Walsh Orthogonal Functions in Geometrical Feature Extraction

Walsh functions are used in designinq a feature extraction algorithm. The "axis-symmetry" property of the Walsh functions is used to decompose geometrical patterns. An axissymmetry (a.s.)-histogram is obtained from the Walsh spectrum of a pattern by adding the squares of the spectrm coefficients that correspond to a given a.s.-number τ and plotting these against τ. Since Walsh transformation is not positionally invariant, the sequency spectrum does not specify the pattern uniquely. This disadvantage is overcome by performing a normalization on the input pattern through Fourier transformation. The a.s.-histogram is obtained from the Walsh spectrum coefficients of the Fourier-normalized rather than the original pattern. Such histogram contains implicit information about symmetries, periodicities, and discontinuities present in a figure. It is shown that a.s.-histograms result in great dimensionality reduction in the feature space, which leads to a computationally simpler classification task, and that patterns which differ only in translations or 90° rotation have equal a.s.-histograms.