EPFL | Biomedical Imaging Group

A Guided Tour of Splines for Medical Imaging

M. Unser

Plenary talk, Twelfth Annual Meeting on Medical Image Understanding and Analysis (MIUA'08), Dundee UK, Scotland, July 2-3, 2008, in press.

Please do not bookmark the In Press papers as content and presentation may differ from the published version.

In a few seconds, you should be redirected to the published version. The preprint version is still available here

Splines, which were invented by Schoenberg more than fifty years ago, constitute an elegant framework for dealing with interpolation and discretization problems. Our purpose in this talk is to motivate their use in medical imaging, emphasizing their ease of use, as well as their fundamental properties. In particular, we will describe efficient digital filtering algorithms for the interpolation and spline-based processing of signals and images. We will show that splines are intimately linked to differentials and identify B-splines as the exact mathematical translators between the discrete and continuous versions of the (scale-invariant) operator. This partly explains why these functions play such a fundamental role in wavelet theory. Splines may also be justified on variational and/or statistical grounds; e.g., they provide Wiener (i.e, MMSE) estimators for fractal processes such as fractional Brownian motion. We will illustrate spline processing with applications in biomedical imaging where its impact has been the greatest so far. Specific tasks include high-quality interpolation, snakes, and various types of image registration. There is now compelling evidence (several independent studies in medical imaging) that splines offer the best cost-performance tradeoff among available interpolation methods.

The associated software is available here.

Selected Reading List for Learning More About Splines

Spline tutorial: M. Unser, "Splines: A Perfect Fit for Signal and Image Processing," IEEE Signal Processing Magazine, vol. 16, no. 6, pp. 22-38, November 1999.
Splines and recursive filtering algorithms: 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.
Splines and sampling theory: M. Unser, "Sampling—50 Years After Shannon," Proceedings of the IEEE, vol. 88, no. 4, pp. 569-587, April 2000.
Splines and approximation theory: T. Blu, M. Unser, "Quantitative Fourier Analysis of Approximation Techniques: Part II—Wavelets," IEEE Transactions on Signal Processing, vol. 47, no. 10, pp. 2796-2806, October 1999.
Splines and linear systems theory: M. Unser, "Cardinal Exponential Splines: Part II—Think Analog, Act Digital," IEEE Transactions on Signal Processing, vol. 53, no. 4, pp. 1439-1449, April 2005.
Splines and wavelet theory: M. Unser, T. Blu, "Wavelet Theory Demystified," IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 470-483, February 2003.
Splines, operators and fractals: M. Unser, T. Blu, "Self-Similarity: Part I—Splines and Operators," IEEE Transactions on Signal Processing, vol. 55, no. 4, pp. 1352-1363, April 2007.
Splines and stochastic processes: M. Unser, T. Blu, "Generalized Smoothing Splines and the Optimal Discretization of the Wiener Filter," IEEE Transactions on Signal Processing, vol. 53, no. 6, pp. 2146-2159, June 2005.

© 2008 MIUA. 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 MIUA. 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.