CN105138860A - Hyperspectral nonlinear demixing method based on boundary projection optimal gradient - Google Patents

Abstract

The invention relates to a hyperspectral nonlinear demixing method based on a boundary projection optimal gradient. According to the hyperspectral nonlinear demixing method, by selecting a special search point, step length is determined by a Lipschitz constant, the optimal convergence rate under boundary constraint is greatly accelerated, and the optimal convergence rate as shown in the specification is achieved. In addition, the boundary projection optimal gradient method can be effectively applied to GBM-based hyperspectral nonlinear demixing, and has the advantages of high convergence rate and no selection sensitivity on initial values.

Description

A kind of EO-1 hyperion nonlinear solution mixing method based on border projection Optimal gradient

Technical field

The present invention relates to high spectrum image solution and mix field, specifically relate to a kind of EO-1 hyperion nonlinear solution mixing method based on border projection Optimal gradient.

Background technology

In the past few decades, high light spectrum image-forming has been the hot research field of remote sensing application, as mixed in target detection, spectrum solution and object matching and classification.Due to reasons such as high light spectrum image-forming sensor and earth's surface changes, mixed pixel is extensively present in high light spectrum image-forming.In this case, the quantitative analysis that EO-1 hyperion solution is mixed for high-spectral data is follow-up is very important, and mixed pixel is resolved into a series of pure spectra feature mixed comprising by spectrum solution, is called end member, and the ratio in each pixel shared by pure end member, be called abundance.The mixture model that spectrum solution is mixed can be linear also can be nonlinear, and this depends on high spectrum image to be studied.

Because linear mixed model (LMM) is relatively simple and be easy to explain, LMM and be widely used in geoscience and remote sensing process field.Such as, but LMM may be not suitable with under many circumstances, when there is multiple scattering effect or interacting closely, nonlinear mixed model (NLMMs) improves a kind of selection to overcome the inherent limitation of LMM.NLMMs and proposing in Hyperspectral imagery processing field, and two large classes can be divided into.The first kind of NLMMs is the natural characteristic based on environment, and they comprise the bilinearity mixture model (BMM) proposed in " IEEEGEOSCIENCEANDREMOTESENSINGLETTERS " the 11st volume the 4th phase in 2014 " ABilinear – BilinearNonnegativeMatrixFactorizationMethodforHyperspec tralUnmixing " based on people such as the model of bidirectional reflectance and OlivierEches.The Equations of The Second Kind of NLMMs be other physically based deformation approach provide model more flexibly, they comprise the neural network model that the people such as GiorgioLicciardi propose in " IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING " the 49th volume o. 11th " Pixelunmixinginhyperspectraldatabymeansofneuralnetworks " in 2011, the rear nonlinear model etc. that the people such as the nuclear model that the people such as YanfengGu propose in " IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING " the 51st volume the 7th phase " Spectralunmixinginmultiple-kernelhilbertspaceforhyperspe ctralimagery " in 2013 and YoannAltmann propose in " IEEETRANSACTIONSONIMAGEPROCESSING " the 23rd volume the 6th phase " Unsupervisedpost-nonlinearunmixingofhyperspectralimagesu singahamiltonianmontecarloalgorithm " in 2014.Rescattering effect between ground and tree crown is taken into account by BMM, but the bilinearity in BMM interacts and usually demonstrates very strong correlativity, it is very responsive to noise that this makes solution sneak out journey, the people such as AbderrahimHalimi adopt a kind of effective mode to overcome the potential hypothesis in BMM at " IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING " the 49th volume o. 11th " Nonlinearunmixingofhyperspectralimagesusingageneralizedb ilinearmodel " middle generalized bilinear model (GBM) proposed in 2011.

Hyperspectral image nonlinear solution based on GBM is mixed adopts dual stage process usually, mainly comprises two steps.The first step is called Endmember extraction, mainly from high spectrum image, extracts pure end member, and the mutual end member of such secondary also can be obtained by the pure end member extracted.Second step is called that abundance is estimated, mainly estimates pure end member and abundance corresponding to the mutual end member of secondary respectively.Certain methods has proposed the abundance for estimating GBM, if the people such as AbderrahimHalimi are at " IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING " the 49th volume o. 11th " Nonlinearunmixingofhyperspectralimagesusingageneralizedb ilinearmodel " the middle bayes method proposed in 2011, half Non-negative Matrix Factorization (semi-NMF) etc. that the people such as the gradient descent method (GDA) that the people such as AbderrahimHalimi propose in " IGARSS " 2011 " Unmixinghyperspectralimagesusingthegeneralizedbilinearmo del " and NaotoYokoya propose in " IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING " the 52nd volume the 2nd phase " Nonlinearunmixingofhyperspectraldatausingsemi-nonnegativ ematrixfactorization " in 2014.But the cost of bayes method is that calculated amount is large, and semi-NMF method easily converges to local extremum, and responsive to initial value, GDA is undertaken separating mixed method by pixel, this hinder we be applied to large high spectrum image solution mixed in.

Summary of the invention

For overcoming relevant art defect, the present invention proposes a kind of EO-1 hyperion nonlinear solution mixing method scheme based on border projection Optimal gradient.

Technical solution of the present invention is as follows:

Based on an EO-1 hyperion nonlinear solution mixing method for border projection Optimal gradient, it is characterized in that, based on the mathematical model that high spectrum image solution is mixed, and the generalized bilinear model conversation of being mixed by EO-1 hyperion nonlinear solution is the optimum obtaining following constraint condition:

$\underset{A,B}{min}f(A,B)=\left|\right|Y-EA-FB|{|}_F^2,$

$s.t.A\≥0,\underset{i=1}{\overset{M}{\Σ}}{A}_{i,j}=1,\∀j=1,...,P,0\≤B\≤C,$

Namely meeting constraint condition condition when, the objective function obtained about the minimum value of A and B, wherein, A is the abundance matrix of the pure end member of high spectrum image; B is the mutual abundance matrix of secondary;

Wherein, f (A, B) represents the function f about A and B, || .|| _frepresenting matrix not Luo Beini crow this Frobenius norm, s.t. represent make condition meet, represent that i is from 1 to M summation, represent for arbitrarily; Y ∈ R ^{d × P}represent that the matrix that EO-1 hyperion mixed pixel is formed, D and P represent the wave band sum of the spectrum dimension of high spectrum image and the sum of all pixels of space dimension respectively, E=[e ₁..., e _m] ∈ R ^{d × M}represent the matrix that the pure end member of high spectrum image is formed, e _i(i=1 ..., M) represent i-th pure end member, M represents the number of the pure end member of high spectrum image, A=[a ₁..., a _p] ∈ R ^{m × P}represent abundance matrix, a _i(i=1 ..., P) represent that the abundance of i-th pixel is vectorial, represent secondary interaction end variable matrix, represent Hadamard's product (i.e. dot product operation), B ∈ R ^{m (M-1)/2 × P}represent the mutual abundance matrix of secondary, C _{(i, j), k}=A _i,ka _j,k(k ∈ 1 ..., and P}), ∈ represents and belongs to;

The minimum value of concrete acquisition A and B comprises the steps:

Step 1.1: initiation parameter: random initializtion A ^land B ^l, make l=0, ε=10 ^-6, A ^land B ^lrepresent the iteration result of the l time of A and B respectively;

Step 1.2: upgrade A ^l+1; Solve following optimization problem by border projection Optimal gradient method and obtain A ^l+1:

$A^{l+1}=\underset{A}{min}f(A,B^l)=\left|\right|\left[\begin{array}{c}Y-F{B}^l\\ \mathrm{\δ}{1}_P^T\end{array}\right]-\left[\begin{array}{c}E\\ \mathrm{\δ}{1}_M^T\end{array}\right]A|{|}_F^2,$

s.t.A≥0,

Wherein δ for control abundance and close to 1 degree, represent that element is the P dimension row vector of 1 entirely, represent that element is the M dimension row vector of 1 entirely;

Step 1.3: upgrade B ^l+1; Solve following optimization problem by border projection Optimal gradient method and obtain B ^l+1:

$B^{l+1}=\underset{B}{\min }f(A^{l+1},B)=\left|\right|Y-{\mathrm{EA}}^{l+1}-FB|{|}_F^2,$

s.t.0≤B≤C,

Step 1.4: differentiate the condition of convergence: if this flow process terminates, and the abundance matrix A and secondary mutual abundance matrix B, the ε that obtain the pure end member of high spectrum image are global convergence thresholdings, if then l=l+1, revolution performs step 2.2.

At above-mentioned a kind of EO-1 hyperion nonlinear solution mixing method based on border projection Optimal gradient, in described step 1.2 and step 1.3, border projection Optimal gradient method is optimized based on following steps:

$\begin{array}{c}\underset{X}{\min }g\left(X\right)=\left|\right|Z-WX|{|}_F^2,\\ s.t.H\≤X\≤U,\end{array},$ Specifically comprise:

Step 1.1: initialization; V ^k=X ^k, α ^k=1, k=0, L=||W ^tw|| ₂, wherein Z, W, H and U represent known matrix, and V represents the constriction point of special selection, and β represents combination coefficient, and L represents Li Puxizi constant, || .|| ₂2 norms of representing matrix, k represents kth time iteration;

Step 1.2: upgrade X ^k;

X ^k＝P[Q],

$Q=V^k-\frac{1}{L}\▿g\left(V^k\right)$

Wherein ▽ is gradient operator, and P [Q] represents each element Q of matrix Q _i,jcarry out as lower boundary projection operation:

$X_{i,j}^k=\left\{\begin{array}{cc}{H}_{i,j},& if{Q}_{i,j}\&le;H_{i,j};\\ Q_{i,j},& if{H}_{i,j}<Q_{i,j}<U_{i,j};\\ U_{i,j},& if{Q}_{i,j}\&GreaterEqual;U_{i,j};\end{array}\right.$

Step 1.3: upgrade β ^k+1; ${\mathrm{\&beta;}}^{k+1}=\frac{1+\sqrt{4{\left({\mathrm{\&beta;}}^k\right)}^2+1}}{2}$

Step 1.4: upgrade V ^k+1; $V^{k+1}=V^k+\frac{{\mathrm{\&beta;}}^k-1}{{\mathrm{\&beta;}}^{k+1}}(X^k-X^{k-1})$

Step 1.5: differentiate the condition of convergence: || T [▽ g (X ^k)] || _f≤ max (10 ^-3, ε) || T [▽ g (X ⁰)] || _f, wherein max (a, b) represents the higher value getting a and b, T [▽ g (X ^k)] represent gradient matrix ▽ g (X ^k) in each element ▽ g (X ^k) _i,jproceed as follows:

${\&dtri;}^{P}g{\left(X^k\right)}_{i,j}=\left\{\begin{array}{cc}\&dtri;g{\left(X^k\right)}_{i,j},& if{X}_{i,j}<X_{i,j}<U_{i,j};\\ \min (0,\&dtri;g{\left(X^k\right)}_{i,j}),& if{X}_{i,j}=H_{i,j};\\ \max (0,\&dtri;g{\left(X^k\right)}_{i,j}),& if{X}_{i,j}=U_{i,j},\end{array}\right.$

Wherein min (a, b) represents the smaller value getting a and b;

If || T [▽ g (X ^k)] || _f> max (10 ^-3, ε) || T [▽ g (X ⁰)] || _f, then k=k+1, revolution performs step 1.2; If || T [▽ g (X ^k)] || _f≤ max (10 ^-3, ε) || T [▽ g (X ⁰)] || _f, this flow process terminates, and solves and obtains X.

Therefore, border projection Optimal gradient method of the present invention is by selecting special Searching point, and the mode that step-length is determined by Li Puxizi constant, greatly accelerates the optimized speed of convergence under boundary constraint, reach optimal convergence rates in addition, border projection Optimal gradient method can be effectively applied to EO-1 hyperion nonlinear solution based on GBM mixed in, it has fast convergence rate, selects the advantages such as insensitive to initial value.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of the embodiment of the present invention.

Fig. 2 is the abundance figure that the MoffettField data solution of the embodiment of the present invention mixes rear vegetation.

Fig. 3 is the abundance figure that the MoffettField data solution of the embodiment of the present invention mixes rear water.

Fig. 4 is the abundance figure that the MoffettField data solution of the embodiment of the present invention mixes rear soil.

Fig. 5 is the abundance figure that the MoffettField data solution of the embodiment of the present invention mixes rear vegetation-water.

Fig. 6 is the abundance figure that the MoffettField data solution of the embodiment of the present invention mixes rear vegetation-soil.

Fig. 7 is the abundance figure that the MoffettField data solution of the embodiment of the present invention mixes rear water-soil.

Embodiment

Below in conjunction with drawings and Examples, the present invention is further detailed.

With reference to accompanying drawing 1, the present invention forms primarily of 2 steps: set up the mathematical model that high spectrum image solution is mixed, the nonlinear solution solved based on GBM mixes model.The True Data that embodiment is chosen is MoffettField high-spectral data collection, mix for the solution based on GBM before it, the image that we choose 50 × 50 carrys out confirmatory experiment effect, and remaining 203 wave bands of wave band of removing water vapor absorption, it mainly comprises three kinds of end members: vegetation, water and soil.In order to verify the validity of put forward the methods, we adopt following index: reconstructed error (RE) and average spectral modeling distance (SMAD), and it is defined as follows:

$RE(Y,Y)=\sqrt{\frac{1}{LP}\underset{i=1}{\overset{P}{\&Sigma;}}\left|\right|y_i-y_i|{|}^2},$

$SMAD(Y,Y)=\underset{i=1}{\overset{P}{\&Sigma;}}{\cos }^{-1}\left(\frac{y_i^T{y}_i}{\sqrt{y_i^T{y}_i}\sqrt{{y_i}^T{y}_i}}\right),$

Wherein y _iand y _irepresent reconstructed pixel and reference pixel respectively, cos ^-1represent arc cosine.

During concrete enforcement, technical solution of the present invention can adopt computer software technology to realize automatic operational scheme.It is as follows that embodiment performs step:

(1) set up the mathematical model that high spectrum image solution is mixed, the generalized bilinear model (GBM) that EO-1 hyperion nonlinear solution is mixed is converted into following optimization problem:

$\underset{A,B}{min}f(A,B)=\left|\right|Y-EA-FB|{|}_F^2,$

$s.t.A\&GreaterEqual;0,\underset{i=1}{\overset{M}{\&Sigma;}}{A}_{i,j}=1,\&ForAll;j=1,...,P,0\&le;B\&le;C,$

Above-mentioned formula represents asks objective function about the minimum value of A and B, and satisfy condition $A\&GreaterEqual;0,\underset{i=1}{\overset{M}{\&Sigma;}}{A}_{i,j}=1,\&ForAll;j=1,...,P;$

Wherein, min minimizes operator, and f (A, B) represents the function f about A and B, || .|| _frepresenting matrix not Luo Beini crow this Frobenius norm, s.t. represent make condition meet, represent that i is from 1 to M summation, represent for arbitrarily.Y ∈ R ^{d × P}represent that the matrix that EO-1 hyperion mixed pixel is formed, D and P represent the wave band sum of the spectrum dimension of high spectrum image and the sum of all pixels of space dimension respectively, E=[e ₁..., e _m] ∈ R ^{d × M}represent the matrix that the pure end member of high spectrum image is formed, e _i(i=1 ..., M) represent i-th pure end member, M represents the number of the pure end member of high spectrum image, A=[a ₁..., a _p] ∈ R ^{m × P}represent abundance matrix, a _i(i=1 ..., P) represent that the abundance of i-th pixel is vectorial, represent secondary interaction end variable matrix, represent Hadamard's product (i.e. dot product operation), B ∈ R ^{m (M-1)/2 × P}represent the mutual abundance matrix of secondary, C _{(i, j), k}=A _i,ka _j,k(k ∈ 1 ..., and P}), ∈ represents and belongs to;

(2), the nonlinear solution solved based on GBM mixes model, and obtain abundance matrix A and the mutual abundance matrix B of secondary of the pure end member of high spectrum image, it is as follows that it specifically solves flow process:

(2.1): initiation parameter:

Random initializtion A ^land B ^l, make l=0, ε=10 ^-6, A ^land B ^lrepresent the iteration result of the l time of A and B respectively;

(2.2): upgrade A ^l+1;

Solve following optimization problem by border projection Optimal gradient method and obtain A ^l+1:

$A^{l+1}=\underset{A}{min}f(A,B^l)=\left|\right|\left[\begin{array}{c}Y-F{B}^l\\ \mathrm{\&delta;}{1}_P^T\end{array}\right]-\left[\begin{array}{c}E\\ \mathrm{\&delta;}{1}_M^T\end{array}\right]A|{|}_F^2,$

s.t.A≥0,

Wherein δ for control abundance and close to 1 degree, represent that element is the P dimension row vector of 1 entirely, represent that element is the M dimension row vector of 1 entirely;

(2.3): upgrade B ^l+1;

Solve following optimization problem by border projection Optimal gradient method and obtain B ^l+1:

$B^{l+1}=\underset{B}{\min }f(A^{l+1},B)=\left|\right|Y-{\mathrm{EA}}^{l+1}-FB|{|}_F^2,$

s.t.0≤B≤C,

(2.4): differentiate the condition of convergence:

If this flow process terminates, and the abundance matrix A and secondary mutual abundance matrix B, the ε that obtain the pure end member of high spectrum image are global convergence thresholdings, if then l=l+1, revolution performs step 2.2.

Border projection Optimal gradient method described in step (2), it is the optimization problem for solving following boundary constraint:

$\underset{X}{\min }g\left(X\right)=\left|\right|Z-WX|{|}_F^2,$

s.t.H≤X≤U,

Its solution procedure comprises following sub-step:

Step 1): initialization;

V ^k=X ^k, α ^k=1, k=0, L=||W ^tw|| ₂, wherein Z, W, H and U represent known matrix, and V represents the constriction point of special selection, and β represents combination coefficient, and L represents Li Puxizi constant, || .|| ₂2 norms of representing matrix, k represents kth time iteration;

Step 2): upgrade X ^k;

X ^k＝P[Q],

$Q=V^k-\frac{1}{L}\&dtri;g\left(V^k\right)$

Wherein ▽ is gradient operator, and P [Q] represents each element Q of matrix Q _i,jcarry out as lower boundary projection operation:

$X_{i,j}^k=\left\{\begin{array}{cc}{H}_{i,j},& if{Q}_{i,j}\&le;H_{i,j};\\ Q_{i,j},& if{H}_{i,j}<Q_{i,j}<U_{i,j};\\ U_{i,j},& if{Q}_{i,j}\&GreaterEqual;U_{i,j};\end{array}\right.$

Step 3): upgrade β ^k+1;

${\mathrm{\&beta;}}^{k+1}=\frac{1+\sqrt{4{\left({\mathrm{\&beta;}}^k\right)}^2+1}}{2}$

Step 4): upgrade V ^k+1;

$V^{k+1}=V^k+\frac{{\mathrm{\&beta;}}^k-1}{{\mathrm{\&beta;}}^{k+1}}(X^k-X^{k-1})$

Step 5): differentiate the condition of convergence:

|| T [▽ g (X ^k)] || _f≤ max (10 ^-3, ε) || T [▽ g (X ⁰)] || _f, wherein max (a, b) represents the higher value getting a and b, T [▽ g (X ^k)] represent gradient matrix ▽ g (X ^k) in each element ▽ g (X ^k) _i,jproceed as follows:

${\&dtri;}^{P}g{\left(X^k\right)}_{i,j}=\left\{\begin{array}{cc}\&dtri;g{\left(X^k\right)}_{i,j},& if{X}_{i,j}<X_{i,j}<U_{i,j};\\ \min (0,\&dtri;g{\left(X^k\right)}_{i,j}),& if{X}_{i,j}=H_{i,j};\\ \max (0,\&dtri;g{\left(X^k\right)}_{i,j}),& if{X}_{i,j}=U_{i,j},\end{array}\right.$

Wherein min (a, b) represents the smaller value getting a and b;

If || T [▽ g (X ^k)] || _f> max (10 ^-3, ε) || T [▽ g (X ⁰)] || _f, then k=k+1, revolution performs step 1.2; If || T [▽ g (X ^k)] || _f≤ max (10 ^-3, ε) || T [▽ g (X ⁰)] || _f, this flow process terminates, and solves and obtains X.Border projection Optimal gradient method speed of convergence is

In embodiment, D=203, P=2500, M=3, ε=10 ^-6.To MoffettField high-spectral data collection carry out nonlinear solution mixed after, the abundance figure of the vegetation obtained, water, soil, vegetation-water, vegetation-soil and water-soil is respectively as shown in accompanying drawing 2-7.By contrast algorithm GDA and semi-NMF, RE and SMAD is as shown in table 1.

Table 1 compares REs (10 ^-2) and SMADs (10 ^-2)

Should be understood that, the part that this instructions does not elaborate all belongs to prior art.

Should be understood that; the above-mentioned description for embodiment is comparatively detailed; therefore the restriction to scope of patent protection of the present invention can not be thought; those of ordinary skill in the art is under enlightenment of the present invention; do not departing under the ambit that the claims in the present invention protect; can also make and replacing or distortion, all fall within protection scope of the present invention, request protection domain of the present invention should be as the criterion with claims.

On the whole, propose a kind of EO-1 hyperion nonlinear solution mixing method based on border projection Optimal gradient, border projection Optimal gradient method speed of convergence is can be effectively applied to EO-1 hyperion nonlinear solution based on GBM mixed in, it has fast convergence rate, selects the advantages such as insensitive to initial value.

Specific embodiment described herein is only to the explanation for example of the present invention's spirit.Those skilled in the art can make various amendment or supplement or adopt similar mode to substitute to described specific embodiment, but can't depart from spirit of the present invention or surmount the scope that appended claims defines.

Claims (2)

1. based on an EO-1 hyperion nonlinear solution mixing method for border projection Optimal gradient, it is characterized in that, based on the mathematical model that high spectrum image solution is mixed, and the generalized bilinear model conversation of being mixed by EO-1 hyperion nonlinear solution is the optimum obtaining following constraint condition:

$\underset{A,B}{min}f(A,B)=\left|\right|Y-EA-FB|{|}_F^2,$

$s.t.A\&GreaterEqual;0,\underset{i=1}{\overset{M}{\&Sigma;}}{A}_{i,j}=1,\&ForAll;j=1,...,P,0\&le;B\&le;C,$

Namely meeting constraint condition condition when, the objective function obtained about the minimum value of A and B, wherein, A is the abundance matrix of the pure end member of high spectrum image; B is the mutual abundance matrix of secondary;

Wherein, f (A, B) represents the function f about A and B, || .|| _frepresenting matrix not Luo Beini crow this Frobenius norm, s.t. represent make condition meet, represent that i is from 1 to M summation, represent for arbitrarily; Y ∈ R ^{d × P}represent that the matrix that EO-1 hyperion mixed pixel is formed, D and P represent the wave band sum of the spectrum dimension of high spectrum image and the sum of all pixels of space dimension respectively, E=[e ₁..., e _m] ∈ R ^{d × M}represent the matrix that the pure end member of high spectrum image is formed, e _i(i=1 ..., M) represent i-th pure end member, M represents the number of the pure end member of high spectrum image, A=[a ₁..., a _p] ∈ R ^{m × P}represent abundance matrix, a _i(i=1 ..., P) represent that the abundance of i-th pixel is vectorial, F=[e ₁⊙ e ₂..., e ₁⊙ e _m, e ₂⊙ e ₃..., e ₂⊙ e _m..., e _m-1⊙ e _m] ∈ R ^{d × M (M-1)/2}represent secondary interaction end variable matrix, ⊙ represents Hadamard's product (i.e. dot product operation), B ∈ R ^{m (M-1)/2 × P}represent the mutual abundance matrix of secondary, C _{(i, j), k}=A _i,ka _j,k(k ∈ 1 ..., and P}), ∈ represents and belongs to;

The minimum value of concrete acquisition A and B comprises the steps:

Step 1.1: initiation parameter: random initializtion A ^land B ^l, make l=0, ε=10 ^-6, A ^land B ^lrepresent the iteration result of the l time of A and B respectively;

Step 1.2: upgrade A ^l+1; Solve following optimization problem by border projection Optimal gradient method and obtain A ^l+1:

$\begin{array}{c}{A}^{l+1}=\underset{A}{\min }f(A,B^l)={\left|\right|\left[\begin{array}{c}Y-F{B}^l\\ \mathrm{\&delta;}{1}_P^T\end{array}\right]-\left[\begin{array}{c}E\\ \mathrm{\&delta;}{1}_M^T\end{array}\right]A\left|\right|}_F^2,\\ s.t.A\&GreaterEqual;0,\end{array}$

Wherein δ for control abundance and close to 1 degree, represent that element is the P dimension row vector of 1 entirely, represent that element is the M dimension row vector of 1 entirely;

Step 1.3: upgrade B ^l+1; Solve following optimization problem by border projection Optimal gradient method and obtain B ^l+1:

$\begin{array}{c}{B}^{l+1}=\underset{B}{\min }f(A^{l+1},B)={\left|\right|Y-{\mathrm{EA}}^{l+1}-\mathrm{FB}\left|\right|}_F^2,\\ s.t.0\&le;B\&le;C,\end{array}$

Step 1.4: differentiate the condition of convergence: if this flow process terminates, and the abundance matrix A and secondary mutual abundance matrix B, the ε that obtain the pure end member of high spectrum image are global convergence thresholdings, if then l=l+1, revolution performs step 2.2.

2. a kind of EO-1 hyperion nonlinear solution mixing method based on border projection Optimal gradient according to claim 1, is characterized in that: in described step 1.2, and border projection Optimal gradient method is optimized based on following steps:

$\begin{array}{c}\underset{X}{\min }g\left(X\right)=\left|\right|Z-WX|{|}_F^2,\\ s.t.H\&le;X\&le;U,\end{array},$ Specifically comprise:

Step 1.1: initialization; V ^k=X ^k, α ^k=1, k=0, L=||W ^tw|| ₂, wherein Z, W, H and U represent known matrix, and V represents the constriction point of special selection, and β represents combination coefficient, and L represents Li Puxizi constant, || .|| ₂2 norms of representing matrix, k represents kth time iteration;

Step 1.2: upgrade X ^k;

X ^k＝P[Q],

Wherein be gradient operator, P [Q] represents each element Q of matrix Q _i,jcarry out as lower boundary projection operation:

$X_{i,j}^k=\left\{\begin{array}{cc}{H}_{i,j},& if{Q}_{i,j}\&le;H_{i,j};\\ Q_{i,j},& if{H}_{i,j}<Q_{i,j}<U_{i,j};\\ U_{i,j},& if{Q}_{i,j}\&GreaterEqual;U_{i,j};\end{array}\right.$

Step 1.3: upgrade β ^k+1; ${\mathrm{\&beta;}}^{k+1}=\frac{1+\sqrt{4{\left({\mathrm{\&beta;}}^k\right)}^2+1}}{2}$

Step 1.4: upgrade V ^k+1; $V^{k+1}=V^k+\frac{{\mathrm{\&beta;}}^k-1}{{\mathrm{\&beta;}}^{k+1}}(X^k-X^{k-1})$

Step 1.5: differentiate the condition of convergence: $\left|\right|T\&lsqb;\&dtri;g\left(X^k\right)\&rsqb;|{|}_F\&le;max({10}^{-3},\mathrm{\&epsiv;})||T\&lsqb;\&dtri;g\left(X^0\right)\&rsqb;|{|}_F,$ Wherein max (a, b) represents the higher value getting a and b, represent gradient matrix in each element proceed as follows:

${\&dtri;}^{P}g{\left(X^k\right)}_{i,j}=\left\{\begin{array}{cc}\&dtri;g{\left(X^k\right)}_{i,j},& if{H}_{i,j}<X_{i,j}<U_{i,j};\\ \min \left(0,\&dtri;g{\left(X^k\right)}_{i,j}\right),& if{X}_{i,j}=H_{i,j};\\ \max \left(0,\&dtri;g{\left(X^k\right)}_{i,j}\right),& if{X}_{i,j}=U_{i,j},\end{array}\right.$

Wherein min (a, b) represents the smaller value getting a and b;

If $\left|\right|T\&lsqb;\&dtri;g\left(X^k\right)\&rsqb;|{|}_F>max({10}^{-3},\mathrm{\&epsiv;})||T\&lsqb;\&dtri;g\left(X^0\right)\&rsqb;|{|}_F,$ Then k=k+1, revolution performs step 1.2; If $\left|\right|T\&lsqb;\&dtri;g\left(X^k\right)\&rsqb;|{|}_F\&le;max({10}^{-3},\mathrm{\&epsiv;})||T\&lsqb;\&dtri;g\left(X^0\right)\&rsqb;|{|}_F,$ This flow process terminates, and solves and obtains X.