CN109785242A - A kind of solution mixing method based on the high spectrum image by wave band generalized bilinear model - Google Patents
A kind of solution mixing method based on the high spectrum image by wave band generalized bilinear model Download PDFInfo
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Abstract
The invention discloses a kind of solution mixing method based on the high spectrum image by wave band generalized bilinear model, step includes: 1, establishes by wave band generalized bilinear model；2, it is based on Bayesian MAP criterion and Regularization Theory, establishes the corresponding Optimized model of solution mixing method；3, the Optimized model is solved using alternated process multiplier method.The present invention can not only take into account the Gaussian noise of the different gauss levels of the different-waveband of high spectrum image, the mixed noise being widely present in true high spectrum image can also be taken into account, not only contain Gaussian noise in high spectrum image, also containing impulsive noise, band, dead pixel and dead wire etc., therefore have the advantages that the different gauss level noise robustness to mixed noise and different-waveband in true high spectrum image.
Description
Technical field
The present invention relates to high spectrum image solutions to mix field, specifically, the present invention relates to one kind by wave band generalized bilinear
High spectrum image solution mixes model and method.
Background technique
High-spectrum seems to be collected by imager in several hundred narrow continuous spectral bands.Due to high spectrum
Resolution ratio, therefore the problem of inevitably lead to mixed pixel, so that different substances occupies the same pixel.Mixed pixel
Presence have large effect, such as target identification, sub-pix charting and classification etc. to many applications.High spectrum image solution is mixed
Mixed pixel is decomposed into a series of pure substance (i.e. end member) and corresponding ratio (i.e. abundance).
Linear mixed model (LMM) is a model that is simple and should using extensively, it assumes that only per a branch of incident ray
With a kind of matter interaction, thus each pixel is the linear combination of end member.But when in the presence of close mixing, orographic factor
Or when multiple scattering effect, LMM can fail.Nonlinear mixed model (NLMMs) provides certain methods to overcome the above problem, it
Two classes can be generally divided into.The first kind includes some flexible models based on signal processing, including rear nonlinear model, nerve
Network model and nuclear model etc..Second class includes some models based on physics, including close mixed model, bilinearity hybrid guided mode
Type (BMM) and polyteny mixed model etc..Wherein, BMM only takes into account second order melange effect, it is not intended that high-order mixing effect
It answers.This is because high-order melange effect is not only seldom to the mixed precision contribution of solution is promoted, but also greatly increase computation complexity.It is some
Representative BMM model has proposed.Nascimento model is the extension LMM model with virtual end member, Fan mould
Type (FM) is the truncation Taylor expansion of non-linear mixing function, and generalized bilinear model (GBM) can regard pushing away for LMM and FM as
Extensively.Different algorithms has proposed to mix for GBM solution, and Halimi et al. proposes Bayesian algorithm for estimating GBM model
Abundance and noise variance, they also proposed based on gradient decline (GDA) solution mixing method pixel-by-pixel.In addition, Yokoya etc.
People proposes the optimization method based on half Non-negative Matrix Factorization (semi-NMF) and mixes for GBM solution, and Li et al. people proposes boundary throwing
Shadow Optimal gradient method is mixed for GBM solution.The hypothesis that most of solution mixing methods based on GBM imply be high spectrum image only by
Additive white Gaussian noise (AWGN) pollution, and assume that the noise level of different-waveband is identical.
But for the actual high spectrum image solution based on GBM is mixed, however it remains both sides challenge.One side
Face is the AWGN that each wave band of high spectrum image contains varying strength, is deposited extensively in true high spectrum image
Mixed noise problem, i.e., not only contain Gaussian noise in high spectrum image, also containing impulsive noise, band, dead pixel and
Dead wire etc..
Summary of the invention
To overcome relevant art defect, the invention proposes a kind of high-spectrums based on by wave band generalized bilinear model
The solution mixing method of picture, to can solve different-waveband varying strength Gaussian noise and the mixed noise that is widely present to bloom
The mixed influence of spectrogram picture solution, so that the mixed Complex Noise problem adapted in practical high spectrum image of solution, to improve what solution was mixed
Accuracy and robustness.
The present invention adopts the following technical scheme that in order to solve the technical problem
A kind of the characteristics of solution mixing method based on the high spectrum image by wave band generalized bilinear model of the invention include with
Lower step:
Step 1, it is established using formula (1) by wave band generalized bilinear model:
Yi=(EA)i+(FB)i+Si+Ni (1)
In formula (1), YiWave band corresponding to i-th row in the picture element matrix Y of expression high spectrum image, i=1,2 ..., D, and
Y∈RD×P, D and P respectively indicate the wave band sum of the spectrum dimension of the high spectrum image and the sum of all pixels of space dimension, E=
[e1,e1,…,ej,…,eM]∈RD×MIndicate the end member matrix of the high spectrum image, wherein ejIndicate the end member matrix E
In j-th of end member, j=1,2 ..., M, M indicate the end member in the high spectrum image sum, A=[a1,a2,…,ak,…,
aP]∈RM×PIndicate the abundance matrix of the high spectrum image, wherein akIndicate the rich of ith pixel in the abundance matrix A
Spend vector, k=1,2 ..., P, F=[e1⊙e2,...,e1⊙eM,e2⊙e3,...,e2⊙eM,...,eM-1⊙eM]∈RD ×M(M-1)/2Indicate secondary interaction end variable matrix, wherein ⊙ indicates Hadamard's product, B ∈ RM(M-1)/2×PIndicate secondary interactive abundance square
Battle array, SiIndicate wave band corresponding to the i-th row of the sparse noise matrix S of the high spectrum image, and S ∈ RD×P, NiDescribed in expression
Wave band corresponding to the i-th row of the dense noise matrix N of high spectrum image, and meetIndicate the i-th row institute
The Gaussian noise of corresponding wave band obeys the Gaussian Profile of the varying strength of zero-mean,Indicate wave band corresponding to the i-th row
The variance of Gaussian noise, IpIndicate the unit matrix that diagonal line contains p element；N∈RD×P；
Step 2, it is based on Bayesian MAP criterion and Regularization Theory, establishes the solution mixing method pair using formula (2)
The Optimized model answered:
s.t.A≥0,0≤B≤C
In formula (2), W is diagonal matrix, and diagonal entryMin is to minimize operator, | | | |FIndicate square
This black norm of the not Luo Beini of battle array, | | S | |1=∑i.j|Si,j| indicate the i-th row jth column element absolute value in sparse noise matrix S
Summation, λ indicates that regularization parameter, s.t. indicate constraint condition, and C indicates the upper bound matrix of the secondary interactive abundance matrix B；
Step 3, the Optimized model is solved using alternated process multiplier method, it is rich obtains the abundance matrix A, secondary interaction
Spend matrix B and sparse noise matrix S:
Step 3.1 introduces three auxiliary variable V1、V2And V3, the Optimized model is written over, is obtained such as formula (3) institute
Optimized model after the rewriting shown:
s.t.V1=S, V2=A, V3=B
In formula (3), X=V is enabled1, V2Or V3,Indicate nonnegative quadrant R+Indicative function, Xi,j
The the i-th row jth column element for indicating variable X, works as Xi,jBelong to nonnegative quadrant R+When,It is 0, is otherwise+∞；It is the indicative function of section [0, C], works as Xi,jBelong to [0, Ci,j] when,
It is 0, is otherwise+∞；Ci,jI-th row jth column element of the upper bound matrix of secondary interactive abundance matrix B；
Augmented Lagrangian Functions corresponding to the Optimized model after the rewriting are obtained using formula (4):
In formula (4),μ indicates penalty coefficient,Indicate Lagrange multiplier,
Indicate the scaled matrix of the Lagrange multiplier stacked, Λ1、Λ2And Λ3Respectively indicate sparse noise matrix S, abundance matrix A and
The scaled matrix of the corresponding Lagrange multiplier of secondary interactive abundance matrix B；I indicates unit matrix,G (V, Q) indicates the optimization after rewriteeing
The objective function function of model, and have:
Step 3.2, definition current iteration number are k, and initialize k=0；Initialization
Step 3.3: the abundance matrix A of+1 iteration of kth is updated using formula (6)k+1:
Step 3.4: the secondary interactive abundance matrix B of+1 iteration of kth is updated using formula (7)k+1:
Step 3.5: the sparse noise matrix S of+1 iteration of kth is updated using formula (8)k+1:
In formula (8), enable It is soft contraction operator, andIt indicates
Thresholding, sgn (x) indicate the sign function of x, and max () expression takes larger value function；
Step 3.6: first auxiliary variable V of+1 iteration of kth is updated using formula (9)1 k+1:
Step 3.7: second auxiliary variable of+1 iteration of kth is updated using formula (10)
Step 3.8: the third auxiliary variable of+1 iteration of kth is updated using formula (11)
In formula (11), min () expression takes smaller value function；
Step 3.9: the sparse noise matrix S, abundance matrix A and secondary interaction of+1 iteration of kth are updated using formula (12)
The scaled matrix of the corresponding Lagrange multiplier of abundance matrix B
Step 3.10: the initial error r of+1 iteration of kth is updated using formula (13)k+1It is missed with the antithesis of+1 iteration of kth
Poor dk+1:
Step 3.11: differentiate the condition of convergence:
IfAndIt then indicates to obtain high spectrum image
Abundance matrix A, secondary interactive abundance matrix B and sparse noise S, wherein ε indicate convergence threshold otherwise k+1 is enabled to be assigned to k,
And it turns round and executes step 3.3.
Compared with prior art, the beneficial effects of the present invention are:
1, the present invention is established by wave band generalized bilinear model, it can be by the Gaussian noise of the varying strength of different-waveband
Take into account with a variety of different types of noises.In addition, the present invention, which establishes, understands the corresponding Optimized model of mixing method, and using friendship
For method multiplier method solving optimization model.Model reconciliation mixing method of the invention can be suitably used for Complex Noise problem in practice,
So that the robustness and precision that solution is mixed are higher.
2, firstly, the present invention can not only consider the Gaussian noise of the different gauss levels of the different-waveband of high spectrum image
Enter, moreover it is possible to the mixed noise being widely present in true high spectrum image be taken into account, i.e., do not contained only in high spectrum image
There is Gaussian noise, also containing impulsive noise, band, dead pixel and dead wire etc..It is dense according to noise is divided into the characteristics of noise
Noise and sparse noise, and assume that the noise level of different-waveband is different, it is established on this basis by wave band generalized bilinear mould
Type.Secondly, according to Bayesian MAP criterion, it is assumed that each wave band contains the base of the Gaussian noise of independent and different intensity
On plinth, establish based on the Unified frame mixed by wave band generalized bilinear solution, further according to Regularization Theory by the sparse of sparse noise
Characteristic is taken into account, therefore the solution mixing method that the present invention establishes has to the mixed noise and different waves in true high spectrum image
The advantages of different gauss level noise robustness of section.Finally, due to which alternated process multiplier method has been widely used for solving constraint
Under the conditions of optimization problem, and satisfactory effect is achieved, so also solving using alternated process multiplier method above-mentioned
Optimized model.
Detailed description of the invention
Fig. 1 is the flow chart of the embodiment of the present invention.
Specific embodiment
In the present embodiment, as shown in Figure 1, a kind of mixed side of solution based on the high spectrum image by wave band generalized bilinear model
Method is mainly made of 3 steps: 1, being established by wave band generalized bilinear model；2, Bayesian MAP criterion and canonical are based on
Change theoretical, the corresponding Optimized model of foundation solution mixing method；3, using alternated process multiplier method solving optimization model；Specifically,
It is to carry out as follows:
Step 1, in order to overcome traditional GBM model to the different level Gaussian noise and multiple types noise of different-waveband
Noise in true high spectrum image is divided into two major classes by sensitive issue, the present invention, i.e., dense noise and sparse noise.It is thick
Close noise refers to most high spectrum image by noise pollution, it mainly includes Gaussian noise.Sparse noise refers to sub-fraction
For high spectrum image by sparse noise pollution, it mainly includes impulsive noise, dead pixel, dead wire and band.And assume different-waveband
Gaussian noise levels it is different, established using formula (1) by wave band generalized bilinear model:
Yi=(EA)i+(FB)i+Si+Ni (1)
In formula (1), YiWave band corresponding to i-th row in the picture element matrix Y of expression high spectrum image, i=1,2 ..., D, and
Y∈RD×P, D and P respectively indicate the wave band sum of the spectrum dimension of high spectrum image and the sum of all pixels of space dimension, E=[e1,
e1,…,ej,…,eM]∈RD×MIndicate the end member matrix of high spectrum image, wherein ejIndicate j-th of end member in end member matrix E, j
=1,2 ..., M, M indicate the end member sum in high spectrum image, A=[a1,a2,…,ak,…,aP]∈RM×PIndicate high-spectrum
The abundance matrix of picture, wherein akIndicate the abundance vector of ith pixel in abundance matrix A, k=1,2 ..., P, F=[e1⊙
e2,...,e1⊙eM,e2⊙e3,...,e2⊙eM,...,eM-1⊙eM]∈RD×M(M-1)/2Indicate secondary interaction end variable matrix, wherein
⊙ indicates Hadamard's product, B ∈ RM(M-1)/2×PIndicate secondary interactive abundance matrix, SiIndicate the sparse noise matrix of high spectrum image
Wave band corresponding to the i-th row of S, and S ∈ RD×P, NiCorresponding to the i-th row for indicating the dense noise matrix N of high spectrum image
Wave band, and meetIndicate that the Gaussian noise of wave band corresponding to the i-th row obeys the varying strength of zero-mean
Gaussian Profile,Indicate the variance of the Gaussian noise of wave band corresponding to the i-th row, IpIndicate that diagonal line contains p element
Unit matrix；N∈RD×P；
Step 2, available:It is assumed that the white Gaussian of each wave band
Noise is independent of each other, then is had:
In formula (2), m indicates that constant, W are diagonal matrix, and diagonal entryIt is from formula (2) as can be seen that every
The variance of the Gaussian noise of a wave band is bigger, and the weight of the wave band is smaller.Based on Bayesian MAP criterion, abundance matrix A,
Secondary interactive abundance matrix B and sparse noise matrix S can be converted into following optimization problem:
In formula (3), p (EA+FB+S) indicates the prior distribution of EA+FB+S.It, can be with since sparse noise has system performance
ByNorm characterizes, but is based onThe optimization problem of norm is usually np hard problem, is used thusNorm substitutesModel
Number, and traditional GBM model meets the constraint condition in formula (2), therefore formula (4) is utilized to establish the corresponding optimization mould of solution mixing method
Type:
s.t.A≥0,0≤B≤C,
In formula (4), min is to minimize operator, | | | |FThis black norm of the not Luo Beini of representing matrix, | | S | |1=∑i.j
|Si,j| indicate the summation of the i-th row jth column element absolute value in sparse noise matrix S, λ indicates that regularization parameter, s.t. indicate about
Beam condition, C indicate the upper bound matrix of secondary interactive abundance matrix B；
Step 3, since alternated process multiplier method is widely used in solving the optimization problem under constraint condition, and order is achieved
The satisfied effect of people.Therefore the present invention uses alternated process multiplier method solving optimization model, acquisition abundance matrix A, secondary interaction are rich
Spend matrix B and sparse noise matrix S:
Step 3.1 introduces three auxiliary variable V1、V2And V3, Optimized model is written over, is obtained as shown in formula (5)
Optimized model after rewriting:
s.t.V1=S, V2=A, V3=B
In formula (5), X=V is enabled1, V2Or V3,Indicate nonnegative quadrant R+Indicative function, Xi,j
The the i-th row jth column element for indicating variable X, works as Xi,jBelong to nonnegative quadrant R+When,It is 0, is otherwise+∞；It is the indicative function of section [0, C], works as Xi,jBelong to [0, Ci,j] when,
It is 0, is otherwise+∞；Ci,jI-th row jth column element of the upper bound matrix of secondary interactive abundance matrix B；
Augmented Lagrangian Functions corresponding to Optimized model after being rewritten using formula (6):
In formula (4),μ indicates penalty coefficient,Indicate Lagrange multiplier,
Indicate the scaled matrix of the Lagrange multiplier stacked, Λ1、Λ2And Λ3Respectively indicate sparse noise matrix S, abundance matrix A and
The scaled matrix of the corresponding Lagrange multiplier of secondary interactive abundance matrix B,I indicates unit matrix,G (V, Q) indicates the optimization after rewriteeing
The objective function function of model, and have:
Step 3.2, definition current iteration number are k, and initialize k=0；Initialization
Step 3.3: the abundance matrix A of+1 iteration of kth is updated using formula (8)k+1:
Step 3.4: the secondary interactive abundance matrix B of+1 iteration of kth is updated using formula (9)k+1:
Step 3.5: the sparse noise matrix S of+1 iteration of kth is updated using formula (10)k+1:
In formula (10), enableIt is soft contraction operator, andTable
Show that thresholding, sgn (x) indicate the sign function of x, max () expression takes larger value function；
Step 3.6: first auxiliary variable V of+1 iteration of kth is updated using formula (11)1 k+1:
Step 3.7: second auxiliary variable of+1 iteration of kth is updated using formula (12)
Step 3.8: the third auxiliary variable of+1 iteration of kth is updated using formula (13)
In formula (11), min () expression takes smaller value function；
Step 3.9: the sparse noise matrix S, abundance matrix A and secondary interaction of+1 iteration of kth are updated using formula (14)
The scaled matrix of the corresponding Lagrange multiplier of abundance matrix BWith
Step 3.10: the initial error r of+1 iteration of kth is updated using formula (15)k+1It is missed with the antithesis of+1 iteration of kth
Poor dk+1:
Step 3.11: differentiate the condition of convergence:
IfAndIt then indicates to obtain high spectrum image
Abundance matrix A, secondary interactive abundance matrix B and sparse noise S, wherein ε indicate convergence threshold otherwise k+1 is enabled to be assigned to
K, and turn round and execute step 3.3.In addition, the selection of μ has a large effect to convergence rate, updates μ and make initial error and right
The ratio of the norm of even error is kept in a certain range, and finally all converges on 0.During iteration, it is also necessary to
Know diagonal matrix W in advance, the present invention estimates the noise level of each wave band using HySime algorithm thus.Its core is thought
Want to assume that between the wave band of high spectrum image that there is very strong correlation, and can solve to obtain by more regression theories.
In specific implementation, simulated experiment will be carried out to verify the validity that the present invention proposes algorithm (NU-BGBM), comparison is calculated
Method has staff cultivation least square method (FCLS), gradient descent method (GDA), half Non-negative Matrix Factorization method (semi-NMF) and boundary to throw
Shadow optimum gradient method (BPOGM), and the mixed precision of solution is measured using root-mean-square error (RMSE).In general, RMSE is got over
It is small, it is higher to solve mixed precision.
The spectra database of US Geological Survey (USGS) will be used, it there are 224 wave bands, and wavelength band is at 0.38 μm
To 2.5 μm.6 spectrum are randomly selected as end member, the high spectrum image of synthesis there are 64 × 64 pixels, they do not have pure picture
Member, high spectrum image are divided into 8 × 8 fritter, and the pixel in each fritter is by a kind of filling random in 6 end members, then
Using 9 × 9 spatial low-pass filter, and pixel of the abundance greater than 80% is substituted by the average of all end members, is done so
Purpose is so that without containing Pure pixel in the high spectrum image of synthesis.And two-wire is generated according to by wave band generalized bilinear model
Property abundance matrix.Then different types of noise (1) Gaussian noise is added: the Gauss of zero-mean is added in the HSI of all wave bands
The signal-to-noise ratio of white noise, each wave band is the random value of 10dB to 50dB；(2) impulsive noise.30% arteries and veins is added in wave band 60-70
Rush noise；(3) dead wire: dead wire is added in wave band 120-130.
In order to simulate high spectrum image existing various noises in practice as much as possible, to a kind of noise, two kinds of noises and
It is all tested in the case of three kinds three kinds.Table 1 gives RMSE of the different solution mixing methods under different interference scenarios.It can from table 1
To find out, when only being polluted by Gaussian noise in high spectrum image, the solution of algorithm proposed by the present invention mixes effect and is better than it
It compares algorithm, this is because method only proposed by the present invention can make an uproar the Gauss of high spectrum image different-waveband different level
Sound is taken into account.When high spectrum image is only polluted by impulsive noise or dead wire, the solution that method proposed by the present invention obtains is mixed
Effect will be substantially better than other control methods, this is because traditional solution mixes algorithm to sparse noise-sensitive, the only present invention is mentioned
Method out can take into account the sparse characteristic of sparse noise.When high spectrum image is by two or three of noise pollution, this
Invention propose solution mixing method effect will be better than other control methods, this is because method only proposed by the present invention can will
The Gaussian noise of different-waveband varying strength and a variety of mixed noises are taken into account.Therefore, compared to control methods, this meal is proposed
Method the solution under Various Complex noise conditions is mixed more robust, and the mixed precision of solution can be promoted.
RMSE (10 of the more different solution mixing methods of table 1 under different interference scenarios-2)
| Noise type | FCLS | GDA | semi-NMF | BPOGM | NU-BGBM |
| Gaussian noise | 7.103 | 6.053 | 5.520 | 5.157 | 0.990 |
| Impulsive noise | 7.123 | 6.395 | 6.161 | 5.403 | 0.167 |
| Dead wire | 6.812 | 5.773 | 5.796 | 5.436 | 0.171 |
| Gaussian noise, impulsive noise | 8.411 | 7.781 | 7.651 | 7.065 | 1.004 |
| Gaussian noise, dead wire | 8.084 | 7.185 | 7.197 | 6.996 | 1.003 |
| Impulsive noise, dead wire | 7.941 | 7.254 | 7.469 | 6.962 | 0.296 |
| Gaussian noise, impulsive noise, dead wire | 9.010 | 8.395 | 8.609 | 8.216 | 1.021 |
Claims (1)
1. a kind of solution mixing method based on the high spectrum image by wave band generalized bilinear model, which is characterized in that including following
Step:
Step 1, it is established using formula (1) by wave band generalized bilinear model:
Yi=(EA)i+(FB)i+Si+Ni (1)
In formula (1), YiIndicate wave band corresponding to the i-th row, i=1,2 ..., D, and Y ∈ R in the picture element matrix Y of high spectrum imageD ×P, D and P respectively indicate the wave band sum of the spectrum dimension of the high spectrum image and the sum of all pixels of space dimension, E=[e1,
e1,…,ej,…,eM]∈RD×MIndicate the end member matrix of the high spectrum image, wherein ejIndicate jth in the end member matrix E
A end member, j=1,2 ..., M, M indicate the sum of the end member in the high spectrum image, A=[a1,a2,…,ak,…,aP]∈RM×P
Indicate the abundance matrix of the high spectrum image, wherein akIndicate the abundance vector of ith pixel in the abundance matrix A, k=
1,2 ..., P, F=[e1⊙e2,...,e1⊙eM,e2⊙e3,...,e2⊙eM,...,eM-1⊙eM]∈RD×M(M-1)/2Indicate secondary
Interaction end variable matrix, wherein ⊙ indicates Hadamard's product, B ∈ RM(M-1)/2×PIndicate secondary interactive abundance matrix, SiIndicate the height
Wave band corresponding to the i-th row of the sparse noise matrix S of spectrum picture, and S ∈ RD×P, NiIndicate the thick of the high spectrum image
Wave band corresponding to the i-th row of close noise matrix N, and meetIndicate the height of wave band corresponding to the i-th row
This noise obeys the Gaussian Profile of the varying strength of zero-mean,Indicate the side of the Gaussian noise of wave band corresponding to the i-th row
Difference, IpIndicate the unit matrix that diagonal line contains p element；N∈RD×P；
Step 2, it is based on Bayesian MAP criterion and Regularization Theory, it is corresponding to establish the solution mixing method using formula (2)
Optimized model:
In formula (2), W is diagonal matrix, and diagonal entryMin is to minimize operator, | | | |FRepresenting matrix
Not this black norm of Luo Beini, | | S | |1=∑i.j|Si,j| indicate the total of the i-th row jth column element absolute value in sparse noise matrix S
Indicate that regularization parameter, s.t. indicate constraint condition with, λ, C indicates the upper bound matrix of the secondary interactive abundance matrix B；
Step 3, the Optimized model is solved using alternated process multiplier method, obtains the abundance matrix A, secondary interactive abundance square
Battle array B and sparse noise matrix S:
Step 3.1 introduces three auxiliary variable V1、V2And V3, the Optimized model is written over, is obtained as shown in formula (3)
Optimized model after rewriting:
In formula (3), X=V is enabled1, V2Or V3,Indicate nonnegative quadrant R+Indicative function, Xi,jIt indicates to become
The the i-th row jth column element for measuring X, works as Xi,jBelong to nonnegative quadrant R+When,It is 0, is otherwise+∞；It is the indicative function of section [0, C], works as Xi,jBelong to [0, Ci,j] when,For
0, it is otherwise+∞；Ci,jI-th row jth column element of the upper bound matrix of secondary interactive abundance matrix B；
Augmented Lagrangian Functions corresponding to the Optimized model after the rewriting are obtained using formula (4):
In formula (4),μ indicates penalty coefficient,Indicate Lagrange multiplier,Indicate heap
The scaled matrix of folded Lagrange multiplier, Λ1、Λ2And Λ3Respectively indicate sparse noise matrix S, abundance matrix A and secondary friendship
The mutually scaled matrix of the corresponding Lagrange multiplier of abundance matrix B；I is indicated
Unit matrix,G (V, Q) indicates the objective function function of the Optimized model after rewriteeing, and has:
Step 3.2, definition current iteration number are k, and initialize k=0；Initialize Ak、Bk、Sk、V1 k、
Step 3.3: the abundance matrix A of+1 iteration of kth is updated using formula (6)k+1:
Step 3.4: the secondary interactive abundance matrix B of+1 iteration of kth is updated using formula (7)k+1:
Step 3.5: the sparse noise matrix S of+1 iteration of kth is updated using formula (8)k+1:
In formula (8), enableIt is soft contraction operator, andIndicate door
Limit, sgn (x) indicate the sign function of x, and max () expression takes larger value function；
Step 3.6: first auxiliary variable V of+1 iteration of kth is updated using formula (9)1 k+1:
Step 3.7: second auxiliary variable of+1 iteration of kth is updated using formula (10)
Step 3.8: the third auxiliary variable of+1 iteration of kth is updated using formula (11)
In formula (11), min () expression takes smaller value function；
Step 3.9: sparse noise matrix S, abundance matrix A and secondary interactive abundance using formula (12) update+1 iteration of kth
The scaled matrix of the corresponding Lagrange multiplier of matrix BWith
Step 3.10: the initial error r of+1 iteration of kth is updated using formula (13)k+1With the antithesis error d of+1 iteration of kthk+1:
Step 3.11: differentiate the condition of convergence:
IfAndIt then indicates to obtain the rich of high spectrum image
Matrix A, secondary interactive abundance matrix B and sparse noise S are spent, wherein ε indicates convergence threshold, otherwise, enables k+1 be assigned to k, and return
Turn to execute step 3.3.
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