CN112417188A - Hyperspectral image classification method based on graph model - Google Patents

Hyperspectral image classification method based on graph model Download PDF

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CN112417188A
CN112417188A CN202011436246.5A CN202011436246A CN112417188A CN 112417188 A CN112417188 A CN 112417188A CN 202011436246 A CN202011436246 A CN 202011436246A CN 112417188 A CN112417188 A CN 112417188A
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蒋俊正
黄炟鑫
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Guilin University of Electronic Technology
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Abstract

The invention discloses a hyperspectral image classification method based on a graph model, which is characterized by comprising the following steps of: 1) modeling a graph; 2) resolving an optimization problem; 3) decomposing a Hessian matrix; 4) an approximation of newton step size; 5) carrying out iterative solution; 6) distributed solving; 7) and solving the classification result. The method has low calculation complexity under large-scale data and can finish classification with higher precision.

Description

Hyperspectral image classification method based on graph model
Technical Field
The invention relates to the technical field of hyperspectral images, graph models and machine learning, in particular to a hyperspectral image classification method based on a graph model.
Background
The hyperspectral image data is a special image, is firstly applied to the satellite remote sensing technology, and is widely applied to the fields of military affairs, medicine, geology, agriculture and the like. The common image is subjected to broadband imaging during imaging, the imaging result hardly has spectral characteristics, and the related research is extremely dependent on the spatial resolution of the image. The hyperspectral image is the result of imaging of multiple narrow bands, and compared with a common image, the hyperspectral image carries abundant spectrum information. The research on the hyperspectral data is very extensive, and classification is an important item. The hyperspectral image classification aims at marking pixel points, and the marking result can well reflect the relevant characteristics of the target to be detected. However, the hyperspectral image classification has the problems of small known sample size, high data dimensionality and the like.
The graph is a common data structure and is commonly used in wireless sensor networks, social networks and other scenarios. In addition, graphs are also commonly used for dimensionality reduction of data with higher feature dimensions.
Machine learning is a generic term for a class of task solution methods that mathematically model related tasks and then use tools such as computers to solve the modeled optimization problem to obtain results. Machine learning has been widely used in industry, traffic, IC design, etc. through decades of development.
Disclosure of Invention
The invention aims to provide a hyperspectral image classification method based on a graph model, aiming at the problems of large data size, high data dimensionality, small known sample size and the like in a hyperspectral image classification task. The method has low calculation complexity under large-scale data and can finish classification with higher precision.
The technical scheme for realizing the purpose of the invention is as follows:
a hyperspectral image classification method based on a graph model comprises the following steps:
1) graph modeling: suppose the hyperspectral image data is X ═ X1,x2,…,xm×n]∈Rm×n×dTotal of m × n pixel points, xi∈RdIs the feature vector of the ith pixel,
Figure BDA0002828363860000011
labels on pixels in the data are all from L ═ L1,l2,...,lcAnd all the types of the C are replaced by C,
Figure BDA0002828363860000012
is a tag of a known moiety in X,
Figure BDA0002828363860000013
is an unknown tag, the values are all 0, and
Figure BDA0002828363860000014
feature vector x of different pixel points of hyperspectral image dataiThe graph G is constructed as { V, E, W }, where V is a node set of the graph, corresponding to each pixel in the data and connected by an edge, E is an edge set, the edge is used to describe similarity and adjacency relation between nodes, W is a weight matrix, and an internal element represents a similarity degree corresponding to two nodes and is defined as formula (1):
Figure BDA0002828363860000021
Figure BDA0002828363860000022
is a neighbor set of the node i;
2) the optimization problem is summarized as follows: the one-hot encoding matrix Y of the label Y is defined as shown in formula (2):
Figure BDA0002828363860000023
yifor the ith element in the label Y, the classification problem of the hyperspectral image can be summarized as shown in formula (3) by Y:
Figure BDA0002828363860000024
and (3) solving the formula (3) to obtain a one-hot coding matrix F for classification, wherein in the formula (3), alpha is a weight factor, and L isn＝I-D-1/2WD-1/2Is a normalized graph Laplace matrix, I is a unit matrix, yjAnd fjThe j-th columns of Y and F respectively,
Figure BDA0002828363860000025
is fjD is a degree matrix, defined as shown in formula (4):
Figure BDA0002828363860000026
will D-1/2WD-1/2Is marked as WnThe problem in the formula (3) is composed of c optimization problems, which are independent from each other and similar in solving process, and only one of the optimization problems is analyzed, so that f is equal to fjThe method comprises the following steps:
Figure BDA0002828363860000027
in the formula (5)
Figure BDA0002828363860000028
In order to match the terms of the search,
Figure BDA0002828363860000029
is a regular term;
3) decomposition of the Hessian matrix: the problem in the formula (5) is solved by using a Newton method, and the information of the first order and the second order gradient of the problem is respectively shown in formulas (6) and (7):
Figure BDA00028283638600000210
Figure BDA00028283638600000211
thus obtaining the Hessian matrix H ═ alpha I + L of the objective functionnLet Wn＝Wnd+WnuWherein W isnd＝diag(Wn) From Ln＝I-WnThe method comprises the following steps:
A＝αI+(1+θ)(I-Wnd) (8)，
B＝Wnu+θ(I-Wnd) (9)，
wherein theta is a parameter for controlling decomposition, and the Hessian matrix can be decomposed into a combination of A and B as shown in formula (10):
H＝A-B (10)；
4) approximation of newton step size: when solving by the Newton method, the Newton step length of the k step
Figure BDA0002828363860000031
From equation (10) there is:
Figure BDA0002828363860000032
let P equal Wn-αI，Q＝(I-P)-1Q ≈ I + P by Taylor expansion, the Newton step size of equation (11) is approximated by equation (12):
Figure BDA0002828363860000033
5) and (3) iterative solution: equation (5) is solved according to equation (13) using the approximate newton step size in equation (12):
Figure BDA0002828363860000034
in equation (13), k in the upper right corner of the variable is the number of iterations, and f0＝0，f1When 1 is equal to
Figure BDA0002828363860000035
When the result is true, terminating the iteration;
6) distributed solution: the matching terms in the formula (5) are respectively expressed as
Figure BDA0002828363860000036
The first order gradient information at the ith node is shown in equation (14):
Figure BDA0002828363860000037
wijis WnThe element in the ith row and j column corresponds to the weight value between the node i and the node j on the graph, so that for the node i, the following elements exist:
6-1) with
Figure BDA0002828363860000038
The node in (1) communicates to obtain fkIn that
Figure BDA0002828363860000039
A value of (d) above;
6-2) calculation
Figure BDA00028283638600000310
Figure BDA00028283638600000311
Is ekValue at node i, AiiIs the ith element on the diagonal of A;
6-3) with
Figure BDA0002828363860000041
The node in (1) communicates to obtain e at
Figure BDA0002828363860000042
A value of (d) above;
6-4) calculation
Figure BDA0002828363860000043
Figure BDA0002828363860000044
Is ukThe value at node i;
6-5) with
Figure BDA0002828363860000045
The node in (1) communicates to obtain u is
Figure BDA0002828363860000046
A value of (d);
6-6) calculating the step size
Figure BDA0002828363860000047
Figure BDA0002828363860000048
Is the step length skThe value at node i;
6-7) calculation
Figure BDA0002828363860000049
To fkUpdating the value at the node i;
6-8) on all nodes
Figure BDA00028283638600000410
Are spliced into fk+1Judging the termination condition
Figure BDA00028283638600000411
Whether or not: if the condition is terminated
Figure BDA00028283638600000412
If it is true, fk+1Setting a corresponding column in the one-hot matrix, and starting to solve the next column of the one-hot matrix; if the condition is terminated
Figure BDA00028283638600000413
If not, f is usedk+1Performing the next iteration with the initial value;
7) solving a classification result: obtaining a one-hot coding matrix F through the step 6), and finally obtaining a classification result through a formula (15)
Figure BDA00028283638600000414
Figure BDA00028283638600000415
The technical scheme includes that a graph model is adopted to carry out dimensionality reduction on hyperspectral image data, then a learning task is subjected to problem regression, then a Hessian matrix is approximated through matrix decomposition, and the approximated Hessian matrix is used for solving through a quasi-Newton method.
The method has the advantages of low calculation complexity and high classification precision, and is more suitable for the large-scale hyperspectral image classification problem.
Drawings
FIG. 1 is a schematic diagram of iterative convergence in KSC data for implementing the method;
FIG. 2 is a schematic diagram of iterative convergence in IndianPines data by the implementation method.
Detailed Description
The invention will be further described with reference to the following drawings and examples, but is not limited thereto.
Example (b):
a hyperspectral image classification method based on a graph model comprises the following steps:
1) graph modeling: suppose the hyperspectral image data is X ═ X1,x2,...,xm×n]∈Rm×n×dTotal of m × n pixel points, xi∈RdIs the feature vector of the ith pixel,
Figure BDA00028283638600000416
labels on pixels in the data are all from L ═ L1,l2,...,lcAnd all the types of the C are replaced by C,
Figure BDA00028283638600000417
is a tag of a known moiety in X,
Figure BDA00028283638600000418
is an unknown tag, the values are all 0, and
Figure BDA00028283638600000419
feature vector x of different pixel points of hyperspectral image dataiAccording to the similar characteristics, a graph G is constructed, wherein the graph G is { V, E, W }, V is a node set of the graph, corresponding to each pixel in the data and connected by an edge, E is an edge set, and the edge is used for describing similarity and adjacent relation between nodesW is a weight matrix, and the internal elements represent the degree of similarity between two nodes, and are defined as shown in formula (1):
Figure BDA0002828363860000051
Figure BDA0002828363860000052
is a neighbor set of the node i;
2) the optimization problem is summarized as follows: the one-hot encoding matrix Y of the label Y is defined as shown in formula (2):
Figure BDA0002828363860000053
yifor the ith element in the label Y, the classification problem of the hyperspectral image can be summarized as shown in formula (3) by Y:
Figure BDA0002828363860000054
and (3) solving the formula (3) to obtain a one-hot coding matrix F for classification, wherein in the formula (3), alpha is a weight factor, and L isn＝I-D-1/2WD-1/2Is a normalized graph Laplace matrix, I is a unit matrix, yjAnd fjThe j-th columns of Y and F respectively,
Figure BDA0002828363860000055
is fjD is a degree matrix, defined as shown in formula (4):
Figure BDA0002828363860000056
will D-1/2WD-1/2Is marked as WnThe problem in the formula (3) consists of c optimization problems which are independent from each other and similar in solving process, and only one of the optimization problems is analyzed to enable the problem to be solvedf＝fjThe method comprises the following steps:
Figure BDA0002828363860000057
in the formula (5)
Figure BDA0002828363860000058
In order to match the terms of the search,
Figure BDA0002828363860000059
is a regular term;
3) decomposition of the Hessian matrix: the problem in the formula (5) is solved by using a Newton method, and the information of the first order and the second order gradient of the problem is respectively shown in formulas (6) and (7):
Figure BDA00028283638600000510
Figure BDA00028283638600000511
thus obtaining the Hessian matrix H ═ alpha I + L of the objective functionnLet Wn＝Wnd+WnuWherein W isnd＝diag(Wn) From Ln＝I-WnThe method comprises the following steps:
A＝αI+(1+θ)(I-Wnd) (8)，
B＝Wnu+θ(I-Wnd) (9)，
wherein theta is a parameter for controlling decomposition, and the Hessian matrix can be decomposed into a combination of A and B as shown in formula (10):
H＝A-B (10)；
4) approximation of newton step size: when solving by the Newton method, the Newton step length of the k step
Figure BDA0002828363860000061
From equation (10) there is:
Figure BDA0002828363860000062
let P equal Wn-αI，Q＝(I-P)-1Q ≈ I + P by Taylor expansion, the Newton step size of equation (11) is approximated by equation (12):
Figure BDA0002828363860000063
5) and (3) iterative solution: equation (5) is solved according to equation (13) using the approximate newton step size in equation (12):
Figure BDA0002828363860000064
in equation (13), k in the upper right corner of the variable is the number of iterations, and f0＝0，f1When 1 is equal to
Figure BDA0002828363860000065
When the result is true, terminating the iteration;
6) distributed solution: the matching terms in the formula (5) are respectively expressed as
Figure BDA0002828363860000066
The first order gradient information at the ith node is shown in equation (14):
Figure BDA0002828363860000067
wijis WnThe element in the ith row and j column corresponds to the weight value between the node i and the node j on the graph, so that for the node i, the following elements exist:
6-1) with
Figure BDA0002828363860000068
The node in (1) communicates to obtain fkIn that
Figure BDA0002828363860000069
A value of (d) above;
6-2) calculation
Figure BDA0002828363860000071
Figure BDA0002828363860000072
Is ekValue at node i, AiiIs the ith element on the diagonal of A;
6-3) with
Figure BDA0002828363860000073
The node in (1) communicates to obtain e at
Figure BDA0002828363860000074
A value of (d) above;
6-4) calculation
Figure BDA0002828363860000075
Figure BDA0002828363860000076
Is ukThe value at node i;
6-5) with
Figure BDA0002828363860000077
The node in (1) communicates to obtain u is
Figure BDA0002828363860000078
A value of (d);
6-6) calculating the step size
Figure BDA0002828363860000079
Figure BDA00028283638600000710
Is the step length skThe value at node i;
6-7) calculation
Figure BDA00028283638600000711
To fkAt one sectionUpdating the value at the point i;
6-8) on all nodes
Figure BDA00028283638600000712
Are spliced into fk+1Judging the termination condition
Figure BDA00028283638600000713
Whether or not: if the condition is terminated
Figure BDA00028283638600000714
If it is true, fk+1Setting a corresponding column in the one-hot matrix, and starting to solve the next column of the one-hot matrix; if the condition is terminated
Figure BDA00028283638600000715
If not, f is usedk+1Performing the next iteration with the initial value;
7) solving a classification result: obtaining a one-hot coding matrix F through the step 6), and finally obtaining a classification result through a formula (15)
Figure BDA00028283638600000716
Figure BDA00028283638600000717
Simulation example 1:
in the embodiment, KSC data is adopted for simulation, 176 frequency bands are arranged in the KSC data, 5211 pixel points are total after background removal, the pixel points can be divided into 13 classes and are a multi-class learning task, before simulation, tag values are sampled to generate data sets with different known tag numbers, for multi-item comparison, data sets with 5, 10, 15 and 20 known samples in each class are generated, table 1 shows that the classification performance comparison in the KSC data by adopting the method of the embodiment and the centralized method is performed, the OA parameters and the calculation time are included, and the simulation result of table 1 shows that in the KSC data, compared with the centralized method, the classification accuracy by adopting the method of the embodiment and the centralized method is completely consistent, the calculation speed by adopting the method of the embodiment is slightly higher than that by adopting the centralized method, as shown in figure 1, iterative convergence can be performed within limited times by adopting the method of the embodiment, it is demonstrated that the method of the present example produces better results and is less time consuming.
TABLE 1
Figure BDA00028283638600000718
Figure BDA0002828363860000081
Simulation example 2:
in the embodiment, IndanPines data is adopted for simulation, 200 frequency bands exist in the IndanPines data, 10249 pixel points are obtained after background removal, the pixel points can be divided into 16 classes and are also a multi-class learning task, as in simulation example 1, sampling produced datasets of only 5, 10, 15, 20 known samples for each class, and table 2 shows the comparison of classification performance in indianpins data using the method of the present example and the centralized method, which also includes OA parameters and the time consumption of calculation, the simulation results in table 2 show that, in Indian pines data, compared with the centralized method, the classification accuracy of the method is basically consistent with that of the centralized method, in the simulation example, the calculation speed of the method is far better than that of the method adopting a centralized method, and as shown in fig. 2, iterative convergence can be performed within a limited number of times by adopting the method of the example, which shows that the method of the example is more suitable for a hyperspectral image classification task under large-scale data.
TABLE 2
Number of samples known 5 10 15 20
OA value Using the method of this example 41.98％ 46.69％ 49.71％ 51.67％
Using this example method time consuming(s) 37.09 38.59 39.1 39.83
Using centralized approach OA values 42.02％ 46.7％ 49.72％ 51.68％
Consuming time using a centralized approach(s) 242.9 271 242.5 299.6
。

Claims (1)

1. A hyperspectral image classification method based on a graph model is characterized by comprising the following steps:
1) graph modeling: suppose the hyperspectral image data is X ═ X1,x2,…,xm×n]∈Rm×n×dTotal of m × n pixel points, xi∈RdIs the feature vector of the ith pixel,
Figure FDA0002828363850000011
labels on pixels in the data are all from L ═ L1,l2,...,lcAnd all the types of the C are replaced by C,
Figure FDA0002828363850000012
is a tag of a known moiety in X,
Figure FDA0002828363850000013
is an unknown tag, the values are all 0, and
Figure FDA0002828363850000014
feature vector x of different pixel points of hyperspectral image dataiThe graph G is constructed as { V, E, W }, where V is a node set of the graph, corresponding to each pixel in the data and connected by an edge, E is an edge set, the edge is used to describe similarity and adjacency relation between nodes, W is a weight matrix, and an internal element represents a similarity degree corresponding to two nodes and is defined as formula (1):
Figure FDA0002828363850000015
Figure FDA0002828363850000016
is a neighbor set of the node i;
2) the optimization problem is summarized as follows: the one-hot encoding matrix Y of the label Y is defined as shown in formula (2):
Figure FDA0002828363850000017
yifor the ith element in the label Y, the classification problem of the hyperspectral image can be summarized as shown in formula (3) by Y:
Figure FDA0002828363850000018
and (3) solving the formula (3) to obtain a one-hot coding matrix F for classification, wherein in the formula (3), alpha is a weight factor, and L isn＝I-D-1/2WD-1/2Is a normalized graph Laplace matrix, I is a unit matrix, yjAnd fjThe j-th columns of Y and F respectively,
Figure FDA0002828363850000019
is fjD is a degree matrix, defined as shown in formula (4):
Figure FDA00028283638500000110
will D-1/2WD-1/2Is marked as WnThe problem in the formula (3) is composed of c optimization problems, which are independent from each other and similar in solving process, and only one of the optimization problems is analyzed, so that f is equal to fjThe method comprises the following steps:
Figure FDA00028283638500000111
in the formula (5)
Figure FDA00028283638500000112
In order to match the terms of the search,
Figure FDA00028283638500000113
is a regular term;
3) decomposition of the Hessian matrix: the problem in the formula (5) is solved by using a Newton method, and the information of the first order and the second order gradient of the problem is respectively shown in formulas (6) and (7):
Figure FDA0002828363850000021
Figure FDA0002828363850000022
thus obtaining the Hessian matrix H ═ alpha I + L of the objective functionnLet Wn＝Wnd+WnuWherein W isnd＝diag(Wn) From Ln＝I-WnThe method comprises the following steps:
A＝αI+(1+θ)(I-Wnd) (8)，
B＝Wnu+θ(I-Wnd) (9)，
wherein theta is a parameter for controlling decomposition, and the Hessian matrix can be decomposed into a combination of A and B as shown in formula (10):
H＝A-B (10)；
4) approximation of newton step size: when solving by the Newton method, the Newton step length of the k step
Figure FDA0002828363850000023
From equation (10) there is:
Figure FDA0002828363850000024
let P equal Wn-αI，Q＝(I-P)-1Q ≈ I + P by Taylor expansion, the Newton step size of equation (11) is approximated by equation (12):
Figure FDA0002828363850000025
5) and (3) iterative solution: equation (5) is solved according to equation (13) using the approximate newton step size in equation (12):
Figure FDA0002828363850000026
in equation (13), k in the upper right corner of the variable is the number of iterations, and f0＝0，f1When 1 is equal to
Figure FDA0002828363850000027
When the result is true, terminating the iteration;
6) distributed solution: the matching terms in the formula (5) are respectively expressed as
Figure FDA0002828363850000028
The first order gradient information at the ith node is shown in equation (14):
Figure FDA0002828363850000029
wijis WnThe element in the ith row and j column corresponds to the weight value between the node i and the node j on the graph, so that for the node i, the following elements exist:
6-1) with
Figure FDA0002828363850000031
The node in (1) communicates to obtain fkIn that
Figure FDA0002828363850000032
A value of (d) above;
6-2) calculation
Figure FDA0002828363850000033
Figure FDA0002828363850000034
Is ekValue at node i, AiiIs the ith element on the diagonal of A;
6-3) with
Figure FDA0002828363850000035
The node in (1) communicates to obtain e at
Figure FDA0002828363850000036
A value of (d) above;
6-4) calculation
Figure FDA0002828363850000037
Figure FDA0002828363850000038
Is ukThe value at node i;
6-5) with
Figure FDA0002828363850000039
The node in (1) communicates to obtain u is
Figure FDA00028283638500000310
A value of (d);
6-6) calculating the step size
Figure FDA00028283638500000311
Figure FDA00028283638500000312
Is the step length skThe value at node i;
6-7) calculation
Figure FDA00028283638500000313
To fkUpdating the value at the node i;
6-8) will f on all nodesi k+1Are spliced into fk+1Judging the termination condition
Figure FDA00028283638500000314
Whether or not: if the condition is terminated
Figure FDA00028283638500000315
If it is true, fk+1Setting a corresponding column in the one-hot matrix, and starting to solve the next column of the one-hot matrix; if the condition is terminated
Figure FDA00028283638500000316
If not, f is usedk+1Performing the next iteration with the initial value;
7) solving a classification result: obtaining a one-hot coding matrix F through the step 6), and finally obtaining a classification result through a formula (15)
Figure FDA00028283638500000317
Figure FDA00028283638500000318
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