Omega;}}_2;$

(3) one dimension sparse reconstruction method is utilized to solve the reconstructed value of Y

(4) above-mentioned N number of reconstruction image block is utilized to obtain denoising image.