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B-Spline Signal Processing: Part I—Theory

M. Unser, A. Aldroubi, M. Eden

IEEE-SPS best paper award, IEEE Transactions on Signal Processing, vol. 41, no. 2, pp. 821-833, February 1993.

This paper describes a set of efficient filtering techniques for the processing and representation of signals in terms of continuous B-spline basis functions. We first consider the problem of determining the spline coefficients for an exact signal interpolation (direct B-spline transform). The reverse operation is the signal reconstruction from its spline coefficients with an optional zooming factor m (indirect B-spline transform). We derive general expressions for the z-transforms and the equivalent continuous impulse responses of B-spline interpolators of order n. We present simple techniques for signal differentiation and filtering in the transformed domain. We then derive recursive filters that efficiently solve the problems of smoothing spline and least squares approximations. The smoothing spline technique approximates a signal with a complete set of coefficients subject to certain regularization or smoothness constraints. The least squares approach, on the other hand, uses a reduced number of B-spline coefficients with equally spaced nodes; this technique is in many ways analogous to the application of anti-aliasing lowpass filter prior to decimation in order to represent a signal correctly with a reduced number of samples.

IEEE Signal Processing Society's 1995 Best Paper Award

Erratum

  • Equation (3.21), argument of the sampled B-spline: b1n(n ⁄ 2) should be replaced by b1n([n ⁄ 2]).

Please consult also the companion paper by M. Unser, A. Aldroubi, M. Eden, "B-Spline Signal Processing: Part II—Efficient Design and Applications," IEEE Transactions on Signal Processing, vol. 41, no. 2, pp. 834-848, February 1993. 

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