Over-complete and Sparse Representations for Image Decomposition and Inpainting
May 28th , 2004
Second International Conference on Computational Harmonic Analysis - in conjunction with the 19th Annual Shanks Lecture - Invited Talk.

In this talk we present a novel method for separating images into texture and piecewise smooth parts, and show how this formulation can also lead to image inpainting. Our separation and inpainting process exploits both the variational and the sparsity mechanisms, by combining the Basis Pursuit Denoising (BPDN) algorithm and the Total-Variation (TV) regularization scheme.

The basic idea in this work is the use of two appropriate dictionaries, one for the representation of textures, and the other for the natural scene parts, assumed to be piece-wise-smooth. Both dictionaries are chosen such that they lead to sparse representations over one type of image-content (either texture or piecewise smooth). The use of the BPDN with the two augmented dictionaries leads to the desired separation, along with noise removal as a by-product. As the need to choose a proper dictionary for natural scene is very hard, a TV regularization is employed to better direct the separation process.

This concept of separation via sparse and over-complete representation of the image is shown to have a direct and natural extension to image inpainting. When some of the pixels in known locations in the image are missing, the same separation formulation can be changed to fit the problem of decomposing the image while filling in the holes. Thus, as a by-product of the separation we achieve inpainting. This approach should be compared to a recently published inpainting system by Bertalmio, Vese, Sapiro, and Osher. We will present several experimental results that validate the algorithm’s performance.

Joint work with Jean-Luc Starck from CEA, France, and David Donoho, Statistics - Stanford University.