Résumé:
Wavelets and multiresolution analysis have proven to be a powerful paradigm for image processing, and are very popular for performing image compression and denoising. Nevertheless, for a large class of images, isotropic wavelets bases are not optimal mainly because they fail to capture the directional geometric regularity present in them. The construction of stable bases that take into account the geometry of the image is very difficult. The simplest class of images that have geometric regularity is formed by functions that are regular outside a set of edge curves that are also regular. But for natural images, we need a model that incorporates the fact that the image intensity is not necessarily singular at edge locations, which makes edge detection an ill-posed problem. The Bandelet bases, proposed by Le Pennec and Mallat [Band04], have an optimal approximation rate for this more complex class of geometric images (contrarily to other methods such as finite element approximation, Curvelets, or Contourlets). In this talk we will present the second generation of Bandelets. This new coding scheme introduces for the first time a multiresolution representation of an image's geometric features. Unlike first generation Bandelets, the second generation is a fully discrete construction without any resampling or warping of the original image, which enables fast and robust denoising and compression algorithms. It also avoids segmentation and flow computation, which allows constructing orthonormal bases over the whole image. We will conclude this talk with some insight about the application of second generation Bandelets to 3D mesh compression, including how 3D geometry and classical image processing methods are converging. We will show that algorithms that use geometrically oriented orthogonal bases can overcome the shortcomings of ad-hoc schemes that encode the geometry separately at one resolution.