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Wavelet Techniques for Advanced Biomedical Image Reconstruction

M. Unser

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



Our purpose in this talk is to advocate the use of wavelets for image denoising and reconstruction. We start with a short tutorial on wavelet bases, emphasizing the fact that they provide a concise multiresolution representation of images and that they can be computed most efficiently. We then discuss a simple but remarkably effective image-denoising procedure that essentially amounts to discarding small wavelet coefficients (soft-thresholding); we show that this type of algorithm is the solution of a variational problem that promotes sparse solutions. We argue that the underlying principle of wavelet regularization is a powerful concept that can be used advantageously in a variety of inverse image-reconstruction problems. We discuss an elegant iterative solution for general linear inverse problems (Daubechies, Defrise and De Mol, 2004), and present a new multi-level extension this algorithm that yields much faster convergence. We illustrate the concept with several concrete imaging examples: the fast deconvolution of 3-D fluorescence micrographs, the reconstruction of dynamic PET data, and the reconstruction of magnetic resonance images in non-standard configuration (multi-coil system/non- cartesian k-space sampling).


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