THE MELLIN TRANSFORMATION: THEORY AND DIGITAL FILTER IMPLEMENTATION (HOMOMORPHIC SYSTEM, SAMPLING THEOREM, FOURIER TRANSFORM, SIGNAL PROCESSING)
Abstract
Scaled signal processing is an important task in human perception, where moderately scaled sounds or images are perceived as an identity. As an approach to this scaled signal processing problem, the Mellin transform is discussed. Mathematical motivation of using the Mellin transform for this problem is provided and a continuous homomorphic system, which converts scale information to translation information is identified. Problems with previous applications of the Mellin transform are analyzed and variations of the Mellin transform are studied to solve these problems. Among these variations, a set of generalized Mellin transforms are presented and the orthonormal Mellin transform is selected as one which minimizes these problems in signal processing and pattern recognition problems. A sampling theorem for the orthonormal Mellin transform is developed which provides an optimal representation of the Mellin band limited process. Based on this theorem, an optimal digital approximation of the continuous homomorphic system is derived by a least mean square filter design technique with finite dimensional function spaces. It is shown that the Mellin transform is equivalent to this digital homomorphic system followed by the Fourier transform. To reduce computation and storage requirements of applying this homomorphic system, an optimal approximation of this system is studied. To demonstrate potentials of the Mellin transform and the homomorphic system, experiments are provided with Fourier band limited signals generated statistically.
Degree
Ph.D.
Subject Area
Electrical engineering
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