Cone-adapted continuous shearlet transform and reconstruction formula

The wavelet gave the understanding of many problems in various sciences, engineering and other disciplines. The n-dimensional continuous wavelet transform is able to describe the local regularity of functions and distribution and detect the location of singularity points though it decay at fine scale, it does not provide additional information about the geometry of the set of singularities. Several constructions have been introduced, starting with the wedgelets [41] and ridgelets [11]. Among the most successful constructions proposed in the literature, the curvelets [12] and shearlets [43] achieve this additional flexibility by defining a collection of analyzing functions ranging not only over various scales and locations, like traditional wavelets, but also over various orientations and with highly anisotropic supports. Shearlets were developed by Labate et. al.,[58] in 2005 as the first directional representation system which allows a unified treatment of the continuum and digital world similar to wavelets. The shearlets provide an alternative approach to the curvelets, and exhibit some very distinctive features. Similarly to the curvelets, the shearlets are a multiscale directional system and unlike the curvelets the shearlets form an affine system. That is, they are generated by dilating and translating one single generating function, where the dilation matrix is the product of a parabolic scaling matrix and a shear matrix. The wavelet transform associated with above more general dilation groups is called shearlet transform. similarly to the theory of affine systems, the continuous shearlets are associated with the whole range of scaling,