Digital Shearlet Transforms

Abstract

Over the past years, various representation systems which sparsely approximate functions governed by anisotropic features such as edges in images have been proposed. We exemplarily mention the systems of contourlets, curvelets, and shearlets. Alongside the theoretical development of these systems, algorithmic realizations of the associated transforms were provided. However, one of the most common shortcomings of these frameworks is the lack of providing a unified treatment of the continuum and digital world, i.e., allowing a digital theory to be a natural digitization of the continuum theory. In fact, shearlet systems are the only systems so far which satisfy this property, yet still deliver optimally sparse approximations of cartoon-like images. In this chapter, we provide an introduction to digital shearlet theory with a particular focus on a unified treatment of the continuum and digital realm. In our survey we will present the implementations of two shearlet transforms, one based on band-limited shearlets and the other based on compactly supported shearlets. We will moreover discuss various quantitative measures, which allow an objective comparison with other directional transforms and an objective tuning of parameters. The codes for both presented transforms as well as the framework for quantifying performance are provided in the Matlab toolbox ShearLab.

Image Separation Using Wavelets and Shearlets

“…(see [15,16], also [13]). The digital (non-separable) shearlet transform is then, using the shearlet filters ψ…”

“…However, firstly, this algorithmic realization allows only a limited directional selectivity due to separability and, secondly, compactly supported shearlets generated by separable functions do not form a tight frame which causes an additional computational effort to approximate the inverse of the shearlet transform by iterative methods. These problems have been resolved in [16,13] by using non-separable compactly supported generators, and we now summarize this procedure.…”

“…For the shearlet transform, we employ the digital shearlet transform implemented by 2D convolution with discretized compactly supported shearlets, which was introduced in [16], see also [13]. In the earlier work [15], an faithful digitalization of the continuum domain shearlet transform using compactly supported shearlets generated by separable functions has been developed.…”

No-reference image quality assessment with shearlet transform and deep neural networks

“…The proposed NR-IQA is based on the shearlet transform [18][19][20][21][22][23][24]. This multiscale transform is a multidimensional edition of the traditional wavelet transform [25][26][27], and is capable for addressing anisotropic and directional information at different scales.…”

Compressed Sensing MRI Using Discrete Nonseparable Shearlet Transform and FISTA

“…Here, we use for the first time the Discrete Nonseparable Shearlet Transform (DNST) [10] as a sparsifying transform in MRI. DNST is a compactly supported shearlet transform having excellent localization properties in the spatial domain (creating advantage over the band-limited shearlet transform [11], [12]) and excellent directional selectivity (improving over the contourlets and separable compactly supported shearlets [10]). The DNST uses doubly block circulant matrices for calculating the transform coefficients.…”