Shearlet

In applied mathematical analysis, shearlets are a multiscale framework which allows efficient encoding of anisotropic features in multivariate problem classes.

Originally, shearlets were introduced in 2006[1] for the analysis and sparse approximation of functions

They are a natural extension of wavelets, to accommodate the fact that multivariate functions are typically governed by anisotropic features such as edges in images, since wavelets, as isotropic objects, are not capable of capturing such phenomena.

Shearlets are constructed by parabolic scaling, shearing, and translation applied to a few generating functions.

At fine scales, they are essentially supported within skinny and directional ridges following the parabolic scaling law, which reads length² ≈ width.

Similar to wavelets, shearlets arise from the affine group and allow a unified treatment of the continuum and digital situation leading to faithful implementations.

, they still form a frame allowing stable expansions of arbitrary functions

One of the most important properties of shearlets is their ability to provide optimally sparse approximations (in the sense of optimality in [2]) for cartoon-like functions

In imaging sciences, cartoon-like functions serve as a model for anisotropic features and are compactly supported in

singularity curve with bounded curvature.

-term shearlet approximation obtained by taking the

largest coefficients from the shearlet expansion is in fact optimal up to a log-factor:[3][4] where the constant

This approximation rate significantly improves the best

-term approximation rate of wavelets providing only

Shearlets are to date the only directional representation system that provides sparse approximation of anisotropic features while providing a unified treatment of the continuum and digital realm that allows faithful implementation.

A comprehensive presentation of the theory and applications of shearlets can be found in.

[5] The construction of continuous shearlet systems is based on parabolic scaling matrices as a mean to change the resolution, on shear matrices as a means to change the orientation, and finally on translations to change the positioning.

leaves the integer lattice invariant in case

This indeed allows a unified treatment of the continuum and digital realm, thereby guaranteeing a faithful digital implementation.

is then defined as and the corresponding continuous shearlet transform is given by the map A discrete version of shearlet systems can be directly obtained from

be a function satisfying the discrete Calderón condition, i.e., with

It can be shown that the corresponding discrete shearlet system

are compactly supported providing superior spatial localization compared to the classical shearlets, which are bandlimited.

Although a compactly supported shearlet system does not generally form a Parseval frame, any function

can be represented by the shearlet expansion due to its frame property.

One drawback of shearlets defined as above is the directional bias of shearlet elements associated with large shearing parameters.

This effect is already recognizable in the frequency tiling of classical shearlets (see Figure in Section #Examples), where the frequency support of a shearlet increasingly aligns along the

This causes serious problems when analyzing a function whose Fourier transform is concentrated around the

To deal with this problem, the frequency domain is divided into a low-frequency part and two conic regions (see Figure): The associated cone-adapted discrete shearlet system consists of three parts, each one corresponding to one of these frequency domains.

basically differ in the reversed roles of