CN106709881B - A kind of high spectrum image denoising method decomposed based on non-convex low-rank matrix - Google Patents
A kind of high spectrum image denoising method decomposed based on non-convex low-rank matrix Download PDFInfo
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Abstract
The invention discloses a kind of high spectrum image denoising methods decomposed based on non-convex low-rank matrix, comprise the steps that 1, divide high spectrum image；2, generator matrix；3, non-convex low-rank matrix is decomposed；4, also atomic block；5, high spectrum image evidence is restored, the high spectrum image after finally obtaining denoising.The present invention provides a kind of method for efficiently, rapidly removing various noises for remote sensing hyperspectral image denoising.
Description
Technical field
The present invention is under the jurisdiction of the fields such as computer picture, the information processing technology, remote sensing technology, more particularly to a kind of based on non-
The high spectrum image denoising method that convex low-rank matrix is decomposed.
Background technique
With the fast development of Remote Sensing Digital Technique, high spectrum image is widely used in military surveillance, environmental science, geology
The fields such as exploration, biomedical imaging.Due to the mechanical breakdown of sensor, the influence of image transmitting failure etc. various factors,
High spectrum image inevitably will receive various noises, such as Gaussian noise, impulsive noise, item in acquisition and transmission process
The pollution of line etc. seriously constrains further applying for high spectrum image.Meanwhile the sharp increase of high spectrum image dimension, it leads
" dimension disaster " is caused.Therefore, design it is a kind of can both remove various mixed noises, while can solve the height of " dimension disaster "
Spectrum picture denoising method becomes one of key technology in remote sensing images application.
Traditional EO-1 hyperion denoising method often has following defect: 1) only consider white Gaussian noise, but in EO-1 hyperion
Often containing there are many mixed noises in image；2) single band image information is only used, does not account for enriching in high spectrum image
Atural object spatial character and spectral characteristic.In recent years, flourishing with compressive sensing theory, is decomposed based on low-rank matrix
Concern of the high spectrum image denoising method increasingly by numerous researchers.Such algorithm is mostly with the convex approximate matrix order letter of nuclear norm
Number.However, often there are following several problems in high spectrum image denoising in the low-rank matrix decomposition algorithm based on nuclear norm:
1, high spectrum image spectral information abundant is taken full advantage of, but Gaussian noise and part can only be effectively removed
Sparse noise；
2, based on the convex model of nuclear norm, actual effect is poor in the denoising of practical high spectrum image, and along with high-spectrum
As the growth of dimension, the calculating time significantly extends；
3, it is excessive to frequently can lead to matrix rand estination for the convex model based on nuclear norm, and Banded improvement is caused not can be removed；
4, it based on the low-rank matrix decomposition algorithm of bilateral accidental projection, generally requires to provide the priori such as rank of matrix letter in advance
Breath.
Bibliography:
[1]Dabov K,Foi A,Katkovnik V,et al.Image denoising by sparse 3-D
transform-domain collaborative filtering[J].IEEE Transactions on Image
Processing,2007,16(8):2080-95.
[2]Zhang H,He W,Zhang L,et al.Hyperspectral Image Restoration Using
Low-Rank Matrix Recovery[J].IEEE Transactions on Geoscience and Remote
Sensing,2014,52(8):4729-4743.
[3]He W,Zhang H,Zhang L,et al.Hyperspectral Image Denoising via
Noise-Adjusted Iterative Low-Rank Matrix Approximation[J].IEEE Journal of
Selected Topics in Applied Earth Observations and Remote Sensing,2015,8(6):1-
12.
[4]Xie Y,Qu Y,Tao D,et al.Hyperspectral Image Restoration via
Iteratively Regularized Weighted Schatten,p-Norm Minimization[J].IEEE
Transactions on Geoscience and Remote Sensing,2016,54(8):4642–4659.
Summary of the invention
Goal of the invention: present invention seek to address that the technical issues of high spectrum image denoises in remote sensing images field, provides one
The high spectrum image denoising method that kind is decomposed based on non-convex low-rank matrix, to be effectively removed the existing mixing of high spectrum image
Spectrum and space characteristics in image are saved while noise.
Technical solution: the invention discloses it is a kind of based on non-convex low-rank matrix decompose high spectrum image denoising method,
Core is to utilize non-convex low-rank matrix decomposition technique linearisation separation high spectrum image, comprising the following steps:
Step 1, the original high spectrum image containing mixed noise is inputted, high-spectrum seems by tens or even hundreds of companies
The 3 d image data of continuous band image composition.The size of sub-block is initialized, then the step-length of scanning is divided into original image
The sub-block of overlapping, to retain the local detail of high spectrum image；
Step 2, generator matrix: k-th of wave band vectorization to each sub-block, then to the wave band of all vectorizations according to
Lexicographic order is arranged in a matrix；
Step 3, non-convex low-rank matrix is decomposed: by non-convex low-rank matrix resolution process, containing mixing for what step 2 obtained
A low-rank matrix, a sparse matrix and a Un-structured matrix are resolved into the matrix of noise, linearisation；This process
In, with matrix rank function and sparse matrix l2,1Norm linear and be objective function, with the linearisation containing mixed noise matrix
It is decomposed into constraint condition, then solves this problem using Augmented Lagrange method；
Step 4, atomic block is gone back: by every height column one matrix of synthesis of low-rank matrix, by obtained all matrixes by light
Spectrum dimension is arranged in three-dimensional sub-block, and to overlapping parts of images pixel value equalization processing；
Step 5, it restores high spectrum image: the three-dimensional sub-block that equalization is handled being arranged in identical with original high spectrum image
3 d image data, finally obtain denoising after high spectrum image.
In the present invention, step 1 the following steps are included:
Step 1-1, inputs the original high spectrum image containing mixed noise, this image is by tens or even hundreds of continuous
The 3 d image data d of band image composition,Wherein dh,dw,dsThe height of high spectrum image is respectively represented,
Width and wave band number,Indicate that a dimension is respectively d in real number fieldh,dw,dsThree rank tensors；
Step 1-2, the size for initializing sub-block is p, scanning step s, wherein 1≤s < p.By original high spectrum image
It is divided into the sub-block of overlapping, is amounted toIt is a.
In the present invention, step 2 the following steps are included:
Step 2-1, for i-th of sub-block d of selectioni, Indicate that a dimension is respectively p, p, d in real number fieldsThree rank tensors, by k-th of wave band Rp×pVector turns to Indicate that a dimension is p in real number field2Vector, it is suitable according to dictionary to the wave band of all vectorizations
Sequence is arranged in matrix Di,Indicate that a dimension is respectively p in real number field2,dsMatrix；
Step 2-2 repeats step 2-1, amounts toIt is secondary.
In the present invention, step 3 the following steps are included:
Step 3-1, due to target space characteristics imaging while, to each space pixel by dispersion formed it is several
Ten or even several hundred a narrow-bands are to carry out continuous spectrum covering, therefore clean high spectrum image is shown in spectrum dimension
The characteristic of low-rank.Simultaneously in view of the noise contained in high spectrum image, such as: Gaussian noise, impulsive noise, striped peel off
Point etc., therefore, the present invention consider high spectrum image denoising model below:
Di=Li+Si+Ni,
Wherein Li,Si,NiRespectively low-rank matrix, sparse matrix and Un-structured matrix, Li,Si,
Step 3-2 carries out robustness principal component analysis by following formula:
s.t.Di=Li+Si+Ni,
Wherein, s.t. expression meets condition, and min indicates to minimize function, | | Li||*Indicate low-rank matrix LiNuclear norm,
||Si||1Indicate sparse matrix Si1 norm, λ is regularization parameter.However due to nuclear norm coequally all surprises of minimization
Different value, causing cannot approximate rank function well.The present invention considers that following non-convex rank of matrix is approximate:
Wherein, | | Li||γIndicate low-rank matrix LiNon-convex rank of matrix it is approximate, ∑ represents summation operation, σj(Li) indicate low
Order matrix LiJ-th of singular value, γ be one be greater than 0 constant,Non-convex sparse penalty function is represented, is used in the present invention
The non-convex sparse penalty function of following two:
Or
Simultaneously in view of containing longitudinal stripe in high spectrum image, therefore use sparse matrix S morei2,1 norms beTo remove Banded improvement, wherein (Si)mnIndicate variable SiM row n-th arrange member
Element,Indicate extraction of square root；
Step 3-3, using non-convex low-rank matrix decomposition model:
s.t.Di=Li+Si+Ni,
Above-mentioned model is solved using Augmented Lagrange method.The wherein Augmented Lagrangian Functions l (L of this modeli,Si,
Ni,Λi) it is as follows:
Wherein ΛiIt is Lagrange multiplier, ρ is penalty parameter, | | | |FThe F norm of matrix,<>representing matrix it is interior
Product, then successively iteration more new variables (Li,Si,Ni,Λi) ,+1 Iteration of kth is as follows:
WithRespectively indicate variables LiValue, variable S after k+1 iterationiBy k+1 times
Value, variable N after iterationiValue and variable Λ after k+1 iterationiValue after k+1 iteration, argmin indicate son
Problem minimizes point；
Step 3-4 solves variables L by following formulaiValue after k+1 iteration
WhereinDue to | | Li| | γ is non-convex, easy proofIt is continuous, recessed
, it is smooth, can it is micro- and section [0 ,+∞) non-subtract.Remember m=min { p2,ds, it is then right | | Li||γPoint
Locate first order Taylor expansion, then above formula becomes:
Wherein,Indicate variables LiIn j-th of singular value of kth time iteration,It is non-convex sparse penalty function
Derivative.Then it obtainsClosing solution:
Wherein Ti k=U Σ VTIt is matrix Ti kSingular value decomposition, U, V respectively indicate matrix Ti kLeft unitary matrice and the right tenth of the twelve Earthly Branches
Matrix, VTIndicate the transposition of right unitary matrice, andMax is indicated most
Bigization, diag indicate vector becoming a diagonal matrix, Σii(i, i) element of representing matrix Σ；
；
Step 3-5 solves variable S by following formulaiValue after k+1 iteration
WhereinThenClosing solution jth column be:
Wherein | | | |2Indicate 2 norms of vector；
Step 3-6 solves variable N by following formulaiValue after k+1 iteration
This problem is a least square problem,Closed-form solution it is as follows:
Step 3-7 passes through following formula more new variables ΛiValue after k+1 iteration
Step 3-8 repeats step 3-4 to step 3-7, amounts toIt is secondary.
In the present invention, step 4 the following steps are included:
Step 4-1, to low-rank matrixChoose matrix LiA column, synthesize a p × p matrix, will own
Low-rank matrix is arranged in three-dimensional sub-block by spectrum dimension
Step 4-2 repeats step 4-1, amounts toIt is secondary；
Step 4-3, calculates lap image pixel value, and equalization of the present invention handles lap image pixel value.
In the present invention, step 5 includes:
The sub-block that equalization is handled is arranged in three-dimensional d identical with original high spectrum imageh×dw×dsImage data,
High spectrum image after finally obtaining denoising.
The utility model has the advantages that
1) the EO-1 hyperion denoising method denoising speed in the present invention is fast, more robust property.Original high spectrum image is divided
After the sub-block of overlapping, it can realize that non-convex low-rank matrix is decomposed with parallelization, the runing time of significant ground reduction method, and will be former
Beginning linearity is divided into clean high spectrum image, sparse noise and gaussian random noise.
2) the EO-1 hyperion denoising method in the present invention takes full advantage of high spectrum image spectral information abundant, considers simultaneously
To different types of noise, mixed noise is more efficiently removed.Since original high spectrum image to be divided into the sub-block of overlapping,
And to overlapping partial pixel value equalization processing, the local detail information in high spectrum image is effectively kept.
3) the high spectrum image denoising method in the present invention has stronger scalability and flexibility.It is theoretically of the invention
In method the high spectrum image of arbitrary size can be carried out denoising.
Detailed description of the invention
Fig. 1 is the basic flow chart of high spectrum image denoising method of the present invention.
Fig. 2 (a) is original noise-containing image,
Fig. 2 (b) is the image after algorithm VBM3D denoising,
Fig. 2 (c) is the image after algorithm LRMR denoising,
Fig. 2 (d) is the image after algorithm NAILRMA denoising,
Image after Fig. 2 (e) algorithm WSN-LRMA denoising,
Fig. 2 (f) is the image after algorithm NonLRMA denoising.
Fig. 3 (a) is original noise-containing image,
Fig. 3 (b) is the image after algorithm VBM3D denoising,
Fig. 3 (c) is the image after algorithm LRMR denoising,
Fig. 3 (d) is the image after algorithm NAILRMA denoising,
Fig. 3 (e) is the image after algorithm WSN-LRMA denoising,
Fig. 3 (f) is the image after algorithm NonLRMA denoising.
Fig. 4 (a) is original noise-containing image,
Fig. 4 (b) is the image after algorithm VBM3D denoising,
Fig. 4 (c) is the image after algorithm LRMR denoising,
Fig. 4 (d) is the image after algorithm NAILRMA denoising,
Fig. 4 (e) is the image after algorithm WSN-LRMA denoising,
Fig. 4 (f) is the image after algorithm NonLRMA denoising.
Fig. 5 (a) is original noise-containing image,
Fig. 5 (b) is the image after algorithm VBM3D denoising,
Fig. 5 (c) is the image after algorithm LRMR denoising,
Fig. 5 (d) is the image after algorithm NAILRMA denoising,
Fig. 5 (e) is the image after algorithm WSN-LRMA denoising,
Fig. 5 (f) is the image after algorithm NonLRMA denoising.
Fig. 6 (a) is original noise-containing image,
Fig. 6 (b) is the image after algorithm VBM3D denoising,
Fig. 6 (c) is the image after algorithm LRMR denoising,
Fig. 6 (d) is the image after algorithm NAILRMA denoising,
Fig. 6 (e) is the image after algorithm WSN-LRMA denoising,
Fig. 6 (f) is the image after algorithm NonLRMA denoising.
Fig. 7 (a) is original noise-containing image,
Fig. 7 (b) is the image after algorithm VBM3D denoising,
Fig. 7 (c) is the image after algorithm LRMR denoising,
Fig. 7 (d) is the image after algorithm NAILRMA denoising,
Fig. 7 (e) is the image after algorithm WSN-LRMA denoising,
Fig. 7 (f) is the image after algorithm NonLRMA denoising.
Fig. 8 (a) is original noise-containing image,
Fig. 8 (b) is the image after algorithm VBM3D denoising,
Fig. 8 (c) is the image after algorithm LRMR denoising,
Fig. 8 (d) is the image after algorithm NAILRMA denoising,
Fig. 8 (e) is the image after algorithm WSN-LRMA denoising,
Fig. 8 (f) is the image after algorithm NonLRMA denoising.
Specific embodiment
The present invention is done with reference to the accompanying drawings and detailed description and is further illustrated, but application of the invention
Range is without being limited thereto:
The flow chart of this method is as shown in Figure 1, four big processes can be broadly divided into: being first divided into original high spectrum image
The sub-block of overlapping, and by each sub-block matrixing；Secondly the non-convex low-rank matrix for carrying out parallelization is decomposed；Again by low-rank matrix
It is reduced into three-dimensional data block；Finally by lap pixel value equalization, and it is reduced into the high spectrum image after denoising.
Specifically, as shown in Figure 1, the invention discloses a kind of high spectrum images decomposed based on non-convex low-rank matrix to go
Method for de-noising, mainly including the following steps:
Step 1, divide high spectrum image: high-spectrum seems three be made of tens or even hundreds of continuous band images
Dimensional data image.The size of sub-block is initialized, then the step-length of scanning is divided into original image the sub-block of overlapping, to retain
The local detail of high spectrum image；
Step 2, generator matrix: k-th of wave band vectorization to each sub-block, then to the wave band of all vectorizations according to
Lexicographic order is arranged in a matrix；
Step 3, non-convex low-rank matrix is decomposed: being passed through non-convex low-rank matrix resolution process, is contained mixing for what step 2 obtained
A low-rank matrix, a sparse matrix, a Un-structured matrix are resolved into noise matrix, linearisation.During this, with square
Battle array rank function and sparse matrix l2,1Norm linear and be objective function, is decomposed into the linearisation containing mixed noise matrix
Then constraint condition solves this problem using Augmented Lagrange method；
Step 4, it goes back atomic block: by every height column one matrix of synthesis of low-rank matrix, all matrixes then being pressed into spectrum
Dimension is arranged in three-dimensional sub-block, and to overlapping parts of images pixel value equalization processing；
Step 5, it restores high spectrum image: the sub-block that equalization is handled is arranged in identical with original high spectrum image three
Dimensional data image, the high spectrum image after finally obtaining denoising.
For step 1, divide the specific implementation details following steps of high spectrum image:
Step 1-1, inputs the original high spectrum image containing mixed noise, this image is by tens or even hundreds of continuous
The 3 d image data of band image composition, is denoted asWherein dh,dw,dsThe height of high spectrum image is respectively represented,
Width and wave band number；
Step 1-2 initializes the size p of sub-block, scanning step s, wherein 1≤s < p.Original high spectrum image is divided
At the sub-block of overlapping, amount toIt is a.
For step 2, the specific implementation details following steps of generator matrix:
Step 2-1, for the sub-block of selectionBy k-th of wave band
Rp×pVector turns to(k=1 ..., ds), matrix then is arranged according to lexicographic order to the wave band of all vectorizations
Step 2-2 repeats step 2-1, amounts toIt is secondary.
For step 3, the specific implementation details following steps of non-convex low-rank matrix decomposition:
Step 3-1, due to target space characteristics imaging while, to each space pixel by dispersion formed it is several
Ten or even several hundred a narrow-bands are to carry out continuous spectrum covering, therefore clean high spectrum image is shown in spectrum dimension
The characteristic of low-rank.Simultaneously in view of the noise contained in high spectrum image, such as: Gaussian noise, impulsive noise, striped peel off
Point etc., therefore, the present invention consider high spectrum image denoising model below:
Di=Li+Si+Ni
Wherein Li,Si,Respectively low-rank matrix, sparse matrix and Un-structured matrix；
Step 3-2, traditional robustness principal component analysis can be write as:
s.t.Di=Li+Si+Ni
Wherein, s.t. expression meets condition, and min indicates to minimize function, | | Li||*Representing matrix LiNuclear norm, | | Si
||1Representing matrix Si1 norm, λ is regularization parameter.However due to nuclear norm coequally all singular values of minimization, cause
It cannot approximate rank function well.The present invention considers that following non-convex rank of matrix is approximate:
Wherein, | | Li||γIndicate low-rank matrix LiNon-convex rank of matrix it is approximate, ∑ represents summation operation, σj(Li) statement square
Battle array LiJ-th of singular value, γ be one be greater than 0 constant,Non-convex sparse penalty function is represented, is used in the present invention following
Two non-convex sparse penalty functions:
Simultaneously in view of containing longitudinal stripe in high spectrum image, therefore use sparse matrix S morei2,1 norms beTo remove Banded improvement, wherein (Si)mnIndicate variable SiThe n-th column element of m row,Table
Show extraction of square root；
Step 3-3, using non-convex low-rank matrix decomposition model:
s.t.Di=Li+Si+Ni
Above-mentioned model is solved using Augmented Lagrange method.Wherein the Augmented Lagrangian Functions of this model are:
Wherein ΛiIt is Lagrange multiplier, ρ is penalty parameter, | | | |FThe F norm of matrix,<>representing matrix it is interior
Product.Then successively iteration more new variables (Li,Si,Ni,Λi) ,+1 Iteration of kth is as follows:
Step 3-4, about variables LiSubproblem solves,
WhereinDue to | | Li||γIt is non-convex.
It is easy to proveBe it is continuous, recessed, smooth, can it is micro- and section [0 ,+∞) non-subtract.Remember m=
min{p2,ds, it is then right | | Li||γPointLocate first order Taylor expansion, then subproblem can be write as
WhereinIt is the derivative of non-convex sparse penalty function.Then the closing solution of this subproblem is obtained:
Wherein Ti k=U Σ VTIt is matrix Ti kSingular value decomposition, U, V respectively indicate matrix Ti kA left side
Unitary matrice and right unitary matrice, VTIndicate the transposition of right unitary matrice, andmax
It indicates to maximize, diag indicates vector becoming a diagonal matrix；
Step 3-5, about variable SiSubproblem solves,
WhereinThe jth column of the closing solution of this subproblem are:
Step 3-6, about variable NiSubproblem solves,
This subproblem is a least square problem, and closed-form solution is
Step 3-7 updates Lagrange multiplier,
Step 3-8 repeats step 3-4 to step 3-7, amounts toIt is secondary.
For step 4, also the specific implementation details following steps of atomic block:
Step 4-1, the low-rank matrix that step 3 is generatedChoose matrix LiA column, synthesize p × p square
Then all low-rank matrixes are arranged in three-dimensional sub-block by spectrum dimension by battle array
Step 4-2 repeats step 4-1, amounts toIt is secondary；
Step 4-3, calculates lap image pixel value, and equalization of the present invention handles lap image pixel value.
For step 5, the specific implementation details following steps of original image are gone back:
The sub-block that equalization is handled is arranged in three-dimensional d identical with original high spectrum image by step 5-1h×dw×dsFigure
High spectrum image as data, after finally obtaining denoising.
Embodiment
The Experimental Hardware environment of this implementation is: Intel-Core4i767003.4GHz, 16G memory, video card
NVIDIAGeForce GTX950.Software environment is MATLAB 2015b.Test image is derived from disclosed bloom on network
Compose image data set: Hyperspectral Digital Imagery Collection Experiment (HYDICE) urban
dataset,the Earth Observing-1(EO-1)Hyperion Australia dataset,and the
Airborne Visible/InfraredImaging Spectrometer(AVIRIS)Indian Pines dataset.For
The validity of verification method, the present invention have chosen the advanced denoising method proposed in recent years: video block match three dimensional filter side
Method (Video Block Matching 3-D filtering, VBM3D[1]), low-rank matrix restoration methods (Low-Rank
Matrix Recovery, LRMR[2]), the adaptive iteration low-rank matrix approximation method (Noise-Adjusted of noise
Iterative Low-Rank Matrix Approximation, NAILRMA[3]), the low-rank matrix based on weighting p- norm is close
Like method (Weighted Schattenp-normIterativeLow-Rank Matrix Approximation, WSN-LRMA[4]).Method of the invention: non-convex low-rank matrix decomposition method (Nonconvex Low-Rank Matrix
Approximation, NonLRMA).
Example one: this example selects remote sensing fields typical dataset-HYDICEurban data set, it is a kind of by Gauss
The image data of the pollutions such as noise, impulsive noise, striped, atmosphere, the hydrology is the ideal data for evaluating EO-1 hyperion denoising method.
The size of entire image is 307 × 307, altogether includes 210 wave bands.The method being readily seen in the present invention and other comparative approach
The noise in light noise pollution wave band can be removed, while also saving local detail information (such as Fig. 2 of high spectrum image
(a) shown in~Fig. 2 (f), wherein (a) indicates original containing noise image in Fig. 2, (b)-(f) be respectively by algorithm VBM3D,
Image after LRMR, NAILRMA, WSN-LRMA, NonLRMA denoising, remaining image are not always the case operation)；For moderate
The wave band of noise pollution, part Denoising Algorithm cannot be effectively removed noise, on the contrary, this method not only removes various noises,
The local detail information for saving image well, as shown in Fig. 3 (a)~Fig. 3 (f)；For the wave band of severe noise pollution, in addition to
This method can be in removal noise, while can restore main details information, as shown in Fig. 4 (a)~Fig. 4 (f).
Example two: this example selects remote sensing fields typical dataset-EO-1Hyperion Australia data set, this
Data set was shot on December 4th, 2012, and original image size is 256 × 3858, altogether included 242 wave bands.Due to length
Limitation, this method choose therein 200 × 400 subimage block.Contain longitudinal stripe in this data set, such as Fig. 5 (a)~figure more
5 (f), Fig. 6 (a)~Fig. 6 (f), Fig. 7 (a)~Fig. 7 (f), Fig. 8 (a)~Fig. 8 (f), shown.Being readily seen only this method can be with
It is effectively removed Banded improvement, saves image local detailed information.
Claims (3)
1. a kind of high spectrum image denoising method decomposed based on non-convex low-rank matrix, which comprises the following steps:
Step 1, the original high spectrum image containing mixed noise is inputted, the size of sub-block and the step-length of scanning are initialized, original
Beginning high spectrum image is divided into the sub-block of overlapping；
Step 2, generator matrix: k-th of wave band vectorization to each sub-block, to the wave bands of all vectorizations according to lexicographic order
It is arranged in a matrix；
Step 3, non-convex low-rank matrix is decomposed: by non-convex low-rank matrix resolution process, containing mixed noise for what step 2 obtained
Matrix, linearisation resolve into a low-rank matrix, a sparse matrix and a Un-structured matrix；
Step 4, it goes back atomic block: every height column one matrix of synthesis of low-rank matrix is tieed up obtained all matrixes by spectrum
It is arranged in three-dimensional sub-block, and to overlapping parts of images pixel value equalization processing；
Step 5, it restores high spectrum image: the three-dimensional sub-block that equalization is handled is arranged in identical with original high spectrum image three
Dimensional data image, the high spectrum image after being denoised；
Step 1 the following steps are included:
Step 1-1 inputs the original high spectrum image d containing mixed noise,Wherein dh,dw,dsRespectively
The height of high spectrum image, width and wave band number are represented,Indicate that a dimension is respectively d in real number fieldh,dw,ds
Three rank tensors；
Step 1-2, the size for initializing sub-block is p, scanning step s, wherein 1≤s < p, original high spectrum image is divided
At the sub-block of overlapping, amount toIt is a；
Step 2 the following steps are included:
Step 2-1, for i-th of sub-block d of selectioni,
Indicate that a dimension is respectively p, p, d in real number fieldsThree rank tensors, by k-th of wave band Rp×pVector turns toK=
1,…,ds,Indicate that a dimension is p in real number field2Vector, the wave band of all vectorizations is arranged according to lexicographic order
Matrix Di, Indicate that a dimension is respectively p in real number field2,dsMatrix；
Step 2-2 repeats step 2-1, amounts toIt is secondary；
Step 3 the following steps are included:
Step 3-1 establishes following high spectrum image denoising model:
Di=Li+Si+Ni,
Wherein Li,Si,NiRespectively low-rank matrix, sparse matrix and Un-structured matrix,
Step 3-2 carries out robustness principal component analysis by following formula:
s.t.Di=Li+Si+Ni,
Wherein, s.t. expression meets condition, and min indicates to minimize function, | | Li||*Indicate low-rank matrix LiNuclear norm, | | Si|
|1Indicate sparse matrix Si1 norm, λ is regularization parameter；
Consider that the non-convex rank of matrix being shown below is approximate:
Wherein, | | Li||γIndicate low-rank matrix LiNon-convex rank of matrix it is approximate, ∑ represents summation operation, σj(Li) indicate low-rank square
Battle array LiJ-th of singular value, γ be one be greater than 0 constant,Non-convex sparse penalty function is represented, it is non-convex using following two
Sparse penalty function:
Or
And use sparse matrix Si2,1 norms beTo remove Banded improvement,
Wherein (Si)mnIndicate variable SiThe n-th column element of m row,Indicate extraction of square root；
Step 3-3, using non-convex low-rank matrix decomposition model:
Above-mentioned model is solved using Augmented Lagrange method, wherein the Augmented Lagrangian Functions l (L of this modeli,Si,Ni,
Λi) it is as follows:
Wherein ΛiIt is Lagrange multiplier, ρ is penalty parameter, | | | |FIt is the F norm of matrix, the inner product of<>representing matrix,
Successively iteration more new variables (Li,Si,Ni,Λi) ,+1 Iteration of kth is as follows:
WithRespectively indicate variables LiValue, variable S after k+1 iterationiBy k+1 iteration
Value afterwards, variable NiValue and variable Λ after k+1 iterationiValue after k+1 iteration, argmin indicate subproblem
Minimize point；
Step 3-4 solves variables L by following formulaiValue after k+1 iteration
Wherein
Remember m=min { p2,ds, it is right | | Li||γPointLocate first order Taylor expansion, then above formula becomes:
Wherein,Indicate variables LiIn j-th of singular value of kth time iteration,It is the derivative of non-convex sparse penalty function,
Then it obtainsClosing solution:
Wherein Ti k=U Σ VTIt is matrix Ti kSingular value decomposition, U, V respectively indicate matrix Ti kLeft unitary matrice and right unitary matrice,
VTIndicate the transposition of right unitary matrice, andMax indicates to maximize,
Diag indicates vector becoming a diagonal matrix, Σii(i, i) element of representing matrix Σ；
Step 3-5 solves variable S by following formulaiValue after k+1 iteration
WhereinThenClosing solution jth column be:
Wherein | | | |2Indicate 2 norms of vector；
Step 3-6 solves variable N by following formulaiValue after k+1 iteration
Closed-form solution it is as follows:
Step 3-7 passes through following formula more new variables ΛiValue after k+1 iteration
Step 3-8 repeats step 3-4 to step 3-7, amounts toIt is secondary.
2. the method as described in claim 1, which is characterized in that step 4 the following steps are included:
Step 4-1, to low-rank matrixChoose matrix LiA column, a p × p matrix is synthesized, by all low-ranks
Matrix is arranged in three-dimensional sub-block by spectrum dimension
Step 4-2 repeats step 4-1, amounts toIt is secondary；
Step 4-3, equalization handle lap image pixel value.
3. method according to claim 2, which is characterized in that step 5 includes:
The sub-block that equalization is handled is arranged in three-dimensional d identical with original high spectrum imageh×dw×dsImage data obtains
High spectrum image after denoising.
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