CN106778530A - A kind of hyperspectral image nonlinear solution mixing method based on bilinearity mixed model - Google Patents

Abstract

The invention belongs to technical field of remote sensing image processing, specially a kind of hyperspectral image nonlinear solution mixing method based on bilinearity mixed model.Geometrical property of the present invention according to bilinearity mixed model, the linear contributions for having merged common nonlinear effect end points are expressed as by by the non-linear blending constituent in data, the nonlinear solution of complexity is set to mix the mixed problem of simple linear solution that is converted into, and then the mixed algorithm iteration of combination linear solution estimates correct abundance.The present invention observes the mixed model of pixel from EO-1 hyperion, with reference to its geometry and physical significance, can not only effectively make up the mixed deficiency of linear solution, and can preferably overcome the adverse effect that synteny effect brings.With to noise and the preferable robustness of end member number, the mixed effective means of high-spectrum remote sensing nonlinear solution can be solved as a kind of.The high precision solution based on high-spectrum remote sensing is mixed and the detection and identification of ground target in terms of there is important application value.

Description

A kind of hyperspectral image nonlinear solution mixing method based on bilinearity mixed model

Technical field

The invention belongs to technical field of remote sensing image processing, and in particular to a kind of hyperspectral image nonlinear solution mixing method.

Background technology

Remote sensing technology is the emerging complex art for growing up in the sixties in this century, with space, electron-optical, calculating The science and technology such as machine, geography are closely related, and are one of most strong technological means of research earth resource environment.EO-1 hyperion is distant Sense is the multidimensional information acquiring technology that imaging technique is combined with spectral technique.Its image has that spectral resolution is high, collection of illustrative plates The characteristics of unification, for the extraction and analysis of terrestrial object information provide extremely abundant information.It is relatively low yet with spatial resolution And the complicated variety of atural object distribution, the pixel in image is mostly according to certain way by several different pure substance spectras The mixed pixel of composition.The presence of mixed pixel causes the applications such as traditional classification, target detection of pixel level high spectrum image It is restricted [1].To solve the problem, make analysis and apply to be carried out in sub-pixed mapping level, it is necessary to realize that spectrum solution is mixed, by these Pixel analysis are the substance spectra (end member) and the ratio (abundance) [2] in pixel corresponding region of its constituent.Spectrum solution The mixed detectivity that can greatly improve high-spectrum remote-sensing to sub-pixed mapping level target, has been widely used in geological sciences, environment In the fields such as science, precision agriculture and military affairs.

In the past through describing mixed pixel phenomenon frequently with linear mixed model (Linear Mixture Model, LMM), Think that pixel is formed according to respective abundance linear combination by end member.Although LMM models are simple and with certain physical significance, Because being difficult to the accurate description such as tight mixing of sand ground, mineral Mixed Zone and vegetative coverage and the multi-level of urban area mixing The non-linear mixed effects such as conjunction, can bring larger error to result.Therefore, it is various non-linear mixed to meet application requirement higher Matched moulds type and algorithm are proposed to make up the deficiency [3] of linear solution mixing method, [4].Wherein, including be usually used in closely mixing The Hapke models [5] of scene and between only considering end member two-by-two rescattering effect bilinearity mixed model (Bilinear Mixture Model,BMM).BMM is mainly made up of linear hybrid with the non-linear two parts that mix, and the difference according to the latter can divide It is Fan models (Fan Model, FM) [6], generalized bilinear model (Generalized Bilinear Model, GBM) [7], Multinomial posteriority nonlinear model (Polynomial Post-Nonlinear Model, PPNM) [8] etc..Due to BMM have compared with Good physical significance and model it is relatively easy and frequently as the basis of the mixed algorithm research of current nonlinear solution.It is main when known to end member The related abundance algorithm for estimating wanted includes the FCLS (Fully Constrained Least Squares) that first order Taylor launches, [3], [4] such as gradient descent method (Gradient Descent Algorithm, GDA) and Bayesian methods.But they There is a problem of that computing is excessively complicated and can often be absorbed in local minimum, and between end member and virtual end member strong correlation produce it is conllinear Property effect also results in result over-fitting and more sensitive to noise [9].Therefore, the mixed algorithm of BMM solutions needs further to be changed Enter, to obtain the mixed result of nonlinear solution of higher precision simultaneously in the efficiency for improving algorithm.

Some concepts related to the present invention are described below

Bilinearity mixed model

For high-spectral data X=[x₁,x₂,…,x_m]∈R^n×m, its each column x_j∈R^n×1(j=1,2 ... m) all correspond to one It is individual vectorial with the n pixel of wave band, co-exist in m such pixel.With A=[a₁,a₂,...,a_r]∈R^n×rRepresent end member square Battle array (r is end member number), s_ijIt is end member a_iIn pixel x_jIn abundance, ε_jIt is model error.LMM assume under pixel be end member by The linear combination of abundance, to meet physical significance, abundance need to meet the constraints [2] of non-negative and " and being one "：

BMM then increased the rescattering mixed term between end member two-by-two on the basis of LMM, come for FM, GBM and PPNM Say, pixel is expressed as shown in formula (13) (14) (15)：

Wherein, a_i.*a_k=[a_i,1a_k,1,a_i,2a_k,2,...,a_i,na_k,n]^TIt is the Hadamard products between end member, also referred to as Virtual end member, for describing the rescattering non-linear interaction that light occurs between two kinds of end member materials.In FM, this two The intensity of secondary scattering is equal to the product between correspondence end member abundance, does not also just exist and the end member when not existing certain end member in pixel Related rescattering.But because FM is a relatively stringent model, GBM is just to the friendship between end member two-by-two on the basis of FM A nonlinear parameter γ is introduced in mutually_i,k,j∈ [0,1] make model become more flexibly, and when for,…,r-1, K=i+ ... 1, there is r γ,_i,k,jGBM is exactly LMM when=0, if same γ_i,k,j=1 GBM is just equivalent to FM.PPNM compares preceding two Considering end member and the rescattering of itself kind of model more, and with a parameter b_jNon-linear effects in regulation and control pixel.

The content of the invention

It is an object of the invention to propose a kind of high precision and the mixed side of the low hyperspectral image nonlinear solution of computational complexity Method.

It is proposed by the present invention that algorithm is mixed based on bilinearity mixed model hyperspectral image nonlinear solution, concretely comprise the following steps：It is first First pass through method of geometry and determine a non-linear end points for concentrating all rescattering effects, bilinearity blended data is approximately thrown Shadow is its linear hybrid part, so as to be converted to the mixed problem of simple linear solution.Due to without considering in optimization with end member number Virtual end member and partial parameters and direct solution abundance that mesh increases and is significantly increased, influence synteny is weakened on solving precision While reduce algorithm complex.Then according to the approximation relation of projection, repaiied using the mixed algorithm iteration of traditional linear solution Simultaneously estimate to obtain last abundance in orthographic projection position.Compared with other outstanding congenic methods, the present invention is to noise and end member number Mesh has preferable robustness, not only increases abundance estimated accuracy and reduces operation time.Particular content is described below：

First, the analysis of bilinearity mixed model geometrical property

Each pixel x in high spectrum image_jA point in n-dimensional space can be considered to be, under LMM hypothesis, by Need to meet the constraints of non-negative and " and being one ", end member collection [a in abundance₁,a₂,...,a_r] (r is to constitute r-1 n-dimensional subspace ns End member number) in a minimum monomorphous Simplex for all data points of parcel_L.Such as three end members just constitute two dimensional surface On a triangle.On the other hand, from for mathematical meaning, the linear hybrid part in BMM is exactly LMM, and abundance is same Sample need to meet non-negative and " and being one " the two constraintss.The non-linear mixed term of so BMM is considered as increased to LMM One is disturbed, and makes the data being located at originally in monomorphous to external deviation, and dividing for similar bulge shape is formed in r n-dimensional subspace ns Cloth.Due to the effect of different nonlinear factors, the data distribution of each model has differences.

Because r n-dimensional subspace ns need to be opened into by wherein r linearly independent vector, it is believed that one in the subspace is located at R-1 ties up sub- simple form Simplex_LOutside end points and known r end member combination define BMM data points dividing within this space Cloth.That is the corresponding data point of FM, GBM and PPNM is actually all located at being made up of true end member and an additional endpoint R dimensions affine hull Affine_BLIn, and Simplex_LIt is Affine_BLIn a r-1 tie up sub- simple form.Arbitrarily pass through non-linear end The straight line of point and BMM data points all will be with Simplex_LIntersecting, subpoint is approximately the linear hybrid part of BMM data points.

2nd, the determination of non-linear end points p

Non-linear end points p decides the projected position of BMM data, and the influence of key is played to the mixed result of last solution.It is actual On, r r-1 dimension hyperplane in r n-dimensional subspace ns is determined with the corresponding BMM data points that they are formed respectively per r-1 end member, These planes characterize rescattering effect when lacking an end member respectively.And the point p exactly intersection points of these hyperplane and with end Unit constitutes least affine bag, is the concentrated reflection of all rescattering effects.Point p is exactly the non-of them to the linear contributions of data Linear hybrid composition.

It is determined that during these planes, the non-linear mixing midpoint in each plane is first calculated according to BMM and known end member ω₁,...,ω_r-1,ω_r(i.e. each end member abundance is identical), is calculated if PPNM for FM and GBM such as formulas (5) by formula (6)：

Then data are dropped into r with principal component analysis (Principle Component Analysis, PCA) algorithm to tie up, Vertex set { ω₁,a₂,...,a_r},{a₁,ω₂,...,a_r..., { a₁,a₂,...,ω_rJust determine that r r-1 dimension is super in the space Plane H₁…H_r.Represent coplanar as 0 with monomorphous volume, non-linear end points p ∈ R^r×1It is the unique of suitable constant linear equation group (7) Solution：

Wherein U^TBe r characteristic vector is constituted before data matrix, and the solution of equation group (7) is present and unique, i.e., these Plane always has unique intersection point p in r dimension spaces.It is determined that after end points p, the optimal of formula (8) is solved using least square method Solution can obtain pixel x_j In center of gravity sit Mark h_j：

While x_j In projection coordinate s '_jFor：

3rd, the amendment of projected position and abundance are estimated

s′_jThe projected position of determination will have one between position corresponding to the linear hybrid part of BMM pixels less Deviation, but it and the true abundance s of pixel_jThere is approximate linear relationship between the corresponding non-linear mixing portions of BMM.Using the pass System can be such that it is constantly approached to real linear hybrid position by correcting the projected position of data repeatedly, and throw simultaneously Shadow coordinate s '_jAlso true abundance will progressively be converged on.

Try to achieve projection coordinate s '_jAfterwards, its non-linear mixing portion is calculated according to each BMMThen solving-optimizing problem (10) obtain making pixel to the most short coefficient lambda of air line distance where it_j：

Wherein, have for FM and GBMPPNM is Try to achieve the least square solution of formula (10)Then projected position can be modified toRevised point y_jBy than initial projection more approaching to reality linear hybrid position.It is last just available traditional Linear abundance algorithm for estimating solves new projection y_jAbundance, FCLS [10] algorithm is used here, and iterate most with this After converge on correct abundance.And after abundance is tried to achieve, the nonlinear parameter linear programming method in BMM is directly tried to achieve.

Specific steps according to being used in the algorithm that the above, the present invention are used are summarized as follows：

Input：High-spectral data X ∈ R^n×mWith end member matrix A ∈ R^n×r；

Output：Abundance matrix S ∈ R^r×m；

Step 1 determines non-linear end points p：

1.1：By formula (5), PPNM calculates each self-corresponding r non-linear mixing midpoint to FM and GBM respectively by formula (6) ω₁,...,ω_r-1,ω_r(i.e. each end member Abundances are equal)；

1.2：Data are dropped into r using PCA algorithms to tie up, r r-1 dimension hyperplane H in the space is determined₁…H_r；

1.3：System of linear equations (7) is solved, non-linear end points p is its unique solution.

Step 2 amendment projection and estimation abundance：

2.1：The optimal solution of (8) is calculated with least square method and approximate abundance s ' obtained by formula (9)_j；

2.2：For each observation pixel x_j, j=1 ..., m performs following operation：

2.2a) initialize iterations t=0 and abundance

2.2b) calculateThe non-linear mixed vectors of BMM of correspondence subpoint

2.2c) solve optimization problem (10) and obtain coefficientAnd calculate new projected position

2.2d) solved with linear abundance algorithm for estimatingAbundance s^(t+1)；

Abundance matrix S=S 2.2e) is exported if reaching maximum iteration or meeting convergence precision^(t+1), otherwise return to step Rapid 2.2b)

Step 3：Output result S=(s₁,s₂,…,s_m) and try to achieve nonlinear parameter γ and b with linear programming

In the present invention：

Step 2.2c) in coefficientIt is least square solution：

Step 2.2d) it is middle using FCLS Algorithm for Solving s^(t+1)。

Step 2.2e) in maximum iteration be 200.

Step 2.2e) in, by judging the relative change between each iteration and preceding an iterationIt is It is no to be less than 10^-8To judge whether convergence.

The beneficial effects of the present invention are：Its mixed model that pixel is observed from EO-1 hyperion, with reference to its geometry and thing Reason meaning, can not only effectively make up the mixed deficiency of linear solution, and can preferably overcome the unfavorable shadow that synteny effect brings Ring, the mixed effective means of high-spectrum remote sensing nonlinear solution can be solved as a kind of.Based on high-spectrum remote sensing High precision solution is mixed and detection and the identification aspect of ground target have important application value.

Emulation and the experiment of true high-spectrum remote sensing show, compared with analogous algorithms, are taken in the inventive method Algorithm has more preferable abundance estimated accuracy and relatively low computational complexity, with to noise and the preferable robust of end member number Property, this has important practical significance for the complicated high-spectral data for the treatment of.

Brief description of the drawings

The geometry distribution of Fig. 1 model datas.Wherein, (a) LMM, (b) FM, (c) GBM, (d) PPNM.

Fig. 2 Method And Principle schematic diagrames.Wherein, the determination of (a) non-linear end points, (b) FM data projections and distortion.

The amendment of Fig. 3 distortions.

5 endmember spectras in Fig. 4 USGS library of spectra.

Fig. 5 algorithm GAEB-FCLS convergences.

The true high-spectrum remote sensings of Fig. 6.Wherein, (a) AVIRIS Cuprite and subregion (b) HYDICE WashingtonDC and subregion.

The abundance figure that each algorithm of Fig. 7 AVIRIS data is estimated.Wherein, (a) FCLS, (b) KFCLS are from left to right often classified as, (c) Fan-FCLS, (d) PPNM-GDA, (e) GBM-GDA, (f) GAEB-FCLS (FM), (g) GAEB-FCLS (GBM), (h) GAEB-FCLS(PPNM)；The montmorillonite that often row is respectively from top to bottom, three kinds of abundance of material of desert earth's surface and alunite.

The abundance figure of Fig. 8 HYDICE data algorithms GAEB-FCLS.Wherein, often row is successively a point FM from top to bottom, GBM and Estimated result under PPNM；Each column corresponds respectively to 5 kinds of abundance of end member：(a) water body, (b) roof, (c) trees, (d) road, (e) meadow.

Specific embodiment

Below, specific embodiment of the invention is illustrated with as a example by analogue data and actual remote sensing image data respectively.

The mixed algorithm GEAB- of the high-spectrum remote sensing nonlinear solution based on bilinearity mixed model used in the present invention FCLS is represented.

1st, analogue data experiment

By GAEB-FCLS algorithms and linear abundance algorithm for estimating FCLS [10] in this section, the data based on Gaussian kernel are driven Dynamic nonlinear solution mixing method KFCLS [12] (nuclear parameter is obtained by cross-validation method between 0.01-300) and FM, GBM with The corresponding traditional derivation algorithm of tri- models of PPNM：Fan-FCLS [6], GBM-GDA [7] and PPNM-GDA [8] carry out performance Compare.And root-mean-square error RMSE (Root Mean Square Error) and the reconstructed error RE of image using abundance (Reconstructed Error) two factors evaluate the precision that each algorithm abundance is estimated：

Wherein, s_iAnd x_iTrue abundance and data andWithIt is the abundance and the data by model reconstruction estimated, m is picture First number, n is wave band number, and r is end member number.

First, the spectrum that 10 kinds of different atural objects are have chosen from US Geological Survey (USGS) library of spectra is given birth to as end member Into analogue data, Fig. 4 is listed including atural objects such as vegetation, sand grounds in five interior endmember spectras.The data of the library of spectra 224 spectral bands in 0.38 μm of -2.5 mum wavelength interval are covered, spectral resolution is 10nm.The generating mode of data is such as Under：The data abundance for obeying Dirichlet distributions [11] is first randomly generated, then according to three kinds of bilinearity mixed model FM, GBM and PPNM construct three class BMM data respectively.The wherein nonlinear parameter γ of GBM data random values in [0,1], and Parameter b spans are [- 0.3,0.3] in PPNM data.Three groups of experiments are all provided with per class data, respectively by changing noise With the performance that run time under end member number and relatively more different pixel numbers carrys out comprehensively evaluation algorithms.Each experiment is identical Under the conditions of independent operating 10 times.

Test the impact analysis of 1 noise：The com-parison and analysis noise robustness of each algorithm in this experiment.Using pixel number It is 5 three class BMM data of end member composition in 2000, Fig. 4, respectively adds the height of 20dB, 30dB, 40dB, 50dB and 60dB This white noise simultaneously considers muting situation.Table 1, table 2 and table 3 list each algorithm under tri- kinds of models of FM, GBM and PPNM successively For the abundance estimated accuracy of different signal to noise ratio noise datas.All in all, with the reduction of data SNR, all algorithms pair The mixed result precision of the solution of three class data is also gradually deteriorated, and this algorithm compares other algorithms abundance estimated accuracy in all cases It is always best.The result of linear algorithm FCLS will be worse than the mixed algorithm of other four kinds of nonlinear solutions, although KFCLS algorithms are reconstructed Error is both less than FCLS but RMSE is poor, shows the problem of over-fitting.Because PPNM models have preferably with respect to FM and GBM Robustness [8], therefore PPNM-GDA algorithms are better than Fan-FCLS in the result of three class data and two kinds of BMM solutions of GBM-GDA are mixed Algorithm.Although the RE of PPNM-GDA is smaller during 20dB noises, the RMSE of abundance is but worse than the algorithm of this paper, also shows one Fixed over-fitting.The experiment shows that GAEB-FCLS is superior to being had by the data of different degrees of influence of noise compared to other algorithms Property.

Each arithmetic result under the data of the FM of table 1 difference noises compares (× 10^-2)

Each arithmetic result under the data of the GBM of table 2 difference noises compares (× 10^-2)

Each arithmetic result under the data of the PPNM of table 3 difference noises compares (× 10^-2)

Test the impact analysis of 2 end member numbers：Each algorithm of com-parison and analysis is in different end member number conditions in this experiment The precision of the lower mixed result of solution.The pixel number of data is 2000, adds the white Gaussian noise of 50dB, end member number to take 3,5 Hes respectively 8.Each algorithm mixes knot for the solution of different end member number datas under table 4, table 5 and table 6 compare tri- kinds of models of FM, GBM and PPNM Really.From the point of view of the RMSE of abundance, the arithmetic result of participation contrast is substantially all and is reduced with the increase of metadata number, but The trend that the RE of KFCLS results but shows significant increase with end member number and reduces, further shows that crossing for it is intended Conjunction problem.For three class BMM data, the RMSE and RE of the mixed result of algorithm GAEB-FCLS solutions are smaller than other algorithms, still table Reveal best solution mixcibility energy.

Each arithmetic result under the data of the FM of table 4 difference end member numbers compares (× 10^-2)

Each arithmetic result under the data of the GBM of table 5 difference end member numbers compares (× 10^-2)

Each arithmetic result under the data of the PPNM of table 6 difference end member numbers compares (× 10^-2)

Test 3 pixel numbers and Convergence Properlies：This experiment analyzes the convergence of proposed method first, then Each algorithm is tested with different pixel numbers and end member number and compares run time.Fig. 5 depicts GAEB-FCLS algorithms to 2000 The convergence curve of the class data of the FM of individual pixel, GBM and PPNM tri-.It can clearly be seen that the RMSE of abundance is with iteration dullness Drop, and algorithm is basic in 40 iteration or so convergence.Wherein, the result precision under two kinds of models of PPNM and FM is almost converged on 0, in addition, though because the faint projection disturbance of GBM nonlinear parameters γ makes convergence precision slightly poorer to other two kinds of models, but according to So smaller value has been converged to quickly.It is 5 and 50dB white Gaussian noises in end member number for the run time of relatively more each algorithm Under the conditions of, capture unit number is 1000,2000,4000,6000 and 8000 and then compares average result respectively, as shown in table 7.This Outward, be also compares in table 8 in 2000 pixels, 50dB white Gaussian noises, end member number takes 3 to 10 Riming time of algorithm.From knot From the point of view of fruit, as the run time for increasing all algorithms of pixel number is consequently increased, and contrast other the two kinds mixed precision of solution compared with The calculating time of good nonlinear solution mixed algorithm GBM-GDA and PPNM-GDA, GAEB-FCLS reduces more.With end member number Increase, the run time substantial increase of two kinds of GDA algorithms especially PPNM-GDA, and the time of GAEB-FCLS is not only less than Above both and also influenceed less by end member number.As can be seen here, set forth herein GAEB-FCLS algorithms be avoided that traditional BMM The mixed algorithm of solution is increased the problem for causing computation complexity very big by end member number.

Riming time of algorithm compares (s) under the different pixel numbers of table 7

Riming time of algorithm compares (s) under the different end member numbers of table 8

2nd, true remote sensing images experiment

This section employs airborne visible ray/Infrared Imaging Spectrometer (Airborne Visible/Infrared first Imaging Spectrometer, AVIRIS) it is taken at the Nevada, USA Cuprite mining areas bloom on June 19th, 1997 Spectrogram picture is come the abundance estimated result analyzing and evaluate each algorithm.The image size is 512 × 614 (shown in such as Fig. 6 (a)), There are 224 wave bands in 0.4-2.5 μm of range of wavelengths, spectral resolution is 10nm, and spatial resolution is 20m.Picked before solution is mixed Except influence of noise is larger and water vapor absorption wave band (1-2,104-113,148-167,221-224) remaining 188 wave bands afterwards.Separately Outward, also using HYDICE Washington country square Washington DC Mall images (Fig. 6 (b)) as second experimental data, It has 210 wave bands between 0.4-2.4 μm, same removal part signal to noise ratio is relatively low and water vapor absorption wave band (1-4,76,87, 101-111,136-153,198-210) remaining 191 wave bands afterwards.It is used for abundance after extracting end member with VCA algorithms [11] in experiment Estimate.By that will solve the end member material abundance figure of mixed result, truly distribution will be contrasted with atural object, and calculate the reconstruct mistake of image RE is differed to evaluate each algorithm solution mixcibility energy.

From the first width AVIRIS images intercept size for 50 × 80 subregion as test data, such as Fig. 6 (a) institutes Show.Understood using HySime algorithms [13] and according to historic survey and on-site inspection result [14] [15], the sub-district of AVIRIS Mainly include montmorillonite, 3 kinds of end member materials of desert earth's surface and alunite in domain, and end member is extracted with VCA algorithms.Fig. 7 is Each algorithm solution of AVIRIS images corresponding abundance figure of mixed result, compared with other algorithms, two kinds of calculations of PPNM-GDA and GAEB-FCLS Method preferably inverting can obtain alunite and be really distributed close to its atural object.9 kinds of Image Reconstructions to these algorithms of table are missed Difference is compared, and the RE of GAEB-FCLS mixed results of solution under different BMM hypothesis mixed less than corresponding nonlinear solution will be calculated Method.And for second HYDICE data, then in an experiment by its size for 100 × 100 sub-district area image is surveyed for algorithm Examination (shown in such as Fig. 6 (b)).Using with AVIRIS data identical methods, it is known that be primarily present water body in the scene, roof, tree Wood, road and 5 kinds of meadow atural object.As it can be observed in the picture that algorithm GAEB-FCLS is respectively under tri- kinds of model hypothesis of FM, GBM and PPNM Abundance inversion chart, can correctly reflect 5 kinds of true distributions of atural object.From the point of view of table 9, the KFCLS with over-fitting problem Algorithm is minimum for the RE of HYDICE images, but algorithm GAEB-FCLS mixes the result RE of algorithm with the nonlinear solution under corresponding BMM Compared to all relatively small.

The algorithm reconstructed error contrast (× 10 of the True Data of table 9^-2)

In summary, for simulation and actual high-spectral data, algorithm proposed by the present invention is similar relative to other For algorithm, the convergence rate all with preferable estimated accuracy and faster can rapidly and precisely realize high spectrum image Solution mix.

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Claims (5)

1. a kind of hyperspectral image nonlinear solution mixing method based on bilinearity mixed model, it is characterised in that：It is mixed in bilinearity On the basis of matched moulds type, introducing a non-linear end points makes the mixed problem projection transform of the nonlinear solution of high-spectrum remote sensing be letter Single linear solution is mixed, and the problem is iteratively solved while then mixing algorithm repairing orthographic projection position using linear solution；Specific step It is rapid as follows：

(1) the non-linear end points of bilinearity mixed model is determined

Based on different spectral mixing models, high-spectrum remote sensing X=[x₁,x₂,…,x_m]∈R^n×mIn each pixel x_j∈ R^n×1(j=1,2 ... can m) be expressed as：

$x_j^{LMM}=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{a}_i{s}_{ij}+{\mathrm{\ϵ}}_j---\left(1\right)$

$x_j^{FM}=\underset{i=1}{\overset{r}{\Σ}}{a}_i{s}_{i,j}+\underset{i=1}{\overset{r-1}{\Σ}}\underset{k=i+1}{\overset{r}{\Σ}}(a_i.*a_k)s_{i,j}{s}_{k,j}+{\mathrm{\ϵ}}_j---\left(2\right)$

$x_j^{GBM}=\underset{i=1}{\overset{r}{\Σ}}{a}_i{s}_{i,j}+\underset{i=1}{\overset{r-1}{\Σ}}\underset{k=i+1}{\overset{r}{\Σ}}(a_i.*a_k){\mathrm{\γ}}_{i,k,j}{s}_{i,j}{s}_{k,j}+{\mathrm{\ϵ}}_j---\left(3\right)$

$x_j^{PPNM}=\underset{i=1}{\overset{r}{\Σ}}{a}_i{s}_{i,j}+b_j\underset{i=1}{\overset{r}{\Σ}}\underset{k=1}{\overset{r}{\Σ}}(a_i.*a_k)s_{i,j}{s}_{k,j}+{\mathrm{\ϵ}}_j---\left(4\right)$

Formula (1) corresponds respectively to linear mixed model LMM and three kinds of bilinearity mixed models (BMM) to formula (4)：FM, GBM and PPNM；Wherein, A=[a₁,a₂,...,a_r]∈R^n×rEnd member matrix is represented, r is end member number, s_ijIt is end member a_iIn pixel x_jIn Abundance, ε_jIt is noise error；Abundance all meets non-negative and " and being 1 " constraint：

$s_{i,j}\≥0,\underset{i=1}{\overset{r}{\Σ}}{s}_{i,j}=1---\left(5\right)$

Nonlinear parameter γ_i,k,j(0≤γ_i,k,j≤ 1) and b_jFor regulating and controlling the non-linear effects degree in pixel；

R non-linear mixing midpoint in bilinearity mixed model is calculated first, it is then individual super flat to determine r to Data Dimensionality Reduction Face, finally calculates the intersection point of these hyperplane, that is, obtain non-linear end points, and idiographic flow is：

Input high-spectral data X ∈ R^n×mWith end member matrix A ∈ R^n×r；

Step 1.1：By formula (6), PPNM calculates each self-corresponding r non-linear mixing midpoint to FM and GBM respectively by formula (7) ω₁,...,ω_r-1,ω_r(i.e. each end member Abundances are equal)：

${\mathrm{\ω}}_q=\frac{1}{r-1}\underset{i=1}{\overset{r}{\Σ}}{a}_i+\frac{1}{{(r-1)}^2}\underset{i=1}{\overset{r-1}{\Σ}}\underset{k=i+1}{\overset{r}{\Σ}}(a_i.*a_k),(i\≠q,k\≠q,q=1,...,r)---\left(6\right)$

${\mathrm{\ω}}_q=\frac{1}{r-1}\underset{i=1}{\overset{r}{\Σ}}{a}_i+\frac{1}{{(r-1)}^2}\underset{i=1}{\overset{r}{\Σ}}\underset{k=1}{\overset{r}{\Σ}}(a_i.*a_k),(i\≠q,k\≠q,q=1,...,r)---\left(7\right)$

Step 1.2：Data are dropped into r using PCA algorithms to tie up, vertex set { ω is determined₁,a₂,...,a_r, { a₁,ω₂,..., a_r..., { a₁,a₂,...,ω_rThe r r-1 dimensions hyperplane H that opens within this space₁…H_r；

Step 1.3：System of linear equations (8) is solved, non-linear end points p is its unique solution：

$\det \left(\left[\begin{array}{ccccc}1& 1& ...& 1& 1\\ U^T{\mathrm{\ω}}_1& U^T{a}_2& ...& U^T{a}_r& p\end{array}\right]\right)=0,...,\det \left(\left[\begin{array}{ccccc}1& 1& ...& 1& 1\\ U^T{a}_1& U^T{a}_2& ...& U^T{\mathrm{\ω}}_r& p\end{array}\right]\right)=0---\left(8\right)$

Wherein, U^TIt is the matrix that r characteristic vector before data is constituted, det represents determinant；

Output nonlinear end points p；

(2) amendment projected position and iterative estimate abundance

Projection coordinate s ' is calculated first with p points_j, then substitute into model and calculate corresponding non-linear mixing portion, then determine to make picture Unit is finally new with FCLS Algorithm for Solving so as to obtain revised new projected position to the most short coefficient of air line distance where it The abundance of projected position, and carried out until convergence with this iteration, idiographic flow is：

Input：High-spectral data X ∈ R^n×mWith end member matrix A ∈ R^n×r, non-linear end points p；

Output：Abundance matrix S ∈ R^r×m；

Step 2.1：Barycentric coodinates s ' must be projected with least square method solving-optimizing problem (9)_j

$\begin{array}{cccc}\min & \frac{1}{2}\left|\right|U^T{x}_j-\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{U}^T{a}_i{h}_{i,j}-{\mathrm{ph}}_{r+1,j}|{|}^2,& s.t.& \underset{i=1}{\overset{r+1}{\Σ}}{h}_{i,j}=1\end{array}---\left(9\right)$

Here,

Step 2.2：For each observation pixel x_j, j=1 ..., m performs following operation：

2.2a) initialize iterations t=0 and abundance

2.2b) calculateThe non-linear mixed vectors of BMM of correspondence subpoint

For GBM and FM, it is defined as

For PPNM, it is defined as：

2.2c) solve optimization problem (10) and obtain coefficientAnd new projected position is calculated with formula (11)

$\begin{array}{cc}min& \frac{1}{2}\left|\right|x_j-\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{a}_i{s}_{i,j}^{\left(t\right)}-{\mathrm{\λ}}_j^{\left(t\right)}{\hat{x}}_j^{\left(t\right)}|{|}^2\end{array}---\left(10\right)$

$y_j^{\left(t\right)}=x_j-{\mathrm{\λ}}_j^{\left(t\right)}{\hat{x}}_j^{\left(t\right)}---\left(11\right)$

2.2d) solved with linear abundance algorithm for estimatingAbundance s^(t+1)；

Abundance matrix S=S 2.2e) is exported if reaching maximum iteration or meeting convergence precision^(t+1), otherwise return to 2.2b)；

Step 2.3：Output result S=(s₁,s₂,…,s_m), and try to achieve nonlinear parameter γ and b with linear programming.

2. solution mixing method according to claim 1, it is characterised in that：Step 2.2c) in coefficientIt is least square solution：

${\mathrm{\λ}}_j^{\left(t\right)}={(x_j-{\mathrm{\Σ}}_{i=1}^r{a}_i{s}_{i,j}^{\left(t\right)})}^T{\hat{x}}_j^{\left(t\right)}/\left({\hat{x}}_j^{\left(t\right)T}{\hat{x}}_j^{\left(t\right)}\right).$

3. solution mixing method according to claim 1, it is characterised in that：Step 2.2d) it is middle using FCLS Algorithm for Solving s^(t+1)。

4. solution mixing method according to claim 1, it is characterised in that：Step 2.2e) in maximum iteration be 200.

5. solution mixing method according to claim 4, it is characterised in that：Step 2.2e) in, by judge each iteration with it is preceding Relative change between an iterationWhether 10 are less than^-8To judge whether convergence.