CN112907473B - Quick hyperspectral image pixel unmixing method based on multi-core projection NMF - Google Patents
Quick hyperspectral image pixel unmixing method based on multi-core projection NMF Download PDFInfo
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Abstract
The invention discloses a multi-core projection NMF-based rapid hyperspectral image pixel unmixing method, which comprises the following steps: 1) Constructing a multi-core model, and initializing weight coefficient beta= [ beta ] of a kernel function 1 β 2 L β m ]The method comprises the steps of carrying out a first treatment on the surface of the 2) Determining the number of end members by using a PCA method, and initializing a base matrix W by using a VCA method; 3) Updating the weight coefficient beta of the kernel function according to the basis matrix W; 4) Updating the multi-core model K; 5) Updating the base matrix W; 6) When the maximum iteration times are met or the convergence threshold is met, taking the current base matrix W as a decomposed base matrix, and finishing the rapid hyperspectral image pixel unmixing based on the multi-core projection NMF; otherwise, turning to the step 3), the method can accurately realize the unmixed hyperspectral image pixels, and the processing time is shorter.
Description
Technical Field
The invention belongs to the technical field of image processing, and relates to a multi-core projection NMF-based rapid hyperspectral image pixel unmixing method.
Background
The spectrum of the hyperspectral image has the characteristics of high spectrum resolution, fine expression ground information, integrated patterns and the like, so that the hyperspectral image is widely applied to agriculture, mineral products, military and other aspects. However, due to the limitations of imaging principles and optical devices, the hyperspectral image has the problem of low spatial resolution, so that the existence of mixed pixels in the hyperspectral image is common, and obstacles are brought to the high-precision ground object classification, the image resolution improvement and the like of the hyperspectral image.
The pixels are basic units for forming the remote sensing image, and are called hybrid pixels if the pixels contain a plurality of coverage types. While the basic constituent unit of each mixed pixel is called an end member, and the proportion of each corresponding end member in the pixel where it is located is called abundance. Because of the limited spatial resolution of the remote sensor and the complex diversity of the nature ground features, the mixed pixels are commonly existing in the remote sensing image, and become a big obstacle for the remote sensing image to acquire high-precision information. Therefore, the process of decomposing the mixed pixels into different end members in the pixels and obtaining the information of each component and the proportion (abundance) thereof is pixel unmixing.
Non-negative matrix factorization (Non-negative Matrix Factorization, NMF) is one of the currently widely used methods of unmixing hyperspectral images, approximately decomposing hyperspectral data into a product form of a base matrix and a coefficient matrix. Because NMF algorithm is not convex and is easy to fall into a local optimal solution, the efficiency of the algorithm is low and the result accuracy is not high. Compared with the common NMF algorithm, projection non-negative matrix factorization (PNMF) maps data into a low-dimensional projection space by searching a projection matrix, and converts two variables of a base matrix and a coefficient matrix in the NMF into a variable represented by the projection matrix, so that the iteration speed of the parameter algorithm is improved. Since NMF and PNMF have low precision in unmixing low-dimensional linear inseparable data, a scholars put forward Kernel projection non-negative matrix factorization (Kernel PNMF, KPMF) and project original low-dimensional spatial linear inseparable data into high-dimensional space by a Kernel method to realize linear inseparable. Since different kernel functions have different characteristics and different applicable scenes, the KPNMF has a problem of kernel function selection.
The multi-kernel learning (Multiple Kernel Learning, MKL) method adopts a form of combining a plurality of kernel functions, and avoids the problem of kernel function selection by setting weight parameters for different kernel functions. Compared with a single-core method, the MKL method can realize automatic selection and weighting of the core function, and obtains better mapping capability. The multi-core non-negative matrix method (Multiple Kernel NMF, MKNMF) introduces the MKL method into the NMF, thereby improving the algorithm precision. The MKNMF algorithm has poor time performance because of the need to solve kernel functions in addition to the basis and coefficient matrices.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, and provides a multi-core projection NMF-based rapid hyperspectral image pixel unmixing method which can accurately realize hyperspectral image pixel unmixing and has short processing time.
In order to achieve the above purpose, the method for rapidly unmixing the hyperspectral image pixels based on the multi-core projection NMF comprises the following steps:
1) Constructing a multi-core model, and initializing weight coefficient beta= [ beta ] of a kernel function 1 β 2 L β m ]；
2) Determining the number of end members by using a PCA method, and initializing a base matrix W by using a VCA method;
3) Using the formula g=tr (K) -2Tr (WW T K)+Tr(WW T WW T K)+Tr(λW T ) Updating the weight coefficient beta of the kernel function according to the basis matrix W;
4) According to the formulaUpdating the multi-core model K;
5) According to the formulaUpdating the base matrix W;
6) When the maximum iteration times are met or the convergence threshold is met, taking the current base matrix W as a decomposed base matrix, and finishing the rapid hyperspectral image pixel unmixing based on the multi-core projection NMF; otherwise, go to step 3).
The multi-core model constructed in the step 1) is as follows:
wherein k is j Is the basic kernel function, s is the total number of kernel functions, beta j Weight coefficients that are kernel functions; setting a basic kernel function as a Gaussian kernel function, wherein the basic kernel function has the following expression:
using the MKL method, formula (4) is optimized as:
let λ= (λ) ij )，λ ij To restrict W i,j Lagrange multiplier of 0 or more, there is
G＝Tr(K)-2Tr(WW T K)+Tr(WW T WW T K)+Tr(λW T ) (9)
Both ends of the step (9) are simultaneously led to W to obtain
From the Kuhn-Tucker condition lambda ij W ij =0, multiplying both sides of formula (10) by W ij Obtaining
2(KW) ij W ij -(WW T K+KWW T W) ij W ij ＝0 (11)
Then W is iterated from equation (12)
In the iterative process, when phi (x) -WW T And when phi (x) is less than or equal to epsilon, finishing iteration, wherein epsilon is a set threshold value.
The specific process of determining the number of the end members by using the PCA method in the step 2) is as follows:
for hyperspectral images, the ith end member z i The variance contribution ratio of (2) is:
thus, for a given ρ ε (0, 1), there is a k >0 such that
Searching the first k end members so that the information proportion of the hyperspectral image can be more than or equal to rho, wherein k is the number of the end members.
The invention has the following beneficial effects:
according to the multi-core projection NMF-based rapid hyperspectral image pixel unmixing method, when the multi-core projection NMF-based rapid hyperspectral image pixel unmixing method is specifically operated, the hyperspectral data are mapped to a high-dimensional space by introducing a multi-core learning method, so that the linear separability is realized, the unmixing precision is improved, the number of end members is determined by utilizing a PCA method, and the end group matrix W is initialized by utilizing a VCA method, so that the iteration efficiency is improved, the unmixing processing time is shortened, the operation is convenient and simple, and the popularization and the application are facilitated.
Drawings
FIG. 1 is a chart of six spectral choices of the USGS spectral library;
FIG. 2 is a graph of abundance of each end member of the six spectral images of FIG. 1;
FIG. 3 is a graph comparing end member spectral curves extracted by 3 algorithms in a verification test with actual spectral curves;
FIG. 4 is a graph comparing abundance after unmixed with true abundance in 3 algorithms in a validation experiment;
FIG. 5 is a comparison of abundance maps and reference abundance maps after unmixing Urban images by 3 algorithms in a validation experiment;
Detailed Description
The invention is described in further detail below with reference to the attached drawing figures:
nonnegative matrix factorization and nuclear projection nonnegative matrix factorization
NMF is a radical decomposition method. For any given non-negative matrix V, NMF aims to find two non-negative matrices W and H, so that v≡wh, where V is the original data, W is the decomposed base matrix, and H is the coefficient matrix, it is obvious that for NMF algorithm, the number of parameters W and H required is greater than the number of formulas, there are obvious non-convexities, there are a lot of local optimal values, the obtained result is not unique, and the accuracy of the unmixed result is low. Meanwhile, as the calculation amount of NMF is large, the iterative convergence speed is slow, and when the processed data scale is large, the operation efficiency of NMF algorithm is low.
To improve NMF efficiency, yuan et al propose a projected non-negative matrix factorization (PNMF) algorithm. The basic idea of PNMF algorithm is: for n<m non-negative matrix V.epsilon.R n×m Then an attempt is made to find an m-dimensional low-dimensional subspace and an m×m projection matrix P whose objective function is
Wherein, the basis matrix W is more than or equal to 0, and the projection matrix P=WW T 。
Compared with the common NMF algorithm, the PNMF algorithm has only one variable W, so that the calculation amount of each iteration is reduced. Since PNMF methods remain non-convex, PNMF methods are still not achieving global minima. When nonlinear data is processed, there is a problem of performance degradation.
Utilizing nuclear mapping
φ:x∈V L →φ(x)∈F M (2)
Nonlinear data can be mapped from a low-dimensional space to a high-dimensional space in a nonlinear manner to achieve the goal of linear separability of the data. The KPNMF method maps the original nonlinear data to a high-dimensional space, and then decomposes the data by using a nuclear non-negative matrix decomposition mode:
φ:x∈V L →φ(x)∈F M (3)
wherein k=phi (x) T Phi (x) is a function of the corresponding kernel map.
However, how to properly select the kernel function for the multi-dimensional data set with large difference of data characteristic space distribution such as hyperspectral image is still a problem for KPNMF.
In order to solve the problems, the method for rapidly unmixing the hyperspectral image pixels based on the multi-core projection NMF comprises the following steps:
1) Constructing a multi-core model, and initializing weight coefficient beta= [ beta ] of a kernel function 1 β 2 L β m ]；
2) Determining the number of end members by using a PCA method, and initializing a base matrix W by using a VCA method;
3) Using the formula g=tr (K) -2Tr (WW T K)+Tr(WW T WW T K)+Tr(λW T ) Updating the weight coefficient beta of the kernel function according to the basis matrix W;
4) According to the formulaUpdating the multi-core model K;
5) According to the formulaUpdating the base matrix W;
6) When the maximum iteration times are met or the convergence threshold is met, taking the current base matrix W as a decomposed base matrix, and finishing the rapid hyperspectral image pixel unmixing based on the multi-core projection NMF; otherwise, go to step 3).
1. Overall design
Constructing a multi-core model, wherein the multi-core model is as follows:
wherein k is j Is the basic kernel function, s is the total number of kernel functions, beta j Is a weight coefficient of the kernel function. Therefore, under the multi-core framework, the problem of kernel function selection of kernel mapping can be solved by utilizing the feature mapping capability of different basic kernel functions. The kernel function of multi-kernel learning has a plurality of combination modes, the kernel function is composed of different parameters of a single kernel function, and aiming at hyperspectral unmixing, a basic kernel function is set as a Gaussian kernel function, and the form is as follows:
in conjunction with the MKL method, formula (4) is optimized as:
let k j ＝Tr[(E-WW T )(E-WW T ) T K j ]，β＝[β 1 β 2 L β s ]，k＝[k 1 k 2 L k s ]The following steps are:
and solving the equation (7) by utilizing linear programming to obtain the weight coefficient beta of the kernel function.
Substituting the weight coefficient beta of the kernel function into the formula (6) to solve W to obtain
Let λ= (λ) ij )，λ ij To restrict W i,j Lagrange multiplier of 0 or more, there is
G＝Tr(K)-2Tr(WW T K)+Tr(WW T WW T K)+Tr(λW T ) (9)
Both ends of the step (9) are simultaneously led to W to obtain
From the Kuhn-Tucker condition lambda ij W ij =0, multiplying both sides of formula (10) by W ij Obtaining
2(KW) ij W ij -(WW T K+KWW T W) ij W ij ＝0 (11)
Then W is iterated from equation (12)
In the iterative process, when |phi (x) -WW T And when phi (x) is less than or equal to epsilon, finishing iteration, wherein epsilon is a set threshold value.
2. PCA-based end member number determination and VCA-based end member matrix initialization
2a) A common method for determining the number of end members includes: principal component analysis (Principal Components Analysis, PCA), pseudo-dimensionality (Virtual Dimensionality, VD), minimum error signal subspace identification (Hyspectral Signal Identification By Minimum Error, hySime), eigenvalue likelihood maximization (Eigenvalue Likelihood Maximization, ELM), and the like. The invention selects PCA method to determine the number of end members.
For hyperspectral images, the ith end member z i The variance contribution ratio of (2) is:
thus, for a given ρ ε (0, 1), there is a k >0 such that
Searching the first k end members so that the information proportion of the hyperspectral image can be more than or equal to rho, wherein k is the number of the end members.
2b) The method comprises the steps of initializing an end member matrix by adopting a vertex component analysis method (Vertex Component Analysis, VCA), and sequentially extracting the end member by repeatedly searching an orthogonal subspace and calculating a projection vector 2 norm of an image matrix in the orthogonal subspace on the basis of a linear spectrum mixed model by the VCA method.
The initial end members are:
in practice, when k end members have been extracted, k-dimensional space E corresponding to the k end members k The method comprises the following steps:
E k ＝[e 1 ,e 2 ,L,e k ] (16)
then E k Orthogonal projection matrix of (a)The method comprises the following steps:
the (k+1) th end member is:
wherein,is->A random vector in (a), i.e
Verification test
In order to verify the performance of the invention, a unmixed experiment is carried out on the basis of the invention, and experimental comparison and analysis are carried out from two aspects of unmixed precision and time performance.
The simulation data is synthesized by selecting a portion of the spectrum. A portion of the spectral synthesis simulated hyperspectral image was selected from the 7 th edition spectral library splib07a published by the united states address survey agency (United States Geological Survey, USGS). Firstly, extracting spectral characteristics of all minerals from an AVIRIS 1995 spectral library in splib07a, and after removing spectral characteristics which are severely interfered, arranging according to the spectral angular distance between any two spectrums from large to small to form an experimental spectral library. Any six spectra of the materials were then extracted from the library of experimental spectra, as shown in fig. 1. Finally, a simulated hyperspectral image is generated based on the linear spectral mixture model. The simulated hyperspectral image has a size 169×169, including 224 spectral features, satisfying the non-negative sum of abundance as a constraint, and fig. 2 is an abundance map corresponding to six spectral end members, and includes 36 square regions, each region having a size 13×13. Wherein the six squares of the first row are pure pixel areas, and each square corresponding area only comprises one spectrum end member. The second row to the sixth row mix pixel areas, mixed by equal ratios of two to six unequal numbers of end members. The background area is mixed by random proportions of the six end members. To simulate sensor noise and other possible errors, an independent co-distributed gaussian white noise was added with an SNR of 20dB.
For the unmixing accuracy, the spectral angular distance (Spectral Angel Distance, SAD) and root mean square error (Root Mean Square Error, RMSE) were used for evaluation, respectively. SAD is used for measuring the similarity between a real end member and an end member extracted after unmixing, and the smaller the SAD value is, the smaller the difference between the spectrum obtained by unmixing and the spectrum of a real ground object is. The RMSE is used for measuring the similarity between the real abundance of the ground objects and the calculated abundance after unmixing, and the smaller the RMSE value is, the closer the ground object abundance obtained by unmixing is to the real ground object abundance.
Wherein A is the real end member spectrum in the spectrum library, and A' is the end member spectrum extracted after unmixing.
Wherein X is the real abundance of the ground object and is the estimated abundance after unmixing.
For the time performance of the invention, under the condition of the same computer hardware configuration, the running time required by obtaining the expected experimental result from the same data set through the algorithm is used as the evaluation index of the algorithm. The shorter the run time, the better the time performance is reflected.
In order to ensure the robustness of the experimental result, each algorithm is calculated for 10 times, and the average value is taken for evaluation.
Fig. 3 is a comparison of NMF, MKNMF and the end member spectral curves extracted after unmixing of the simulated hyperspectral image according to the invention and their spectral curves in the database, corresponding in turn to the spectral curves of end member 1 to end member 6 by reference numerals.
Fig. 4 is a comparison of an abundance map obtained by unmixing a simulated hyperspectral image with a real abundance map obtained by an NMF algorithm, a MKNMF algorithm and the present invention, wherein line 1 corresponds to the real abundance map of the simulated hyperspectral image, and lines 2 to 4 are the NMF algorithm, the MKNMF algorithm and the abundance map obtained by the present invention, respectively.
As can be seen from fig. 4, the NMF algorithm can only extract the 2 nd, 3 rd and 5 th end members of the 6 end members effectively, while the MKNMF algorithm and the invention can extract the 6 spectra effectively, and the obtained end member spectra are closer to the real spectra, and the end member extraction effect is obviously better than that of the NMF algorithm.
In order to better evaluate the unmixed results, indexes such as SAD and RMSE obtained after unmixing of each algorithm, algorithm running time and the like are listed in a table 1, and the optimal value of each item is thickened, and as shown in the table 1, the MKNMF is similar to the SAD and RMSE obtained by the invention due to the adoption of the MKL method and the same kernel function, and the SAD value and SAD mean value of each end member obtained after unmixing are obviously smaller than those of the NMF algorithm, and the end member extraction precision is higher. The method has the advantage that the unmixed precision of the algorithm on the hyperspectral image can be effectively improved by the MKL method.
For the run time of the algorithm, although the invention employing the MKL method increased 18.79% compared to the NMF algorithm, the increase in 43.73% compared to the MKNMF algorithm was smaller in magnitude. The invention can improve the unmixing precision and simultaneously has better time performance.
TABLE 1
The true hyperspectral image data pel unmixing experiment is based on the Urban public dataset. Urban is a hyperspectral image acquired by the HYDICE sensor in 1997. The size after clipping is 256×256, and each pixel corresponds to a region of 2×2 square meters. There are 210 bands, ranging from 400 to 2500nm, with a spectral resolution of 10nm. After removing the bands (1-4, 76, 87, 101-111, 136-153, 198-210) affected by dense steam and atmospheric effects, 162 bands are reserved. The area mainly comprises six types of features of Asphalt pavement (Asphalt Road), grass, tree, roof, metal and earth (Dirt).
Fig. 5 is a comparison of the abundance maps obtained after unmixing the Urban images by NMF, MKNMF, and the present invention, and their reference abundance maps. For the abundance pattern in which the 1 st action refers to the real ground object, the 2 nd line to the 4 th line sequentially correspond to the NMF, the NMF and the abundance pattern after unmixing of the invention. Column 1 to column 6 in the figure correspond to the abundance diagrams of six end members of asphalt pavement, grassland, tree, roof, metal and soil. As can be seen from fig. 5, the NMF algorithm can only separate asphalt pavement, grasslands, trees and roofs, but has poor metal and soil separation effect. And the MKNMF can effectively separate six ground objects, and has better unmixing effect than NMF.
Table 2 shows the results of hyperspectral data experiments in Urman area, the optimal values of each item are bolded, and for the unmixing precision, SAD and RMSE of the invention are optimal except for the Metal feature, which shows that the invention has good unmixing precision. For time performance, compared with NMF algorithm, the method is increased by 15.26%, but is obviously faster than MKNMF algorithm with similar unmixing precision.
From the comprehensive comparison of the unmixing precision and the algorithm running time, the real data experiment verifies the simulation data experiment result, which shows that compared with the NMF algorithm, the invention can obviously improve the unmixing precision of the hyperspectral image, and has better time performance while guaranteeing the unmixing precision.
TABLE 2
Claims (4)
1. The quick hyperspectral image pixel unmixing method based on the multi-core projection NMF is characterized by comprising the following steps of:
1) Constructing a multi-core model, and initializing weight coefficient beta= [ beta ] of a kernel function 1 β 2 L β m ]；
2) Determining the number of end members by using a PCA method, and initializing a base matrix W by using a VCA method;
3) Using the formula g=tr (K) -2Tr (WW T K)+Tr(WW T WW T K)+Tr(λW T ) Updating the weight coefficient beta of the kernel function according to the basis matrix W;
4) According to the formulaUpdating the multi-core model K;
5) According to the formulaUpdating the base matrix W;
6) When the maximum iteration times are met or the convergence threshold is met, taking the current base matrix W as a decomposed base matrix, and finishing the rapid hyperspectral image pixel unmixing based on the multi-core projection NMF; otherwise, go to step 3).
2. The method for unmixing fast hyperspectral image pixels based on multi-core projection NMF as claimed in claim 1, wherein the multi-core model constructed in step 1) is:
wherein k is j Is the basic kernel function, s is the total number of kernel functions, beta j Weight coefficients that are kernel functions;
setting a basic kernel function as a Gaussian kernel function, wherein the basic kernel function has the following expression:
using the MKL method, formula (4) is optimized as:
3. the method for unmixing fast hyperspectral image pixels based on multi-core projection NMF according to claim 1, characterized in that λ= (λ ij )，λ ij To restrict W i,j Lagrange multiplier of 0 or more, there is
G＝Tr(K)-2Tr(WW T K)+Tr(WW T WW T K)+Tr(λW T ) (9)
Both ends of the step (9) are simultaneously led to W to obtain
From the Kuhn-Tucker condition lambda ij W ij =0, multiplying both sides of formula (10) by W ij Obtaining
2(KW) ij W ij -(WW T K+KWW T W) ij W ij ＝0 (11)
Then W is iterated from equation (12)
In the iterative process, when |phi (x) -WW T And when phi (x) is less than or equal to epsilon, finishing iteration, wherein epsilon is a set threshold value.
4. The method for rapidly unmixing hyperspectral image pixels based on multi-core projection NMF according to claim 1, wherein the specific process of determining the number of end members by PCA method in step 2) is as follows:
for hyperspectral images, the ith end member z i The variance contribution ratio of (2) is:
thus, for a given ρ ε (0, 1), there is a k >0 such that
Searching the first k end members so that the information proportion of the hyperspectral image can be more than or equal to rho, wherein k is the number of the end members.
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