In the literature, the optimum discrete interpolation approximation is presented. However, this approximation is the optimum for the union of the set of band-limited signals and the set of the corresponding approximation errors. In this paper, under several assumptions, we present two optimum extended discrete interpolation approximations such that the set of the corresponding approximation errors is included in the set of signals if we ignore some negligible component of error. In this paper, we assume that the decimated sampling interval T satisfies T M, where M is the number of paths of the filter bank. The maximally distinct or under sampled filter banks treated in this paper have FIR analysis filters with or without a continuous pre-filter and FIR synthesis filters with a band-limited continuous D/A filter. The first discussion is useful for designing a kind of down-converter which transforms HDTV signal with wide band-width to SDTV signal with narrow band-width, for example. In this discussion, we assume that the SDTV signal is limited in |ω|π/T and the Fourier spectrum of the HDTV signal has wider band but is approximately included in the corresponding narrow band of the SDTV signal. In the well-known scalable coding of signals, if the spectrum of a signal with higher resolution is not included approximately in the spectrum of the corresponding signal with lower resolution, the quality of the quantized output signal with lower resolution will become quite low practically. As shown in [3], however, scalable coding has received much attention recently in the fields of HDTV/SDTV compatibility. Therefore, it will be natural to consider that the spectrum of HDTV signal with higher resolution is similar to and is included approximately in the corresponding spectrum of SDTV signal with lower resolution. The analysis filters are FIR filters with a continuous pre-filter approximately band-limited in |ω|π/T. To improve the quality of the SDTV signal, the whole spectrum component of the HDTV signal is used in the presented down-converter. Another discussion is a general theory of approximation for filter banks using the prescribed analysis filters. In this discussion, although some modification for the band-width is introduced in the process of analysis, the final band-width of the receiver is limited in |ω| π. The FIR analysis filters do not have pre-filter. The condition imposing on the set of signals is more general than the corresponding condition in the first optimum approximation theory. Finally, we present the optimum transmultiplexer TR. In general, under the condition that the receiver filters are prescribed, a transmultiplexer has approximation error between the original signal and the transferred signal. However, the presented TR minimizes approximately the supreme value of arbitrary positive measures of approximation error that can be defined, totally or separately, with respect to all the channels. Note that, in the presented discussion, we can prescribe the degree of FIR filters used in TR, strictly.

In the literature [9], the optimum discrete interpolation of one-dimensional signals is presented which minimizes various measures of approximation error simultaneously. In the discussion, the ratio λ of the weighted norm of the approximation error and that of the corresponding input signal plays an essential role to determine the structure of the set of signals. However, only the upper bound of λ is provided in [9]. In this paper, we will present more exact and systematic discussion of the optimum discrete interpolation of one-dimensional signals which minimizes various measures of approximation error at the same time. In this discussion, we will prove that the exact value of λ is identical with the upper limit, for ω (|ω| π), of the largest eigen value of a matrix including the weighting function W(ω) and the Fourier transforms of the optimum interpolation functions. Further, we will give a sufficient condition for W(ω) under which the ratio λ is equal to one, where the approximation error, if it is interpolated by sinc, is included in the set of band-limited signals defined by W(ω). Finally, as application of the presented approximation, we will propose a direction to interactive signal processing on Internet and a transmultiplexer system included in it. The transmultiplexer system included in this discussion can realize flexible arrangement of sub-bands which is inevitable in realizing the above proposal on interactive signal processing.

In the main part of this paper, we present a systematic discussion for the optimum interpolation approximation in a shift-invariant wavelet and/or scaling subspace. In this paper, we suppose that signals are expressed as linear combinations of a large number of base functions having unknown coefficients. Under this assumption, we consider a problem of approximating these linear combinations of higher degree by using a smaller number of sample values. Hence, error of approximation happens in most cases. The presented approximation minimizes various worst-case measures of approximation error at the same time among all the linear and the nonlinear approximations under the same conditions. The presented approximation is quite flexible in choosing the sampling interval. The presented approximation uses a finite number of sample values and satisfies two conditions for the optimum approximation presented in this paper. The optimum approximation presented in this paper uses sample values of signal directly. Hence, the presented result is independent from the so-called initial problem in wavelet theory.

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.

Extended interpolatory approximations are discussed for some classes of n-dimensional stochastic signals expressed as the orthogonal expansions with respect to a given set of orthonormal functions. We assume that the norm of the weighted mutual correlation function of the signal is smaller than a given positive number. The presented approximation has the minimum measure of approximation error among all the linear and nonlinear statistical approximations using the similar measure of error and the same generalized moments of these signals.

In the literatures [5] and [10], a systematic discussion is presented with respect to the optimum interpolation of multi-dimensional signals. However, the measures of error in these literatures are defined only in each limited block separately. Further, in these literatures, most of the discussion is limited to theoretical treatment and, for example, realization of higher order linear phase FIR filter bank is not considered. In this paper, we will present the optimum interpolation functions minimizing various measures of approximation error simultaneously. Firstly, we outline necessary formulation for the time-limited interpolation functions ψm(t) (m=0,1,. . . ,M-1) realizing the optimum approximation in each limited block separately, where m are the index numbers for analysis filters. Secondly, under some assumptions, we will present analytic or piece-wise analytic interpolation functions φm(t) minimizing various measures of approximation error defined at discrete time samples n=0, 1, 2,. . . . In this discussion, φm(n) are equal to ψm(n) n=0, 1, 2,. . . . Since ψm(t) are time-limited, φm(n) vanish outside of finite set of n. Hence, in designing discrete filter bank, one can use FIR filters if one wants to realize discrete synthesis filters which impulse responses are φm(n). Finally, we will present one-dimensional linear phase M channel FIR filter bank with high attenuation characteristic in each stop band. In this design, we adopt the cosine-sine modulation initially, and then, use the iterative approximation based on the reciprocal property.

We present a systematic theory for the optimum sub-band interpolation using a given analysis or synthesis filter bank with the prescribed coefficient bit length. Recently, similar treatment is presented by Kida and quantization for decimated sample values is contained partly in this discussion [13]. However, in his previous treatment, measures of error are defined abstractly and no discussion for concrete functional forms of measures of error is provided. Further, in the previous discussion, quantization is neglected in the proof of the reciprocal theorem. In this paper, linear quantization for decimated sample values is included also and, under some conditions, we will present concrete functional forms of worst case measures of error or a pair of upper bound and lower limit of those measures of error in the variable domain. These measures of error are defined in Rn, although the measure of error in the literature [13] is more general but must be defined in each limited block separately. Based on a concrete expression of measure of error, we will present similar reciprocal theorem for a filter bank nevertheless the quantization for the decimated sample values is contained in the discussion. Examples are given for QMF banks and cosine-modulated FIR filter banks. It will be shown that favorable linear phase FIR filter banks are easily realized from cosine-modulated FIR filter banks by using reciprocal relation and new transformation called cosine-sine modulation in the design of filter banks.

In this paper, we consider an extended form of the optimum sub-band interpolation for a family of band limited signals. It is assumed that this family of signals is a certain subset of the set of the signals whose Fourier spectrums have weighted L2 norms smaller than a given positive number. We use the sample values of the output signals of the finite number of linear systems excited, at the same time, by a band limited signal to be restored approximately. The proposed method minimizes the measure of error which is equal to the envelope of the approximation errors in the frequency domain. It is proved that the presented method is the optimum, in a certain sense, among all the linear and the nonlinear approximations using the same sample values of the signal.

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.

A systematic theory of the optimum multi-path interpolation using parallel filter banks is presented with respect to a family of n-dimensional signals which are not necessarily band-limited. In the first phase, we present the optimum spacelimited interpolation functions minimizing simultaneously the wide variety of measures of error defined independently in each separate range in the space variable domain, such as 8 8 pixels, for example. Although the quantization of the decimated sample values in each path is contained in this discussion, the resultant interpolation functions possess the optimum property stated above. In the second phase, we will consider the optimum approximation such that no restriction is imposed on the supports of interpolation functions. The Fourier transforms of the interpolation functions can be obtained as the solutions of the finite number of linear equations. For a family of signals not being band-limited, in general, this approximation satisfies beautiful orthogonal relation and minimizes various measures of error simultaneously including many types of measures of error defined in the frequency domain. These results can be extended to the discrete signal processing. In this case, when the rate of the decimation is in the state of critical-sampling or over-sampling and the analysis filters satisfy the condition of paraunitary, the results in the first phase are classified as follows: (1) If the supports of the interpolation functions are narrow and the approximation error necessarily exists, the presented interpolation functions realize the optimum approximation in the first phase. (2) If these supports become wide, in due course, the presented approximation satisfies perfect reconstruction at the given discrete points and realizes the optimum approximation given in the first phase at the intermediate points of the initial discrete points. (3) If the supports become wider, the statements in (2) are still valid but the measure of the approximation error in the first phase at the intermediate points becomes smaller. (4) Finally, those interpolation functions approach to the results in the second phase without destroying the property of perfect reconstruction at the initial discrete points.

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.