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The classic example is the Back Propagation (BP) algorithm (15,16). The BP algorithm minimizes the output error over all classes for a given set of training data. It achieves this by measuring the output error and adjusting the ANN’s link weights progressively backward through each layer to reduce the error. , the algorithm cannot be proven to result in a global error minimum)]. Other convergence algorithms, such as Radial Basis Functions, have been used and are faster than BP. One parameter that must be set for ANNs is the number of hidden layer nodes.

Separability Metrics for Classification (6,12) Metric Formula* L1 ϭ ͉Ȑi Ϫ Ȑj͉ City block Normalized city block Euclidean NL1 ϭ n ib Divergence Transformed divergence Bhattacharyya JeffriesMatusita Normalizes for class variance jb ib jb L2 ϭ ʈȐi Ϫ Ȑjʈ ϭ [(Ȑi Ϫ Ȑj)T(Ȑi Ϫ Ȑj)]1/2 ͫ͸ n (mib Ϫ mjb) ANG ϭ acos ͫ ͬ ͩ Results in linear decision boundaries 1/2 bϭ1 Mahalanobis Results in piecewise linear decision boundaries ͸ (͉m␴ ϩϪ␴m)/2͉ bϭ1 ϭ Angular Remarks MH ϭ (Ȑi Ϫ Ȑj)T 2 ȐTiȐj ʈȐiʈ ʈȐjʈ ͚ͩ ͚ ͪ i ϩ 2 Ϫ1 j Ϫ1 ͪ Normalizes for topographic shading ͬ 1/2 (Ȑi Ϫ Ȑj) Ϫ1 Assumes normal distributions; normalizes for class covariance; zero if class means are equal D ϭ tr͓(͚i Ϫ ͚j)(͚i Ϫ ͚j )͔ Ϫ1 Ϫ1 ϩ tr ͓(͚i ϩ ͚j )(Ȑi Ϫ Ȑj)(Ȑi Ϫ Ȑj)T ͔ Zero if class means and covariances are equal; does not converge for large class separation Dt ϭ 2͓1 Ϫ eϪD/8͔ Asymptotically converges to one for large class separation B ϭ MH ϩ ln ͫ ͚͚ ͚͚ ͬ ͉( i ϩ j)/2͉ (͉ i͉͉ j͉)1/2 JM ϭ ͓2(1 Ϫ eϪB )͔1/2 Zero if class means and covariances are equal; does not converge for large class separation Asymptotically converges to one for large class separation In the formulae, mib is the mean value for class i and band b, ␴ib is the standard deviation for class i and band b, Ȑi is the mean vector for class i, ͚i is the covariance matrix for class i, and n is the number of spectral bands.

The success of this approach depends on precise calibration of the remotely sensed data and careful compensation or corrections for atmospheric, solar, and topographic effects. The other approach depends on exploiting the unique mathematical characteristics of very high dimensional data. This approach does not necessarily require corrected data. FEATURE EXTRACTION The multispectral image data provided by a remote sensing instrument can be analyzed directly. However, in some cases, 86 INFORMATION PROCESSING FOR REMOTE SENSING it may be beneficial to analyze features extracted from the original data.

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Brownian motion in magnetic fields by Kurlsnogu


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