A simple shearlet-based 2D Radon inversion with an application to computed tomography

Abstract

We find a new and simple inversion formula for the 2D Radon transform (RT) with a straight use of the shearlet system and of well-known properties of the RT. Since the continuum theory of shearlets has a natural translation to the discrete theory, we also obtain a computable algorithm that recovers a digital image from noisy samples of the 2D Radon transform which also preserves edges. A very well-known RT inversion in the applied harmonic analysis community is the biorthogonal curvelet decomposition (BCD). The BCD uses an intertwining relation of differential (unbounded) operators between functions in Euclidean and Radon domains. Hence the BCD is ill-posed since the inverse is unstable in the presence of noise. In contrast, our inversion method makes use of an intertwining relation of integral transformations with very smooth kernels and compact support away from the origin in the Fourier domain, i.e. bounded operators. Therefore, we obtain, at least, the same asymptotic behavior of mean-square error as the BCD (and its shearlet version) for the class of cartoon-like functions. Numerical simulations show that our inverse surpasses, by far, the inverse based on the BCD. Our algorithm uses only fast transformations.