Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is three-fold: We firstly develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implicates that shearlet theory provides a unified treatment of both the continuum and digital realm. Secondly, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying Matlab toolbox called ShearLab is provided. And, thirdly, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform.

Recently, compressed sensing techniques in combination with both wavelet and directional representation systems have been very effectively applied to the problem of image inpainting. However, a mathematical analysis of these techniques which reveals the underlying geometrical content is completely missing. In this paper, we provide the first comprehensive analysis in the continuum domain utilizing the novel concept of clustered sparsity, which besides leading to asymptotic error bounds also makes the superior behavior of directional representation systems over wavelets precise. First, we propose an abstract model for problems of data recovery and derive error bounds for two different recovery schemes, namely 1 minimization and thresholding. Second, we set up a particular microlocal model for an image governed by edges inspired by seismic data as well as a particular mask to model the missing data, namely a linear singularity masked by a horizontal strip. Applying the abstract estimate in the case of wavelets and of shearlets we prove that -provided the size of the missing part is asymptotically to the size of the analyzing functions -asymptotically precise inpainting can be obtained for this model. Finally, we show that shearlets can fill strictly larger gaps than wavelets in this model.

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Tight framelets on a smooth and compact Riemannian manifold M provide a tool of multiresolution analysis for data from geosciences, astrophysics, medical sciences, etc. This work investigates the construction, characterizations, and applications of tight framelets on such a manifold M. Characterizations of the tightness of a sequence of framelet systems for L 2 pMq in both the continuous and semi-discrete settings are provided. Tight framelets associated with framelet filter banks on M can then be easily designed and fast framelet filter bank transforms on M are shown to be realizable with nearly linear computational complexity. Explicit construction of tight framelets on the sphere S 2 as well as numerical examples are given.Keywords: tight framelets, affine system, compact Riemannian manifold, quadrature rule, filter bank, FFT, fast spherical harmonic transform, Laplace-Beltrami operator, unitary extension principle 2010 MSC: 42C15, 42C40, 42B05, 41A55, 57N99, 58C35, 94A12, 94C15, 93C55, 93C95 Introduction and motivationIn the era of information technologies, the rapid development of modern high-tech devices, for example, a super computer, PC, smart phone, wearable and VR/AR device, is driven internally by Moore's Law [55] which contributes to the exponential growth of the computational power, while externally stimulated by the tremendous need of both the public and individual parties in processing massive data from finance, economy, geology, bio-information, cosmology, medical sciences and so on. It has been noticed that Moore's Law is slowing down due to the constrains of the physical law [19] but the volume of data is dramatically increasing. Dealing with Big Data is becoming a crucial part of an individual person, party, government and country.Real-world data often inherit high-dimensionality such as data from a surveillance system, seismology, climatology. High-dimensional data are typically concentrated on a low-dimensional manifold [60,67], for instance, the sphere in remote sensing and CMB data [6], more complex surfaces in brain imaging [68], and discrete graph data from social and traffic networks [61]. Analysis and learning tools on manifolds hence play an increasingly important role in machine learning and statistics.The key to successful manifold learning lies in that data on a manifold may exhibit high complexity on one hand while they are highly sparse at a certain domain via an appropriate multiscale representation system on the other hand. Sparsity within such representations, stemming from computational harmonic analysis, enables efficient analysis and processing of high-dimensional and massive data.Multiresolution analysis in general are designed for data in the Euclidean space R d , d ě 1, for example, a signal in R, an image in R 2 and a video in R 3 . Multiscale representation systems in R d including wavelets, framelets, curvelets, shearlets, etc., which are capable of capturing the sparsity of data, have been well-developed and widely used, see e.g. [7,11,14,17,21,49,50]. The core of the...

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.