96.

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Page 184

the mean is given by σi/(N0.5), where σi is the standard deviation for band i, and N is the number of samples. The standard error for the covariance is given by Covij/((N/2)0.5) (Papoulis, 1991). Then the total error Q is expressed by (Bosdogianni et al., 1997):

(4.51)

The least-squares solution of Equation (4.51) can be obtained by Equation (4.46). Bosdogianni et al. (1997) do not use the technique of least squares. Instead, a method of exhaustive search is applied to find the solutions. Their results show that the proposed method allows the introduction of more end members into the model and offers more stable estimates.

Applications of linear spectral unmixing are numerous. Examples are Puyou-Lacassies et al. (1994), who examine the validity of the unmixing procedure for coarse spectral resolution images; Mustard (1993), who uses hyperspectral AVIRIS data to map the proportions of grass, soil and bedrock; Kalluri et al. (1997) examine mixture modelling algorithms and assess their applicability to MODIS data; Bork et al. (1999) provide a critical analysis of the spectral unmixing method, while Wessman et al. (1997) use the method to map fire and grazing patterns in tallgrass prairie. Borel and Gerstl (1994) propose the use of nonlinear spectral mixing models, as do Ray and Murray (1996). van der Meer (1999a) describes a method of iterative unmixing, in which the patterns present in the ‘error image’ (which is generated from the root mean square errors at each pixel position (Equation 4.52) are used to redefine the end member vectors, or to select end members. The procedure is repeated until the root mean square error image shows little or no systematic pattern. The root mean square error (RMSE) at any pixel position (i, j) in the image is derived from:

(4.52)

In this equation, k is the number of spectral bands or features, Rijm is the observed value of the pixel in band m at row i, column j of the image and R′ijm is the value at the same pixel position computed from the mixture equations. RMSE values for all pixel positions (i, j) can be scaled and viewed to determine whether any systematic patterns are present in the residuals, van der Meer (1999a) also discusses the use of goodness of fit

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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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