## Second Generation Bandelets and their Application to Image and 3D Meshes Compression

### Gabriel Peyré (CMAP, Ecole Polytechnique)

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.