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Unifying Formulation of Continuous/Discrete Signal Processing

M. Unser

Series of invited lectures, Information Engineering Department, University of Siena, Italian Republic, July 23-27, 2007.



Our intent is to provide a general introduction to splines and wavelets, which are the basic mathematical tools for performing continuous/discrete signal processing.

Course Content

  • July 23: Lecture 1—Splines and interpolation

    We explain the basic properties of polynomial splines and describe their use in the context of image processing. In particular, we introduce efficient interpolation/approximation algorithms that are based on recursive digital filtering. We illustrate the concepts with concrete applications examples, including geometric transformation of images, feature detection using differentials, and registration.

  • July 24: Lecture 2—Think analog, act digital

    We introduce an extended Hilbert-space framework that provides the exact link between the traditional—discrete and analog—formulations of signal processing. In contrast to Shannon's sampling theory, the approach uses exponential B-spline functions that are compactly supported and better suited for numerical computations. The framework is ideally suited for hybrid signal processing because it can jointly represent the effect of the various (analog or digital) components of the system.

  • July 25: Lecture 3—Wavelet theory demystified

    Next, we consider the possibility of coarsening the spline grid and introduce the important concept of multiresolution analysis. This leads us quite naturally to the study of wavelet bases which provide a powerful way of decomposing signals into their elementary constituents across scale (multi-resolution decomposition). We emphasize the fundamental role of fractional splines and rely on their remarkable properties to explain a number of advanced aspects of wavelet theory (e.g., vanishing moments, regularity, approximation order).

  • July 26: Test

  • July 27: Lecture 4—Wavelets applications in bioimaging

    We end the series of lectures with a discussion of recent, promising applications of wavelets in bioimaging. These include wavelet-domain denoising, deconvolution of fluorescence micrographs, statistical analysis of functional MRI data, and data fusion for extended depth-of-field.

References

Additional Reading

  • M. Unser, M. Unser, "Wavelet Games," Wavelet Digest, vol. 11, no. 4, April 1, 2003.


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