Hyperbolic shearlets

Wavelets with composite dilations extend the traditional wavelet approach by allowing for the construction of waveforms defined not only at various scales and locations but also according to various orthogonal transformations. The shearlets, which yield optimally sparse representations for a large class of 2D and 3D data is the most widely known example of wavelets with composite dilations. However, many other useful constructions are obtained within this framework. In this paper, we examine the hyperbolic shearlets, a variant of the shearlet construction obtained as a system of well localized waveforms defined at various scales, locations and orientations, where the directionality is controlled by orthogonal transformations producing a sort of shearing along hyperbolic curves. The effectiveness of this new representation is illustrated by applications to image denoising. Our results compare favorably against similar denoising algorithms based on wavelets, curvelets and other sophisticated multiscale representations.