The Optimum Approximation of Multi-Dimensional Signals Based on the Quantized Sample Values of Transformed Signals

Takuro KIDA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E78-A   No.2   pp.208-234
Publication Date: 1995/02/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Digital Signal Processing
interpolation approximation,  quantized sample values,  the optimum approximation,  filter banks,  

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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.