297.

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data set is converted to two difference images, one being shifted by one pixel horizontally and the other being shifted by the same amount vertically. The variance-covariance matrices of the two difference images are calculated and pooled to give CΔ. Finally, the eigenvalues and eigenvectors of are obtained, and MAP images derived using procedures equivalent to those described above for the calculation of principal component images (Equation 2.5).

2.1.4 Maximum noise fraction transformation

This procedure, which is sometimes known as noise-adjusted principal components, requires estimates of the variance-covariance matrices of the signal, CS, and noise, CN, components and it uses the signal to noise ratio to determine the ordering of the MNF components. Thus, the MNFs produce components that successively maximise the signal to noise ratio, just as PCA generates components that successively maximise the variance. The noise variance can be estimated in a number of ways. For example, the differences between adjacent pixels can be used, as described in Section 2.1.3. In this case, MNF and MAF produce the same eigenvectors and, consequently, the same orthogonally transformed images. Other methods of estimating CN are considered at a later stage.

The variance-covariance (dispersion) matrix of the full data set is C, and it is axiomatic that C=CS+CN. Neilsen (1994) shows that the required eigenvalues and eigenvectors are the solution of the generalised eigenproblem equation det(CN−λCS)=0. Martin and Wilkinson (1971) show how this equation is solved by reduction to standard form. The code that they supply is available from libraries such as Netlib (http://netlib2.cs.utk.edu/) in the eispack package. The computed eigenvectors are used to derive component images, as described in Section 2.1.2.

Clearly, the nub of the issue is the estimation of CN. Neilsen (1994) lists five methods, partly derived from Olsen (1993). These are:

1 Simple differencing, using the method of Section 2.1.3

2 Use of a simultaneous autoregressive (SAR) model involving the pixel of interest and its neighbours to the W, NW, N and NE

3 Determine the difference between the value of the pixel of interest and the local mean, which is computed for a rectangular window

4 As method (4) but the local median is used in place of the local mean

5 Compute the residual from a local quadratic surface based on the pixel values neighbouring the pixel of interest.

Neilsen (1994) also considers the problem, mentioned above, of periodic noise, such as banding due to differences in sensor calibrations. Banding, which is seen as horizontal striping, is obviously autocorrelated, so will be

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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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