A quick guide to wedgelets

We give a short introduction to wedgelet approximations, and describe some of the features of the implementation available at the website www.wedgelets.de. Here we only give a short account aiming to provide a first understanding of the algorithm and its features, and refer to [1, 2] for details. Wedgelet approximations Wedgelet approximations were introduced by Donoho [1], as a means to efficiently approximate piecewise constant images. Generally speaking, these approximations are obtained by partitioning the image domain adaptively into disjoint sets, followed by computing an approximation of the image on each of these sets. Optimal approximations are defined by means of a certain functional weighing approximation error against complexity of the decomposition. The optimization can be imagined as a game of puzzle: The aim is to approximate the image by putting together a number of pieces from a fixed set, possibly using a minimal number of pieces. As can be imagined, the efficient computation of such an optimal approximation is a critical issue, depending on the particular class of partitions under considerations. Donoho proposed to use wedges, and to study the associated wedgelet approximations. For the sake of notational convenience, we fix that images are understood as elements of the function space R , where I = {0, . . . , 2 − 1} × {0, . . . , 2 − 1}. Other image sizes can be treated by suitable adaptation, at the cost of a more complicated notation. The wedgelet approach can be described by a two-step procedure: 1. Decompose the image domain I into a disjoint union of wedge-shaped sets, I = ⋃ w∈P w. 2. On each set w ∈ P, approximate the image by a constant.