292.

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from the data under analysis. The process of PCA can be divided into three steps:

1 Calculation of the variance-covariance (or correlation) matrix of multiband images (e.g. in the case of a four-band image, the covariance matrix has dimension 4×4),

2 Extraction of the eigenvalues and eigenvectors of the matrix, and

3 Transformation of the feature space coordinates using these eigenvectors.

Let M and X denote the multiband image mean and individual pixel value vectors, respectively. The first step, derivation of the covariance matrix C, is expressed as:

(2.2)

where n is the number of pixels. If the correlation matrix is used, each entry in C should further be divided by the product of the standard deviations of the features represented by the corresponding row and column. For instance, let c12 denote the entry of 1st row and 2nd column (i.e. the covariance between image band 1 and 2) in matrix C, then the corresponding correlation r12 is obtained by

(2.3)

where σ1 and σ2 are the standard deviations of image band 1 and 2, respectively. The correlations of other entries can be computed in a similar manner.

The second step, calculation of the eigenvectors of C, is achieved by solving the following equation:

(2.4)

where is the eigenvector corresponding to the eigen-value λi, k is the total number of feature space dimension, and I is the identity matrix (i.e. a matrix with diagonal entries set to 1 and off-diagonal entries set to 0). All the eigenvalues λ can be determined by solving (C–λI) =0. The new coordinate system is formed by the normalised eigenvectors of the variance-covariance (or correlation) matrix. The mapping location fi of each pixel on the ith principal component is given by:

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Classification Methods for Remotely Sensed Data
Classification Methods for Remotely Sensed Data, Second Edition
ISBN: 1420090720
EAN: 2147483647
Year: 2001
Pages: 354

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