Abstract
Properties of the Haar transform in image processing and pattern recognition are investigated. A lower bound of the performance of the Haar transform relative to that of the Karhunen-Loeve transform for first-order Markov processes is found. It is proved that the Haar transform is inferior to the Walsh-Hadamard transform for such processes. A unique condition is presented which, if satisfied by the elements of a matrix, will make the Karhunen-Loeve transform of the matrix and the Haar transform equivalent. Some fast algorithms are given to realize the diagonal elements of a Haar transformed matrix.
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