B-Spline Signal Processing: Part II—Efficient Design and Applications

M. Unser, A. Aldroubi, M. Eden

IEEE Transactions on Signal Processing, vol. 41, no. 2, pp. 834–848, February 1993.

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This paper describes a class of recursive filtering algorithms for the efficient implementation of B-spline interpolation and approximation techniques. In terms of simplicity of realization and reduction of computational complexity, these algorithms compare favorably with conventional matrix approaches. A filtering interpretation (lowpass filter followed by an exact polynomial spline interpolator) of smoothing spline and least squares approximation methods is proposed. These techniques are applied to the design of digital filters for cubic spline signal processing. An efficient implementation of a smoothing spline edge detector is proposed. It is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next. In terms of common measures of fidelity (e.g., visual quality, SNR), this data structure appears to be superior to the widely used Gaussian/Laplacian pyramid.

Errata

• Equation (1.5), first line: g(-k) = g(k + 1) should be replaced by g(-k) = g(k + 2).

• Text below Equation (1.5), bottom of the left column of page 835: "Such a signal is also embedded in an infinite sequence of period 2 K - 1" should be replaced by "Such a signal is also embedded in an infinite sequence of period 2 K - 2".

• Unnumbered equation below Equation (1.5), bottom of the left column of page 835: g_p(k) = g(k mod (2 K - 1)) should be replaced by g_p(k) = g(k mod (2 K - 2)).

• Equation (2.3), second line: the factor (z_i ⁄ (1 - z_i²)) should be replaced by (-z_i ⁄ (1 - z_i²)).

• Last paragraph of Section II-D, bottom of the right column of page 836: "The special case of smoothing cubic splines, which is still tractable analytically, is considered in Section IV-C" should be replaced by "The special case of smoothing cubic splines, which is still tractable analytically, is considered in Section IV-B".

• We thank Prof. Howard Weinert for kindly pointing out that the range of validity of equations (4.5)-(4.8) is λ > 1⁄24. For λ ∈ [0, 1⁄24], the roots are real and negative and the impulse response must be adapted accordingly.

Please consult also the companion paper by M. Unser, A. Aldroubi, M. Eden, "B-Spline Signal Processing: Part I—Theory," IEEE Transactions on Signal Processing, vol. 41, no. 2, pp. 821-833, February 1993.

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AUTHOR="Unser, M. and Aldroubi, A. and Eden, M.",
TITLE="\mbox{{B}-{S}pline} Signal Processing: {P}art {II}---{E}fficient
	Design and Applications",
JOURNAL="{IEEE} Transactions on Signal Processing",
YEAR="1993",
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DOI: 10.1109/78.193221