BIOMEDICAL IMAGING GROUP (BIG)
Laboratoire d'imagerie biomédicale (LIB)

A Guided Tour of Splines for Medical Imaging

M. Unser

Minicourse on Mathematics of Emerging Biomedical Imaging III, Paris, French Republic, February 4-6, 2009.

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. 

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