Wavelets and Advanced Biomedical Imaging

Our purpose in this talk is to advocate the use of wavelets for advanced bioimaging. 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 (softthresholding); 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, including MRI and computed tomography. We illustrate our point by presenting a novel wavelet-based deconvolution algorithm for 3-D fluorescence microscopy, as well as some preliminary results for dynamic PET reconstruction. We will also discuss wavelet techniques for the analysis of functional MRI data and optical microscopy (extended depth of field).