Contourlets and Sparse Image Expansions

Recently, the contourlet transform¹ has been developed as a true two-dimensional representation that can capture the geometrical structure in pictorial information. Unlike other transforms that were initially constructed in the continuous-domain and then discretized for sampled data, the contourlet construction starts from the discrete-domain using filter banks, and then convergences to a continuous-domain expansion via a multiresolution analysis framework. In this paper we study the approximation behavior of the contourlet expansion for two-dimensional piecewise smooth functions resembling natural images. Inspired by the vanishing moment property which is the key for the good approximation behavior of wavelets, we introduce the directional vanishing moment condition for contourlets. We show that with anisotropic scaling and sufficient directional vanishing moments, contourlets essentially achieve the optimal approximation rate, O((log M)³M^-2) square error with a best M-term approximation, for 2-D piecewise smooth functions with C² contours. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications.