Shearlet Smoothness Spaces

Abstract

The shearlet representation has gained increasingly more prominence in recent years as a flexible mathematical framework which enables the efficient analysis of anisotropic phenomena by combining multiscale analysis with the ability to handle directional information. In this paper, we introduce a class of shearlet smoothness spaces which is derived from the theory of decomposition spaces recently developed by L. Borup and M. Nielsen. The introduction of these spaces is motivated by recent results in image processing showing the advantage of using smoothness spaces associated with directional multiscale representations for the design and performance analysis of improved image restoration algorithms. In particular, we examine the relationship of the shearlet smoothness spaces with respect to Besov spaces, curvelet-type decomposition spaces and shearlet coorbit spaces. With respect to the theory of shearlet coorbit space, the construction of shearlet smoothness spaces presented in this paper does not require the use of a group structure.

Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups

“…Another recent development in wavelet coorbit theory that heavily relies on the dual action is the paper [26], which embeds the theory of coorbit spaces into the context of decomposition spaces. The latter class of spaces was introduced by Feichtinger and Gröbner [16], and has recently attracted renewed attention, for example in the context of shearlet smoothness spaces [2,29]. The language of decomposition spaces provides a unified framework for the treatment of many different types of smoothness spaces, including α-modulation spaces (and thus the classes of modulation and inhomogeneous Besov spaces), but also shearlet smoothness spaces.…”

On homogeneous decomposition spaces and associated decompositions of distribution spaces

“…[7,8,20,26,37] and references therein. This is in part motivated by applications to signal and image processing of generalized wavelet systems such as -modulation frames, see [6,15,16,29,38], and Shearlet type systems, see [7,[24][25][26]. The second generation curvelet systems are generally based on modified decompositions of the frequency domain as compared to the classical dyadic decompositions.…”

Variational Image Decomposition in Shearlet Smoothness Spaces

“…However, they fail to efficiently deal with multidimensional functions and signals. This limitation is due to their poor directional sensitivity and limited capability in dealing with the anisotropic features which are frequently dominant in multidimensional applications [19]. And curvelets are neither compactly supported nor do they treat the continuum and digital realm uniformly due to the fact that they are based on rotation in contrast to shearing [12][13][14].…”

“…To overcome the limitation, researchers recently considered multiscale and directional representations that can capture the intrinsic geometrical structures such as smooth contours in natural images. In particular, shearlets offer a pyramid of well localized waveforms ranging not only across various scales and locations, but also at various orientations and with highly anisotropic shapes [8][9][10][11]19].…”

“…were given by Demetrio Labate, Lucia Mantovani and Pooran Negi [19]. The construction of these smoothness spaces can be considered as a refinement of the classical construction of Besov spaces.…”