Cardinal exponential splines: part I - theory and filtering algorithms

  • Authors:

  • M. Unser;T. Blu

  • Affiliations:

  • Biomed. Imaging Group, EPFL, Lausanne, Switzerland;-

  • Venue:
  • IEEE Transactions on Signal Processing
  • Year:
  • 2005

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Abstract

Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we propose a complete and self-contained signal processing formulation of exponential splines defined on a uniform grid. We specify the corresponding B-spline basis functions and investigate their reproduction properties (Green function and exponential polynomials); we also characterize their stability (Riesz bounds). We show that the exponential B-spline framework allows an exact implementation of continuous-time signal processing operators including convolution, differential operators, and modulation, by simple processing in the discrete B-spline domain. We derive efficient filtering algorithms for multiresolution signal extrapolation and approximation, extending earlier results for polynomial splines. Finally, we present a new asymptotic error formula that predicts the magnitude and the Nth-order decay of the L2-approximation error as a function of the knot spacing T.