Emergence of Shearlets and Its Applications

Abstract

In recent years, serious effort has been made to design directional representation system for images such as curvelets, ridgelets and shearlets and corresponding transforms. Amongst these transforms the shearlet transforms seems quite interesting since it stems from a square integrable group representations and has the corresponding useful mathematical properties. As we know wavelets are associated with Besov spaces via atomic decompositions, shearlets correspond to certain function spaces known as shearlet co-orbit spaces. Shearlets provide an optimally sparse approximation in the class of piecewise smooth functions with C ² singularity curves namely,

$$ \left\| {f - f_{N} } \right\|_{L2}^{2} \le C_{N}^{ - 2} (\log N )^{ 3} \,{\text{as}}\,N \to \infty $$

where f _N is the non-linear shearlet approximation of a function The main objective of this review paper is to introduce basic elements of shearlet along with our own result regarding denoising of MRI images using shearlet.