On Restoration and Approximation of Multi-Dimensional Signals Using Sample Values of Transformed Signals

Takuro KIDA

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A    No.7    pp.1095-1116
Publication Date: 1994/07/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Surveys of Researches in CAS Fields in the Last Two Decades, )
sampling theorem,  interpolation,  optimum approximation,  

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In this tutorial exposition, we present a discussion for the extended interpolation approximation with respect to a class of 1- or multi-dimensional signals. We will provide some conditions concerning to the convergence of the approximation signal to the original one. An exposition for the optimum interpolation is given with respect to a class of n-dimensional signals whose Fourier spectrums have the weighted L2 norms smaller than a given positive number. In this discussion, in the first phase, we present the outline of the approximation which minimizes the measure of error equal to the envelope of the approximation errors. Initially, it is assumed that the infinite number of interpolation functions with different functional forms are used in the approximation. However, the resultant optimum interpolation functions are expressed as the parallel shifts of the finite number of n-dimensional functions. It should be noted that the optimum interpolation functions presented in this tutorial exposition minimize wide variety of measures of error defined in each separate area in the space variable domain at the same time. Interesting reciprocal relation in the approximation, is discussed. An equivalent expression of the approximation formula in the frequency domain, is provided also. In this paper, we will also introduce the optimum approximation using space-limited analysis filters and interpolation functions with the infinite supports. This approximation satisfies beautiful orthogonal relation and minimizes various measure of error symultaneously including many types of measure of error defined in the frequency domain.