General Hilbert space framework for the discretization of continuous signal processing operators

Michael A. Unser

doi:10.1117/12.217597

1 September 1995 General Hilbert space framework for the discretization of continuous signal processing operators

Michael A. Unser

Proceedings Volume 2569, Wavelet Applications in Signal and Image Processing III; (1995) https://doi.org/10.1117/12.217597
Event: SPIE's 1995 International Symposium on Optical Science, Engineering, and Instrumentation, 1995, San Diego, CA, United States

Abstract

We present a unifying framework for the design of discrete algorithms that implement continuous signal processing operators. The underlying continuous-time signals are represented as linear combinations of the integer-shifts of a generating function p_i with (i equals 1,2) (continuous/discrete representation). The corresponding input and output functions spaces are V(p_i) and V(p₂), respectively. The principle of the method is as follows: we start by interpolating the discrete input signal with a function s₁ (epsilon) V(p₁). We then apply a linear operator T to this function and compute the minimum error approximation of the result in the output space V(p₂). The corresponding algorithm can be expressed in terms of digital filters and a matrix multiplication. In this context, we emphasize the advantages of B-splines; and show how a judicious use of these basis functions can result in fast implementations of various types of operators. We present design examples of differential operators involving very short FIR filters. We also describe an efficient procedure for the geometric affine transformation of signals. The present formulation is general enough to include most earlier continuous/discrete signal processing techniques (e.g., standard bandlimited approach, spline or wavelet-based) as special cases.

Citation Download Citation

Michael A. Unser "General Hilbert space framework for the discretization of continuous signal processing operators", Proc. SPIE 2569, Wavelet Applications in Signal and Image Processing III, (1 September 1995); https://doi.org/10.1117/12.217597

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