EPFL | Biomedical Imaging Group | Hilbert-Space Framework

A General Hilbert Space Framework for the Discretization of Continuous Signal Processing Operators

M. Unser

Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing III, San Diego CA, USA, July 9-14, 1995, vol. 2569, part I, pp. 51–61.

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 φ_i with (i = 1, 2) (continuous/discrete representation). The corresponding input and output functions spaces are V(φ₁) and V(φ₂), respectively. The principle of the method is as follows: we start by interpolating the discrete input signal with a function S₁ ∈ V(φ₁). We then apply a linear operator T to this function and compute the minimum error approximation of the result in the output space V(φ₂). 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.

@INPROCEEDINGS(http://bigwww.epfl.ch/publications/unser9507.html,
AUTHOR="Unser, M.",
TITLE="A General {H}ilbert Space Framework for the Discretization of
	Continuous Signal Processing Operators",
BOOKTITLE="Proceedings of the {SPIE} Conference on Mathematical
	Imaging: {W}avelet Applications in Signal and Image Processing
	{III}",
YEAR="1995",
editor="",
volume="2569",
series="",
pages="51--61",
address="San Diego CA, USA",
month="July 9-14,",
organization="",
publisher="",
note="Part {I}")

© 1995 SPIE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from SPIE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.