Two simple shearlet-based inverses for the multidimensional Radon and John transforms

Abstract

We use well-known intertwining relations of integral operators between the Euclidean space and the manifold of planes/lines to develop new analytical inverses of the Radon/John transforms. The integral operators are convolutions with shearlets as kernels. The shearlets are smooth compactly supported and away from the origin in the Fourier domain, therefore the operators are bounded. We show a series of relations in the form of shearlet coefficients, convolutions and inner products in different spaces. This framework yields potential applications since there is a natural translation from the continuous to the discrete theory of shearlets. We also develop a new biorthogonal shearlet decomposition for the 3D Radon transform inversion. This shades light on the differences between our method of integral (bounded) operators and the biorthogonal curvelet/shearlet decomposition of differential (unbounded) operators.