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Biomedical Imaging Group
Teaching
 EPFL > BIG > Teaching > Advanced Image Processing and Analysis

# Wavelets and applications

M. Unser

Doctorate School

## Goals

Understanding of the design methods and the properties of wavelet bases in continuous and discrete-time (or space). Understanding of the algorithmic aspects of wavelets and related methods. Introduction to advanced wavelet theory using the fundamental connection with splines. Knowledge of some important applications of these tools (e.g., compression and biomedical imaging).

##### Contents

In recent years, techniques developed in different fields—namely, multiscale bases in applied mathematics, subband coding in digital signal processing, multiresolution techniques in computer vision, and splines in approximation theory—have converged to form a unified theory. Wavelets provide an interesting alternative to Fourier and short-time Fourier transform methods, mainly because of self-similarity properties and the fact that good orthonormal bases do exist. The key concept here is successive approximation in a multiresolution setting: a signal can be seen as a "coarse" version plus added "details". This notion is intuitive and yet very powerful; it also leads to interesting applications. This course presents an overview of filter banks and wavelets, their construction and properties, their relation with splines as well as some generalizations. The point of view that is emphasized is expansion into orthogonal and biorthogonal bases. Applications to image/video compression, and biomedical imaging are discussed.

##### Schedule
• Introduction. Historical perspective. The Haar wavelet revisited. Application examples.
• Basic tools. z-transform. Filter bank analysis. Polyphase representation.
• Wavelet bases of l2.
• Image coding.
• Wavelet bases of L2—Part I. Shift-invariant subspaces. Multiresolution analysis.
• Wavelet bases of L2—Part II. Mallat's algorithm. Daubechies wavelets. Biothogonal wavelets.
• Fractional splines.
• Spline foundation of wavelet theory—part I. Polynomials. Vanishing moments. Approximation properties.
• Spline foundation of wavelet theory—part II. Singularities and Regularity,
• JPEG 2000 wavelets.
• Wavelet regularization and non-linear approximation—Part I: Sobolev and Besov spaces. Non-linear approximation
• Wavelet regularization and non-linear approximation—Part II: Denoising. Inverse problems
• Biomedical applications.
##### Recommended textbooks
• G. Strang, T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge, MA, 1996.
• S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999.

© 2009 EPFL •  • 30.11.2009