CN111126452A - Ground feature spectral curve expansion method and system based on principal component analysis - Google Patents

Ground feature spectral curve expansion method and system based on principal component analysis Download PDF

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CN111126452A
CN111126452A CN201911219327.7A CN201911219327A CN111126452A CN 111126452 A CN111126452 A CN 111126452A CN 201911219327 A CN201911219327 A CN 201911219327A CN 111126452 A CN111126452 A CN 111126452A
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刘博�
李立钢
倪伟
张玉皓
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Abstract

The invention discloses a ground feature spectral curve expanding method and system based on principal component analysis, wherein the method comprises the following steps: representing a plurality of spectral curves of a ground object type by a sample matrix, and calculating a mean vector and a covariance matrix of the sample matrix; calculating an eigenvalue and an eigenvector of the covariance matrix; selecting the first p principal component eigenvalues, and forming a new eigenvector matrix by corresponding eigenvectors; generating random samples of normal distribution in an uncorrelated space, and defining standard deviation of each sample set by using characteristic values of variables in a transformation space; and converting the sample sets back to the spectrum correlation space through a new eigenvector matrix to generate an expanded surface feature spectrum curve. The method can quickly generate any number of surface feature (material) spectral curves, and selects the principal component by using principal component analysis, thereby reducing the computational complexity and avoiding the waste of computational resources.

Description

Ground feature spectral 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 feature spectral curve expansion method and system based on principal component analysis.
Background
The feature reflection spectrum refers to the rule that the reflectivity of a feature changes with the incident wavelength, and a curve drawn according to the reflection spectrum of the feature is called a feature reflection spectrum curve.
In a real scene, the shapes of the spectral curves of the same ground objects are basically consistent, but the spectral curves have different states (such as the healthy state and the diseased state of vegetation) and are more or less interfered by the external environment or the measuring instrument during the process of obtaining the reflection spectral curves of the ground objects. In the simulation process, if the influence of the interference is ignored, in the scene where the scene shows different laws (namely textures) in reality, the simulation scene does not have corresponding gray level fluctuation, and the textures close to the reality scene cannot be shown. However, the method of obtaining multiple spectral curves through multiple measurements is not only inefficient, but also has very limited detail information. There is a need for a method that can quickly generate a large number of curves from a small number of curves.
Schott et al, j.r. of the rochess institute of technology, usa, have proposed a method that can generate an arbitrary number of curves, but this method has a high dimensionality of the operational data (the dimensionality of the hyperspectral data may even reach several hundred), and the number of curves that need to be generated in the application is large (often several hundred), so that the amount of computation is rapidly increased, and a large amount of operational resources are occupied. Therefore, it is necessary to develop a curve expansion technique with simpler and faster operation.
Principal Component Analysis (PCA) is a commonly used data Analysis algorithm that transforms raw data into a set of representations linearly independent of each dimension through linear transformation for extracting 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 spectral curve expansion method based on PCA, which considers the transition region of an image by expanding a 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 vividly acquiring the image. Even at the initial stage of texture character modeling, a more realistic effect can be obtained.
In order to achieve the aim, the invention discloses a ground feature spectral curve expanding method based on principal component analysis, which comprises the following steps:
representing a plurality of spectral curves of a ground object type by a sample matrix, and calculating a mean vector and a covariance matrix of the sample matrix;
calculating an eigenvalue and an eigenvector of the covariance matrix;
selecting the first p principal component eigenvalues, and forming a new eigenvector matrix by corresponding eigenvectors;
generating random samples of normal distribution in an uncorrelated space, and defining standard deviation of each sample set by using characteristic values of variables in a transformation space;
and converting the sample sets back to the spectrum correlation space through a new eigenvector matrix to generate an expanded surface feature spectrum curve.
As an improvement of the above method, the plurality of spectral curves of a ground object type represent a sample matrix, and a mean vector and a covariance matrix of the sample matrix are calculated; the method specifically comprises the following steps:
representing the m spectral curves of a known ground object type as a sample matrix X as follows:
Figure BDA0002300358900000021
wherein n represents the number of bands of each spectral curve;
mean vector of X
Figure BDA0002300358900000022
He-XieThe variance matrix Σ is:
Figure BDA0002300358900000023
Figure BDA0002300358900000024
wherein, mukIs the mean of the kth points on all spectral curves, k being 1, 2.. n; sigmai,jIs the covariance of the ith and jth spectral means of the species, i 1, 2.
As an improvement of the above method, the calculating the eigenvalue and the eigenvector of the covariance matrix specifically includes:
calculating a vector consisting of eigenvalues of a covariance matrix sigma
Figure BDA0002300358900000031
And a matrix of feature vectors
Figure BDA0002300358900000032
Figure BDA0002300358900000033
Figure BDA0002300358900000034
Wherein λ is1≥λ2≥λ3…λnIs not less than 0, and for this type of terrain, lambdaiIs that
Figure BDA0002300358900000035
The ith column of feature vectors in
Figure BDA0002300358900000036
The characteristic value of (2).
As an improvement of the above method, the selecting the first p principal component eigenvalues and forming a new eigenvector matrix by using the corresponding eigenvectors thereof specifically includes:
selecting the first p principal component characteristic value compositions
Figure BDA0002300358900000037
Figure BDA0002300358900000038
Forming a new eigenvector matrix
Figure BDA0002300358900000039
Figure BDA00023003589000000310
As an improvement of the above method, random samples of normal distribution are generated in the uncorrelated space, and the standard deviation of each sample set is defined using the eigenvalues of the variables in the transform space; the method specifically comprises the following steps:
a set of gaussian distributed random numbers y is generatedkThe distribution of which satisfies N (0, λ) respectivelyk) Wherein k is 1,2,3 …, n; form the following vector
Figure BDA00023003589000000311
Figure BDA00023003589000000312
As an improvement of the above method, the sample sets are converted back to the spectrum correlation space through a new eigenvector matrix to generate an extended surface feature spectrum curve, specifically:
calculating a fitted curve
Figure BDA00023003589000000313
Figure BDA0002300358900000041
Then its distribution is Nn
Figure BDA0002300358900000042
Since the random variables exhibit the same spectral characteristics as the basis set
Figure BDA0002300358900000043
To represent an extended spectral curve.
The invention also provides a ground feature spectral curve expanding system based on principal component analysis, which comprises:
the mean value and covariance calculation module is used for representing a plurality of spectral curves of a ground object type to a sample matrix and calculating a mean value vector and a covariance matrix of the sample matrix;
the eigenvalue and eigenvector calculation module is used for calculating the eigenvalue and eigenvector of the covariance matrix;
the new characteristic vector matrix construction module is used for selecting the first p principal component characteristic values and constructing a new characteristic vector matrix by the corresponding characteristic vectors;
a sample set generating module, configured to generate a normally distributed random sample in an uncorrelated space, and define a standard deviation of each sample set by using a characteristic value of a variable in a transformed space;
and the extended surface feature spectral curve generation module is used for converting the sample sets back to the spectral correlation space through the new eigenvector matrix to generate an extended surface feature spectral curve.
The invention has the advantages that:
1. the method can quickly generate any number of surface feature (material) spectral curves, and the PCA is used for selecting the principal component, so that the calculation complexity is reduced, and the waste of calculation resources is avoided;
2. according to the method, the principal components are selected by adopting the PCA algorithm, so that the dimensionality of data in the operation process is reduced, and the calculation complexity can be effectively reduced compared with that of the original method; the hyperspectral data reaches hundreds of wave bands, and excessive computing resources can be avoided by selecting the main components;
3. the method of the invention considers the transition region of the image by expanding the curve set, embodies the 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 vividly acquiring the image; even at the initial stage of texture character modeling, a more realistic effect can be obtained.
Drawings
FIG. 1 is a flow chart of a feature spectral curve extending method based on principal component analysis according to the present invention.
FIG. 2 is a diagram showing the principal components and accuracy of gold (gold) selection;
FIG. 3(a) is a raw spectral plot of aluminum;
FIG. 3(b) is an expanded set of curves;
FIG. 4(a) is a plot of the raw spectrum of titanium;
FIG. 4(b) is an expanded set of curves;
FIG. 5(a) is a raw spectral curve of stainless steel;
FIG. 5(b) is an expanded set of curves;
FIG. 6(a) is a raw spectral plot of molybdenum;
fig. 6(b) is a set of curves after expansion.
Detailed Description
The technical solution of the present invention will be 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 multi-variable analysis for a given land cover category. The method entails generating an average vector for each land cover category and a covariance matrix containing the variables for each spectral point. The zero center point is obtained by subtracting the mean vector and then converting it to a spectrally uncorrelated space. The creation of a new curve involves generating random samples of a normal distribution in the uncorrelated space, using the eigenvalues of the variables in the transform space to define the standard deviation for each set of samples. 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 feature spectral curve extending method based on principal component analysis, including:
step 101, sample matrix zero equalization;
according to a set of spectral curves of actual materials (here, the metals aluminum, titanium, stainless steel and molybdenum are taken as examples), the distribution is assumed to conform to a normal distribution N47
Figure BDA0002300358900000051
(i.e., an average value of
Figure BDA0002300358900000052
A multidimensional normal distribution with covariance ∑), where the dimensionality of the multidimensional normal distribution is due to the example data being 47 bands per spectral curve.
The known data can be represented as a sample matrix, in this case 3 x 47 in size:
Figure BDA0002300358900000053
the mean vector is:
Figure BDA0002300358900000054
zero-averaging each row of the sample matrix, i.e. subtracting the average of each column as:
Figure BDA0002300358900000055
102, calculating a covariance matrix of a sample matrix;
Figure BDA0002300358900000061
as shown in the formula, the matrix is a real symmetric matrix, the elements on the main diagonal of the matrix represent the variance of the objects, and the rest elements represent the covariance between the objects.
103, calculating an eigenvalue and an eigenvector of the covariance matrix;
Figure BDA0002300358900000062
Figure BDA0002300358900000063
λ here1≥λ2≥λ3…λ47Not less than 0, and for such types of ground objects (material substances), lambdaiIs that
Figure BDA0002300358900000064
The eigenvalue of the i-th column eigenvector in (a).
104, selecting a proper amount of main components;
comprehensively considering the calculation complexity and the calculation accuracy, selecting proper (the first k) principal components (eigenvalues), and forming a new eigenvector matrix by using the corresponding k eigenvectors as column vectors
Figure BDA0002300358900000065
Figure BDA0002300358900000066
Figure BDA0002300358900000067
And 105, generating any number of curves according to the selected principal components.
Passing the original data centered at 0 through its new eigenvector matrix
Figure BDA0002300358900000068
The transformation is:
Figure BDA0002300358900000069
therein
Figure BDA00023003589000000610
Is an original spectrum curve, and obtains a data set with irrelevant spectrum
Figure BDA0002300358900000071
These spectrally uncorrelated data distributions satisfy N396
Figure BDA0002300358900000072
Covariance matrix
Figure BDA0002300358900000073
The following forms:
Figure BDA0002300358900000074
covariance matrix of the spectrally uncorrelated data
Figure BDA0002300358900000075
Is used to generate a distribution satisfying N47
Figure BDA0002300358900000076
Is used to generate the multi-dimensional random variable. To accomplish this, a set of Gaussian-distributed random numbers y is generatediThe distribution satisfies N (0, λ)i) (where i ═ 1,2,3, …,47), the following vector is formed:
Figure BDA0002300358900000077
the fitted curve can be back-calculated according to equation (1) above:
Figure BDA0002300358900000078
to obtainOne distribution is N47
Figure BDA0002300358900000079
A plurality of random variables, which can be used to represent a spectral curve.
Setting the accuracy of the principal component to be more than 90% aiming at the reflectivity curve of the metal gold, and analyzing the calculation complexity of equation (2): as shown in fig. 2, for the reflectivity of gold (gold), there are only two principal components extracted from the covariance matrix of its sample matrix, but the accuracy of these two principal components is already over 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 spectral curves at a time, so that the calculation complexity can be effectively reduced by adopting the method compared with the original method. Moreover, the hyperspectral data reaches hundreds of wave bands, excessive computing resources can be occupied without selecting the main components, and the computing resources are wasted.
Table 1: dimension comparison using operation matrices before and after PCA
Figure BDA00023003589000000710
In order to verify the method of the invention, a plurality of materials of metal aluminum, titanium, stainless steel and molybdenum are selected as experimental objects, the known spectral curve of each object is 3, and each spectral curve has 47 wave bands. As is clear from the relevant materials, it is not uncommon to generate hundreds of spectral curves in practical applications, and the effect achieved by generating 1000 curves with principal component accuracy of 90% is shown in fig. 3(a), 3(b), 4(a), 4(b), 5(a), 5(b), 6(a) and 6(b), as an example.
Finally, it should be noted that the above embodiments are only used for illustrating the technical solutions of the present invention and are not limited. Although the present invention has been described in detail with reference to the embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (7)

1. A feature spectral curve extending method based on principal component analysis, the method comprising:
representing a plurality of spectral curves of a ground object type by a sample matrix, and calculating a mean vector and a covariance matrix of the sample matrix;
calculating an eigenvalue and an eigenvector of the covariance matrix;
selecting the first p principal component eigenvalues, and forming a new eigenvector matrix by corresponding eigenvectors;
generating random samples of normal distribution in an uncorrelated space, and defining standard deviation of each sample set by using characteristic values of variables in a transformation space;
and converting the sample sets back to the spectrum correlation space through a new eigenvector matrix to generate an expanded surface feature spectrum curve.
2. The principal component analysis-based feature spectral curve extending method according to claim 1, wherein the plurality of spectral curves of a feature type are represented by a sample matrix, and a mean vector and a covariance matrix of the sample matrix are calculated; the method specifically comprises the following steps:
representing the m spectral curves of a known ground object type as a sample matrix X as follows:
Figure FDA0002300358890000011
wherein n represents the number of bands of each spectral curve;
mean vector of X
Figure FDA0002300358890000012
And the covariance matrix Σ is:
Figure FDA0002300358890000013
Figure FDA0002300358890000014
wherein, mukIs the mean of the kth points on all spectral curves, k being 1, 2.. n; sigmai,jIs the covariance of the ith and jth spectral means of the species, i 1, 2.
3. The principal component analysis-based surface feature spectral curve extending method according to claim 2, wherein the eigenvalues and eigenvectors of the covariance matrix are calculated, specifically:
vector formed by calculating eigenvalues of covariance matrix sigma
Figure FDA0002300358890000021
And a matrix of feature vectors
Figure FDA0002300358890000022
Figure FDA0002300358890000023
Figure FDA0002300358890000024
Wherein λ is1≥λ2≥λ3…λnIs not less than 0, and for this type of terrain, lambdaiIs that
Figure FDA0002300358890000025
Ith column of feature vectors ei t＝[ei,1,ei,2,…ei,n]The characteristic value of (2).
4. The principal component analysis-based surface feature spectral curve extending method according to claim 3, wherein the selecting the first p principal component eigenvalues and forming a new eigenvector matrix by using their corresponding eigenvectors specifically comprises:
selecting the first p principal component characteristic value compositions
Figure FDA0002300358890000026
Figure FDA0002300358890000027
Forming a new eigenvector matrix
Figure FDA0002300358890000028
Figure FDA0002300358890000029
5. The principal component analysis-based surface feature spectral curve extending method according to claim 4, wherein random samples of normal distribution are generated in a non-correlated space, and the standard deviation of each sample set is defined using the eigenvalues of variables in a transformed space; the method specifically comprises the following steps:
a set of gaussian distributed random numbers y is generatedkThe distribution of which satisfies N (0, λ) respectivelyk) Wherein k is 1,2,3 …, n; form the following vector
Figure FDA00023003588900000210
Figure FDA0002300358890000031
6. The principal component analysis-based surface feature spectral curve expansion method according to claim 1, wherein the sample sets are converted back to the spectrum correlation space through a new eigenvector matrix to generate an expanded surface feature spectral curve, specifically:
calculating a fitted curve
Figure FDA0002300358890000032
Figure FDA0002300358890000033
Then it is distributed as
Figure FDA0002300358890000034
Since the random variables exhibit the same spectral characteristics as the basis set
Figure FDA0002300358890000035
To represent an extended spectral curve.
7. A feature spectral curve extension system based on principal component analysis, the system comprising:
the mean value and covariance calculation module is used for representing a plurality of spectral curves of a ground object type to a sample matrix and calculating a mean value vector and a covariance matrix of the sample matrix;
the eigenvalue and eigenvector calculation module is used for calculating the eigenvalue and eigenvector of the covariance matrix;
the new characteristic vector matrix construction module is used for selecting the first p principal component characteristic values and constructing a new characteristic vector matrix by the corresponding characteristic vectors;
a sample set generating module, configured to generate a normally distributed random sample in an uncorrelated space, and define a standard deviation of each sample set by using a characteristic value of a variable in a transformed space;
and the extended surface feature spectral curve generation module is used for converting the sample sets back to the spectral correlation space through the new eigenvector matrix to generate an extended surface feature spectral curve.
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