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CN111126452B - Feature spectrum curve expansion method and system based on principal component analysis - Google Patents

Feature spectrum curve expansion method and system based on principal component analysis Download PDF

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CN111126452B
CN111126452B CN201911219327.7A CN201911219327A CN111126452B CN 111126452 B CN111126452 B CN 111126452B CN 201911219327 A CN201911219327 A CN 201911219327A CN 111126452 B CN111126452 B CN 111126452B
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principal component
ground object
curve
spectrum
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CN111126452A (en
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刘博�
李立钢
倪伟
张玉皓
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National Space Science Center of CAS
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Abstract

The invention discloses a ground object spectrum curve expansion method and a system based on principal component analysis, wherein the method comprises the following steps: representing a sample matrix by a plurality of spectrum curves of a ground object type, and calculating a mean vector and a covariance matrix of the sample matrix; calculating eigenvalues and eigenvectors of the covariance matrix; selecting the first p principal component eigenvalues, and forming a new eigenvector matrix by the eigenvectors corresponding to the principal component eigenvalues; generating normally distributed random samples in an uncorrelated space, and defining standard deviation of each sample set by using characteristic values of variables in a transformation space; the sample sets are converted back into a spectrum correlation space through a new eigenvector matrix to generate an extended ground object spectrum curve. The method can quickly generate any number of ground object (material) spectrum curves, and the principal component is selected by using principal component analysis, so that the calculation complexity is reduced, and the waste of calculation resources is avoided.

Description

Feature spectrum curve expansion method and system based on principal component analysis
Technical Field
The invention relates to the technical field of remote sensing image simulation, in particular to a ground object spectrum curve expansion method and system based on principal component analysis.
Background
The ground object reflection spectrum refers to a law that the reflectivity of a ground object changes with the incident wavelength, and a curve drawn according to the reflection spectrum of the ground object is called a ground object reflection spectrum curve.
In a real scene, the shape of the spectrum curve of the same kind of ground object is basically consistent, but the spectrum curve has different states (such as the healthy state and the diseased state of vegetation, and the like), and the spectrum curve of the ground object is more or less interfered by the external environment or a measuring instrument. In the simulation process, if the influence of the interferences is ignored, in the scene where the scene shows different laws (namely textures) in reality, the simulation scene does not have corresponding gray scale fluctuation, and the textures similar to the reality scene are not reflected. However, the mode of obtaining a plurality of spectrum curves through multiple measurements is low in efficiency, and the obtained detailed information is very limited. There is a need for a method that can quickly generate a large number of curves from a small number of curves.
The j.r.schott et al, the institute of robusts, usa, proposed a method for generating any number of curves, but this method, because the dimension of the operation data is generally high (the dimension of the hyperspectral data may even reach several hundred), requires a relatively large number of curves to be generated in the application (typically several hundred), leads to a rapid increase in the calculation amount and occupies a large amount of operation resources. It is necessary to develop a curve expansion technique that is simpler and faster to operate.
Principal component analysis (Principal Component Analysis, PCA) is a commonly used data analysis algorithm that transforms raw data into a set of linearly independent representations of each dimension by linear transformation, which is used to extract the principal feature components of the data, often for dimensionality reduction of high-dimensional data. In short, the data in the high-dimensional space is projected onto the low-dimensional space to realize the dimension reduction of the data.
Disclosure of Invention
The invention aims to overcome the technical defects, and provides a feature spectrum curve expansion method based on PCA, which considers the transition region of an image through an expansion curve set, embodies rich detail information of the image, increases the sense of reality of a scene image and provides a convenient and effective way for rapidly and realistically acquiring the image. Even in the initial stage of texture character shaping, a more realistic effect can be obtained.
In order to achieve the above purpose, the invention discloses a ground object spectrum curve expansion method based on principal component analysis, which comprises the following steps:
step 101, representing m spectral curves of a known ground object type as a sample matrix X as follows:
Figure GDA0004146994900000021
wherein n represents the number of wave bands of each spectrum curve;
step 102, mean vector of X
Figure GDA0004146994900000022
And covariance matrix Σ is:
Figure GDA0004146994900000023
Figure GDA0004146994900000024
wherein mu k Is the mean of the kth point on all spectral curves, k=1, 2, … n; sigma (sigma) i,j Is the covariance of the ith and jth spectral means of the feature class, i=1, 2, … n, j=1, 2, … n;
step 103, calculating a vector composed of eigenvalues of covariance matrix sigma
Figure GDA0004146994900000025
And a matrix of feature vectors>
Figure GDA0004146994900000026
Figure GDA0004146994900000027
Wherein lambda is 1 ≥λ 2 ≥λ 3 ...λ n 0, and for this type of terrain lambda i Is that
Figure GDA0004146994900000028
The ith column feature vector in (a)
Figure GDA0004146994900000029
Is a characteristic value of (2);
104, selecting the characteristic value composition of the p main components
Figure GDA00041469949000000210
Figure GDA00041469949000000211
Figure GDA0004146994900000031
Step 105, generating a group of Gaussian-distributed random numbers y k The distribution of which satisfies N (0, lambda) k ) Wherein k=1, 2,3 …, n; constitutes the following vectors
Figure GDA0004146994900000032
Figure GDA0004146994900000033
Calculating a fitted curve
Figure GDA0004146994900000034
Figure GDA0004146994900000035
It is distributed as
Figure GDA0004146994900000036
Is +.>
Figure GDA0004146994900000037
To represent an extended spectral curve.
The invention also provides a ground feature spectrum curve expansion system based on principal component analysis, which comprises:
the mean and covariance calculation module is used for representing a sample matrix by a plurality of spectrum curves of a ground object type and calculating a mean vector and a covariance matrix of the sample matrix;
the eigenvalue and eigenvector calculation module is used for calculating eigenvalues and eigenvectors of the covariance matrix;
the new feature vector matrix construction module is used for selecting the first p principal component feature values and constructing a new feature vector matrix from the feature vectors corresponding to the principal component feature values;
a sample set generating module, configured to generate normal distributed random samples in an uncorrelated space, and define standard deviations of each sample set using feature values of variables in a transformation space;
and the extended ground object spectrum curve generation module is used for converting the sample sets back to a spectrum related space through a new eigenvector matrix to generate an extended ground object spectrum curve.
The invention has the advantages that:
1. the method can quickly generate any number of ground object (material) spectrum curves, and PCA is used for selecting main components, so that the calculation complexity is reduced, and the waste of operation resources is avoided;
2. according to the method, the principal component is selected by adopting the PCA algorithm, so that the dimension of data in the operation process is reduced, and compared with the original method, the calculation complexity can be effectively reduced; the hyperspectral data reach hundreds of wave bands, and excessive computing resources can be prevented from being occupied by selecting main components;
3. the method considers the transition region of the image by expanding the curve set, reflects rich detail information of the image, increases the sense of reality of the scene image, and provides a convenient and effective way for rapidly and realistically acquiring the image; even in the initial stage of texture character shaping, a more realistic effect can be obtained.
Drawings
FIG. 1 is a flow chart of the method for expanding the spectrum curve of the ground object based on principal component analysis.
FIG. 2 is a diagram showing the principal components and accuracy of gold (gold);
FIG. 3 (a) is an original spectrum of aluminum;
FIG. 3 (b) is an expanded curve set;
FIG. 4 (a) is a raw spectral plot of titanium;
FIG. 4 (b) is an expanded curve set;
FIG. 5 (a) is a raw spectral plot of stainless steel;
FIG. 5 (b) is an expanded curve set;
FIG. 6 (a) is a raw spectral plot of molybdenum;
fig. 6 (b) is an expanded curve set.
Detailed Description
The technical scheme of the invention is described in detail below with reference to the accompanying drawings.
The method of the present invention can generate any number of spectral curves from a smaller set of curves that contain the desired multiple variable analysis for a given class of land cover. The method requires the generation of an average vector for each land cover class and a covariance matrix containing the variables for each spectral point. The zero center point is obtained by subtracting the mean vector and then converted to the spectrally uncorrelated space. The creation of a new curve involves generating normal distributed random samples in an uncorrelated space, the standard deviation of each sample set being defined using the eigenvalues of the variables in the transformation space. These sample sets (vectors) are then converted back into the spectral correlation space where they exhibit the same spectral characteristics as the base set.
As shown in fig. 1, the present invention provides a method for extending a spectrum curve of a ground object based on principal component analysis, including:
step 101, zero-averaging a sample matrix;
from the spectral curves of a set of real materials (here exemplified by the metals aluminum, titanium, stainless steel and molybdenum), it is assumed that their distribution corresponds to a normal distribution
Figure GDA0004146994900000041
(i.e. an average value of +.>
Figure GDA0004146994900000042
Covariance is a multidimensional normal distribution of Σ), where the dimension of the multidimensional normal distribution is because the data of the example is 47 bands per spectral curve.
The known data can be represented as a matrix of samples, the size of which is 3 x 47:
Figure GDA0004146994900000051
the mean vector is:
Figure GDA0004146994900000052
zero-equalizing each row of the sample matrix, namely subtracting the average value of each column to be:
Figure GDA0004146994900000053
102, calculating a covariance matrix of a sample matrix;
Figure GDA0004146994900000054
as can be seen from the formula, the matrix is a real symmetric matrix, the elements on the main diagonal represent the variances of the objects, and the rest of the elements represent the covariances between the objects.
Step 103, calculating eigenvalues and eigenvectors of the covariance matrix;
Figure GDA0004146994900000055
Figure GDA0004146994900000056
lambda here 1 ≥λ 2 ≥λ 3 ...λ 47 0, lambda for the type of ground object (material substance) i Is that
Figure GDA0004146994900000057
The feature value of the i-th column feature vector in (a).
Step 104, selecting proper amount of main components;
comprehensively considering calculation complexity and calculation accuracy, selecting proper number (first k) of principal components (eigenvalues), and respectively forming corresponding k eigenvectors as column vectors to form a new eigenvector matrix
Figure GDA0004146994900000058
Figure GDA0004146994900000061
Figure GDA0004146994900000062
Step 105, generating any number of curves according to the selected principal components.
Passing 0-centered raw data through its new eigenvector matrix
Figure GDA0004146994900000063
The transformation is as follows:
Figure GDA0004146994900000064
therein, wherein
Figure GDA0004146994900000065
Is the original spectral curve, resulting in a spectrally uncorrelated dataset +.>
Figure GDA00041469949000000614
The data distribution of these spectral independence satisfies +.>
Figure GDA0004146994900000066
Covariance matrix->
Figure GDA0004146994900000067
There are the following forms:
Figure GDA0004146994900000068
covariance matrix of the spectrum uncorrelated data
Figure GDA0004146994900000069
Is used to generate a distribution meeting->
Figure GDA00041469949000000610
Is a multi-dimensional random variable of (a). To accomplish this, a set of gaussian distributed random numbers y are generated i The distribution satisfies N (0, lambda) i ) (where i=1, 2,3, …, 47), the following vectors are composed:
Figure GDA00041469949000000611
the fitted curve can be back calculated according to equation (1) above:
Figure GDA00041469949000000612
obtain a distribution of
Figure GDA00041469949000000613
Which may be used to represent a spectral curve.
Aiming at the reflectivity curve of the metal gold, the accuracy of the main component is set to be more than 90%, and the calculation complexity of the equation (2) is analyzed: as shown in fig. 2, for gold (gold) reflectivity, there are only two principal components extracted from the covariance matrix of its sample matrix, but the accuracy of these two principal components has exceeded 90%, and the computational complexity before and after using the PCA algorithm is shown in table 1. In addition, in practical application, it is common to generate hundreds or even thousands of spectrum curves at a time, so compared with the original method, the method can effectively reduce the computational complexity. And hyperspectral data reach hundreds of wave bands, excessive computing resources can be occupied when the main component is not selected, and the computing resources are wasted.
Table 1: dimension comparison using operation matrix before and after PCA
Figure GDA0004146994900000071
In order to verify the method of the invention, a plurality of materials of aluminum, titanium, stainless steel and molybdenum are selected as experimental objects, and each object has 3 known spectrum curves, and each spectrum curve has 47 wave bands. As is clear from the related materials, in practical applications, it is not uncommon to generate several hundred spectral curves, so taking the principal component accuracy of 90% as an example, 1000 curves are generated, and the effects achieved are shown in fig. 3 (a), 3 (b), 4 (a), 4 (b), 5 (a), 5 (b), 6 (a) and 6 (b).
Finally, it should be noted that the above embodiments are only for illustrating the technical solution of the present invention and are not limiting. Although the present invention has been described in detail with reference to the embodiments, it should be understood by those skilled in the art that modifications and equivalents may be made thereto without departing from the spirit and scope of the present invention, which is intended to be covered by the appended claims.

Claims (2)

1. The method for expanding the spectrum curve of the ground object based on the principal component analysis is characterized by comprising the following steps:
step 101, representing m spectral curves of a known ground object type as a sample matrix X as follows:
Figure FDA0004156698010000011
wherein n represents the number of wave bands of each spectrum curve;
step 102, mean vector of X
Figure FDA0004156698010000012
And covariance matrix Σ is:
Figure FDA0004156698010000013
Figure FDA0004156698010000014
wherein mu k Is the mean of the kth point on all spectral curves, k=1, 2, … n; sigma (sigma) i,j Is the covariance of the ith and jth spectral means of the feature class, i=1, 2, … n, j=1, 2, … n;
step 103, calculating a vector composed of eigenvalues of covariance matrix sigma
Figure FDA0004156698010000015
And a matrix of feature vectors>
Figure FDA0004156698010000016
Figure FDA0004156698010000017
Figure FDA0004156698010000018
Wherein lambda is 1 ≥λ 2 ≥λ 3 …λ n 0, and for this type of terrain lambda i Is that
Figure FDA0004156698010000019
The ith column feature vector e in (a) i t ＝[e i,1 ,e i,2 ,…e i,n ]Is a characteristic value of (2);
104, selecting the characteristic value composition of the p main components
Figure FDA00041566980100000110
Figure FDA00041566980100000111
Forming a new eigenvector matrix
Figure FDA0004156698010000021
Figure FDA0004156698010000022
Step 105, generating a group of Gaussian-distributed random numbers y k The distribution of which satisfies N (0, lambda) k ) Wherein k=1, 2,3 …, n; constitutes the following vectors
Figure FDA0004156698010000023
Figure FDA0004156698010000024
Calculating a fitted curve
Figure FDA0004156698010000025
Figure FDA0004156698010000026
It is distributed as
Figure FDA0004156698010000027
Is +.>
Figure FDA0004156698010000028
To represent an extended spectral curve.
2. A ground object spectrum curve expansion system based on principal component analysis, the system comprising:
the mean and covariance calculation module is used for representing a sample matrix by a plurality of spectrum curves of a ground object type, and calculating a mean vector and a covariance matrix of the sample matrix, and specifically comprises the following steps:
the m spectral curves for a known surface feature type are represented as a sample matrix X as follows:
Figure FDA0004156698010000029
wherein n represents the number of wave bands of each spectrum curve;
the mean vector of the sample matrix X
Figure FDA00041566980100000210
And covariance matrix Σ is:
Figure FDA00041566980100000211
Figure FDA0004156698010000031
wherein mu k Is the mean of the kth point on all spectral curves, k=1, 2, … n; sigma (sigma) i,j Is the covariance of the ith and jth spectral means of the feature class, i=1, 2, … n, j=1, 2, … n;
a eigenvalue and eigenvector calculation module for calculating eigenvalues and eigenvectors of a covariance matrix, wherein the eigenvalues of the covariance matrix Σ form vectors
Figure FDA0004156698010000032
And a matrix of feature vectors>
Figure FDA00041566980100000310
The method comprises the following steps: />
Figure FDA0004156698010000033
Figure FDA0004156698010000034
Wherein lambda is 1 ≥λ 2 ≥λ 3 ...λ n 0, and for this type of terrain lambda i Is that
Figure FDA0004156698010000035
The ith column feature vector e in (a) i t ＝[e i,1 ,e i,2 ,…e i,n ]Is a characteristic value of (2);
the new feature vector matrix construction module is used for selecting the first p principal component feature values and constructing a new feature vector matrix by the feature vectors corresponding to the principal component feature values, and specifically comprises the following steps:
selecting the characteristic value composition of the first p principal components
Figure FDA0004156698010000036
Figure FDA0004156698010000037
Forming a new eigenvector matrix
Figure FDA0004156698010000038
Figure FDA0004156698010000039
The sample set generating module is configured to generate normal distributed random samples in an uncorrelated space, and define standard deviations of each sample set by using eigenvalues of variables in a transformation space, and specifically includes:
generating a set of gaussian distributed random numbers y k The distribution of which satisfies N (0, lambda) k ) Wherein k=1, 2,3 …, n; constitutes the following vectors
Figure FDA0004156698010000041
Figure FDA0004156698010000042
The extended ground object spectrum curve generating module is used for converting the sample sets back to a spectrum related space through a new eigenvector matrix to generate an extended ground object spectrum curve, and specifically comprises the following steps:
calculating a fitted curve
Figure FDA0004156698010000043
Figure FDA0004156698010000044
It is distributed as
Figure FDA0004156698010000045
Is +.>
Figure FDA0004156698010000046
To represent an extended spectral curve. />
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