Splines: A Unifying Framework for Image Processing
M. Unser
Plenary talk, 2005 Twelfth IEEE International Conference on Image Processing (ICIP'05), Genova, Italian Republic, September 11-14, 2005.
Our purpose is to justify the use splines in imaging applications, emphasizing their ease of use, as well as their fundamental properties. Modeling images with splines is painless: it essentially amounts to replacing the pixels by B-spline basis functions, which are piecewise polynomials with a maximum order of differentiability. The spline representation is flexible and provides the best cost/quality tradeoff among all interpolation methods: by increasing the degree, one shifts from a simple piecewise linear representation to a higher order one that gets closer and closer to being bandlimited. We will describe efficient digital filter-based algorithms for interpolating and processing images within this framework. We will also discuss the multiresolution properties of splines that make them especially attractive for multi-scale processing.
On the more fundamental front, we will show that splines are intimately linked to differentials; in fact, the B-splines are the exact mathematical translators between the discrete and continuous versions of the operator. This is probably the reason why these functions play such a fundamental role in wavelet theory. Splines may also be justified on variational or statistical grounds; in particular, they can be shown to be optimal for the representation of fractal-like signals.
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, image resizing, tomographic reconstruction and various types of image registration.
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 120 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.
Slides of the presentation (PDF, 9.6 Mb)
@INPROCEEDINGS(http://bigwww.epfl.ch/publications/unser0509.html,
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TITLE="Splines: {A} Unifying Framework for Image Processing",
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address="Genova, Italian Republic",
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