In this paper a special class of functions called smoothlets is presented. They are defined as a generalization of wedgelets and second-order wedgelets. Unlike all known geometrical methods used in adaptive image approximation, smoothlets are continuous functions. They can adapt to location, size, rotation, curvature, and smoothness of edges. The M-term approximation of smoothlets is O(M(-3)). ...

It is by now a well-established fact that the usual two-dimensional tensor product wavelet bases are not optimal for representing images consisting of different regions of smoothly varying greyvalues, separated by smooth boundaries. The chapter starts with a discussion of this phenomenon from a nonlinear approximation point of view, and then proceeds to describe approaches that have been sugges...

In modern world image coding and especially image compression plays a very important role. There are well known and recognized theories concerning this topic such as, for example, Fourier and wavelets theories. Both of these theories allow for representation of images in a sparse way. Unfortunately wavelets, hence very good in catching point discontinuities, cannot properly catch line discontin...

Many algorithms in image processing rely on the computation of sums of pixel values over a large variety of subsets of the image domain. This includes the computation of image moments for pattern recognition purposes, or adaptive smoothing and regression methods, such as wedgelets. In the first part of the paper, we present a general method which allows the fast computation of sums over a large...

The most perceptually important features in images are geometrical, the most prevalent being the smooth contours (“edges”) that separate different homogeneous regions and delineate distinct objects. Although wavelet based algorithms have enjoyed success in many areas of image processing, they have significant shortcomings in their treatment of edges. Wavelets do not parsimoniously capture even ...

We present a heuristic algorithm for the choice of the wedgelet regularization parameter for the purpose of denoising in the case where the noise variance σ is not known. Numerical experiments comparing wavelet thresholding with wedgelet denoising, and with the related schemes quadtree approximation and platelet approximation, allow to assess the respective strengths of the different approaches...

Most wavelet-based image coders fail to model the joint coherent behavior of wavelet coefficients near edges. Wedgelets offer a convenient parameterization for the edges in an image, but they have yet to yield a viable compression algorithm. In this paper, we propose an extension of the zerotree-based Space-Frequency Quantization (SFQ) algorithm by adding a wedgelet symbol to its tree-pruning o...

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 effici...