Multidimensional wavelets and invariance principles for the analysis of bioimages
In this talk, we shall advocate the use of wavelets for the processing and analysis of images in biomicroscopy. We start with a short tutorial on wavelet bases, emphasizing the fact that they provide a concise multiresolution representation of signals and that they can be computed efficiently using filterbanks. We then show how they can be extended to multiple dimensions, either, by using tensor-product-basis functions, or by allowing for some level of redundancy to achieve better invariance with respect to coordinate transformations. In particular, we present a parameric family of multidimensional wavelet transforms that are translation- and rotation-invariant and perfectly reversible (tight frame property). The underlying wavelet templates are steerable—meaning that they can be spatially rotated in any direction—and tunable to some extent, which makes them ideally suited for pattern analysis and key-point detection in 2-D or 3-D. The concepts are illustrated with applications in biological imaging. These include the denoising and deconvolution of fluorescence micrographs, model-based extraction of features (detection of edges, tracing of filaments, localization of junctions), and morphological component analysis.