Bandlets
joint work with Ch. Dossal, S. Mallat and G. Peyré
In this work, which initially corresponds to my doctoral thesis under the direction of S. Mallat, I have built a new image representation adapted to geometrical images: the bandlet representation. We have shown that approximation properties of this adaptive frame led to adaptive nonlinear approximation rates up to logarithmic terms for the geometric images. These results have been translated into compression results both theoretically and numerically.
Taking advantage of the approximation properties of the bandlets, we have proposed a first image denoising algorithm based on on the MDL principle (Compressing is almost estimating). By using G. Peyré’s second-generation strip bases, in collaboration with Ch. Dossal, we have used the techniques from L. Birgé and P. Massart to propose a bandlet model selection algorithm that we have been able to prove to be quasi minimax optimal for geometrical images.