Sampling and interpolation for biomedical imaging

Abstract

This tutorial will explain the modern, Hilbert-space approach for the discretization (sampling) and reconstruction (interpolation) of images (in two or higher dimensions). The emphasis will be on quality and optimality, which are important considerations for biomedical applications.

The main point in the modern formulation is that the signal model need not be bandlimited. In fact, it makes much better sense computationally to consider spline or wavelet-like representations that involve much shorter (e.g. compactly supported) basis functions that are shifted replicates of a single prototype (e.g., B-spline). We will show how Shannon's standard sampling paradigm can be adapted for dealing with such representations. In essence, this boils down to modifying the classical "anti-aliasing" prefilter so that it is optimally matched to the representation space (in practice, this can be accomplished by suitable digital post-filtering). We will also discuss efficient digital-filter-based solutions for high-quality image interpolation. Another important issue will be the assessment of interpolation quality and the identification of basis functions (and interpolators) that offer the best performance for a given computational budget. These concepts will be illustrated with various applications in biomedical imaging: tomographic reconstruction, 3D slicing and re-formatting, estimation of image differentials for feature extraction, and image registration (both rigid-body and elastic).

1. Part I: Basics (2 hours)
1. Introduction: crucial role of interpolation and sampling in imaging
2. Continuous/discrete representations of signals and images
• Splines
• Riesz bases
3. Interpolation revisited
• Digital filtering algorithms
• Biomedical applications
4. Minimum error signal approximation
• Orthogonal projection algorithms
• Image pyramids
5. Imaging applications
2. Part II: Advanced topics (2 hours)
1. Sampling revisited
2. Quantitative approximation theory
3. Interpolation and sampling in the presence of noise

Material

• Copy of slides + reprints

• [1] M. Unser, "Splines: A perfect fit for signal and image processing," IEEE Signal Processing Magazine, vol. 16, no. 6, pp. 22-38, November 1999.
• [2] M. Unser, "Sampling—50 years after Shannon," Proceedings of the IEEE, vol. 88, no. 4, pp. 569-587, April 2000.

About the speaker

Michael Unser is the director of EPFL's Biomedical Imaging Group. Before joining the EPFL as a professor, he was a scientist with the National Institutes of Health, Bethesda USA, from 1985 to 1997. His main research area is biomedical image processing. He has a strong interest in sampling theories, multiresolution algorithms, wavelets, and the use of splines for image processing. He is the author of over 130 published journal papers in these areas.

Dr. Unser is the associate Editor-in-Chief of the IEEE Transactions on Medical Imaging and the Editor-in-Chief of the Wavelet Digest, the electronic newsletter of the wavelet community. He was general co- chair for the first IEEE International Symposium on Biomedical Imaging (ISBI'2002), which was held in Washington, DC, July 7-10, 2002. He currently chairs the technical committee on Bio Imaging and Signal Processing (BISP) of the IEEE-SP Society. Professor Unser is a fellow of the IEEE. He is recipient of the 1995 and 2003 Best Paper Awards and the 2000 Magazine Award from the IEEE Signal Processing Society.