Relating to the problem of suppressing the immanent redundancy contained in an image with out vitiating the quality of the resultant approximation, the interpolation of multi-dimensional signal is widely discussed. The minimization of the approximation error is one of the important problems in this field. In this paper, we establish the optimum interpolatory approximation of multi-dimensional orthogonal expansions. The proposed approximation is superior, in some sense, to all the linear and the nonlinear approximations using a wide class of measures of error and the same generalized moments of these signals. Further, in the fields of information processing, we sometimes consider the orthonormal development of an image each coefficient of which represents the principal featurr of the image. The selection of the orthonormal bases becomes important in this problem. The Fisher's criterion is a powerful tool for this class of problems called declustering. In this paper, we will make some remarks to the problem of optimizing the Fisher's criterion under the condition that the quality of the approximation is maintained.

A systematic theory of the optimum sub-band interpolation using parallel wavelet filter banks presented with respect to a family of n-dimensional signals which are not necessarily band-limited. It is assumed that the Fourier spectrums of these signals have weighted L2 norms smaller than a given positive number. In this paper, we establish a theory that the presented optimum interpolation functions satisfy the generalized discrete orthogonality and minimize the wide variety of measures of error simultaneously. In the following discussion, we assume initially that the corresponding approximation formula uses the infinite number of interpolation functions having limited supports and functional forms different from each other. However, it should be noted that the resultant optimum interpolation functions can be realized as the parallel shift of the finite number of space-limited functions. Some remarks to the problem of distinction of images is presented relating to the generalized discrete orthogonality and the reciprocal property for the proposed approximation.

Extended form of interpolatory approximation is presented for tne n-dimensional (n-D) signals whose generalized spectrums have weighted norms smaller than a given positive number. The presented approximation has the minimum measure of approximation error among all the linear and the nonlinear approximations using the same generalized sample values.