Contourlets

This chapter focuses on the development of a new \true" two- dimensional representation for images that can capture the intrinsic geo- metrical structure of pictorial information. Our emphasis is on the discrete framework that can lead to algorithmic implementations. We propose a double filter bank structure, named the pyramidal directional filter bank, by combining the Laplacian pyramid with a directional filter bank. The result is called the contourlet transform, which provides a flexible multiresolution, local and directional expansion for images. The contourlet trans- form can be designed to satisfy the anisotropy scaling relation for curves, and thus others a fast and structured curvelet-like decomposition sampled signals. As a result, the proposed transform provides a sparse representation for two-dimensional piecewise smooth signals that resemble images. The link between the developed filter banks and the continuous-space constructions is set up precisely in a newly defined directional multiresolution analysis. Finally, we show some numerical experiments demonstrating the potential of the new transform in several image processing tasks.