Understanding the Microlocal Properties of the Shearlet Transform via the Radon and the Wavelet Transforms

Over the last two decades shearlet analysis has played an important role in the field of microlocal analysis. One of the main advantages of the shearlet transform is its efficiency in capturing anisotropic data structures. This effectiveness can be explained by earlier research: in (Bartolucci 2019) it is shown that there is a profound relation between the shearlet transform, the wavelet transform, and the Radon transform. This relation is established through an integral representation.

The first objective of this thesis is to examine the properties of the three transforms as described in the literature and to understand their roles in the resolution of the wavefront set within shearlet analysis. The findings of this literature review are as follows:

(i) The wavelet transform effectively captures isotropic data structures, but falls short in capturing anisotropic features. This limitation motivated the development of the shearlet system. 

(ii) The Radon transform in combination with the Fourier transform can be used to determine whether a point is a regular directed point. 

(iii) The shearlet transform is able to detect the wavefront set. This result was already shown in (Grohs 2010), and has been partially established in a theorem in (Bartolucci 2019) using the integral representation of the shearlet transform in terms of the Radon and wavelet transforms.

The result established in (Bartolucci 2019), as described in (iii), characterizes almost the entire wavefront set, but not the entire wavefront set. Therefore, the second goal of this thesis is to generalize this result. Building on these foundations, we formulate and prove a more general theorem. This new theorem represents a new contribution to the existing literature and offers a deeper theoretical understanding of the shearlet transform’s role in microlocal analysis through its connections with the Radon and wavelet transforms.