The Theory of Wavelets with Composite Dilations

Directional Multiscale Processing of Images Using Wavelets with Composite Dilations

“…The wavelets with composite dilations [26] overcome the limitations of standard wavelets in dealing with the geometry of multivariate functions by including a second set of dilations. Namely, they have the form…”

“…The most notable of these representations are the curvelets [42,8], the contourlets [13] and the shearlets [26,34], which are constructed by defining systems of analyzing waveforms ranging not only at various scales and locations, like traditional wavelets, but also at various orientations, with the number of orientations increasing at finer scales.…”

“…Within the context of improved multidimensional representations, the theory of wavelets with composite dilations, originally developed in [26,27,28], is especially important. This framework is a generalization of the classical theory from which traditional wavelets are derived, and it provides a very flexible setting for the construction of many truly multidimensional variants of wavelets, such as the well known construction of shearlets 1 .…”

“…However, except for some very special cases, no satisfactory method for designing efficient numerical implementations of wavelets with composite dilations was developed so far, due to the difficulty of adapting the standard wavelet implementation algorithms to this more general setting. The goal of this paper is to introduce a 1 Even though the shearlet system is not exactly an example of wavelets with composite dilations, it is closely related to this framework being defined as a union of two such systems [26,34].…”

Optimal $N$-term approximation by linear splines over anisotropic Delaunay triangulations

“…Different geometrical methods were recently developed to design efficient image approximation schemes, where correlations along curves are essentially taken into account to locally capture the geometry of the given image data. Curvelets [6,7] and shearlets [21,22] are prominent examples for non-adaptive redundant function frames with strong anisotropic directional selectivity.…”