The shearlets are an affine system with a single generating mother shearlet function parameterized by a scaling, shear, and translation parameter -- the shear parameter capturing the direction of singularities. The continuous shearlet transform precisely detects the direction of singularities, in the sense of resolving the wavefront set of distributions. This transform can even be regarded as matrix coefficients from a group representation of a special non-abelian group, the shearlet group, thereby providing an extensive mathematical framework for its theory, i.e., for studying the uncertainty principle related to the shearlet group aiming to derive mother shearlet functions which ensure optimal accuracy of the parameters of the associated transform. The associated discrete shearlet transform can be shown to be provide optimally sparse representations for 2-D functions that are smooth away from discontinuities along curves. Another benefit of this approach is that, again thanks to their mathematical structure, these systems provide a Multiresolution analysis similar to the one associated with classical wavelets, which is very useful for the development of fast algorithmic implementations.

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