The Construction of Smooth Parseval Frames of Shearlets

Abstract

Abstract. The shearlet representation has gained increasing recognition in recent years as a framework for the efficient representation of multidimensional data. This representation consists of a countable collection of functions defined at various locations, scales and orientations, where the orientations are obtained through the use of shearing matrices. While shearing matrices offer the advantage of preserving the integer lattice and being more appropriate than rotations for digital implementations, the drawback is that the action of the shearing matrices is restricted to coneshaped regions in the frequency domain. Hence, in the standard construction, a Parseval frame of shearlets is obtained by combining different systems of cone-based shearlets which are projected onto certain subspaces of L 2 (R D ) with the consequence that the elements of the shearlet system corresponding to the boundary of the cone regions lose their good spatial localization property. In this paper, we present a new construction yielding smooth Parseval frame of shearlets for L 2 (R D ). Specifically, all elements of the shearlet systems obtained from this construction are compactly supported and C ∞ in the frequency domain, hence ensuring that the system has also excellent spatial localization.

Introduction to Shearlets

“…Indeed, some basic ideas were already introduced in [41], where it was observed that there exist several ways to extend the shearing matrix to higher dimensions. A new construction yielding smooth Parseval frames of discrete shearlets in any dimensions was recently introduced in [37]. Several other results have also recently appeared, including the extension of the optimally sparse approximation results and the analysis and detection of surface singularities [10,33,34,35,36,54].…”

“…This require to slightly modify the definition of the classical shearlet. Then the boundary shearlets are obtained, essentially, by piecing together the shearlets overlapping the boundary lines ξ 1 = ±ξ 2 which have been projected onto C and C. This modified construction yields smooth Parseval frames of band-limited shearlets and can be found in [37], where also the higher dimensional versions are discussed.…”

Geometric Separation of Singularities Using Combined Multiscale Dictionaries

“…[11] for details). For brevity, in the following we will denote the Parseval frame of 3D shearlets as…”

Discrete shearlet transform on GPU with applications in anomaly detection and denoising

“…It turns out that the 3D system of shearlets is a Parseval frame of L 2 R 3 [6] and it provides essentially optimal approximations for functions of three variables which are C 2 regular away from discontinuities along C 2 surfaces [8].…”

“…Shearlets, in particular, offer a unique combination of very remarkable features: they have a simple and wellunderstood mathematical structure derived from the theory of affine systems [3,6]; they provide optimally sparse representations, in a precise sense, for a large class of images and other multidimensional data where wavelets are suboptimal [7,8]; and the directionality is controlled by shear matrices rather than rotations. This last property, in particular, enables a unified framework for continuum and discrete setting since shear transformations preserve the rectangular lattice and is an advantage in deriving faithful digital implementations [9,10].…”