Optimal recovery of 3D X-ray tomographic data via shearlet decomposition

Abstract

This paper introduces a new decomposition of the 3D X-ray transform based on the shearlet representation, a multiscale directional representation which is optimally efficient in handling 3D data containing edge singularities. Using this decomposition, we derive a highly effective reconstruction algorithm yielding a near-optimal rate of convergence in estimating piecewise smooth objects from 3D X-ray tomographic data which are corrupted by white Gaussian noise. This algorithm is achieved by applying a thresholding scheme on the 3D shearlet transform coefficients of the noisy data which, for a given noise level ε, can be tuned so that the estimator attains the essentially optimal mean square error rate O(log(ε -1)ε 2/3), as ε→0. This is the first published result to achieve this type of error estimate, outperforming methods based on Wavelet-Vaguelettes decomposition and on SVD, which can only achieve MSE rates of O(ε1/2) and O(ε 1/3), respectively.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1007/s10444-012-9276-x

Keywords

Computed tomography, Inverse problems, Radon transform, Shearlets, Wavelets, X-ray tomography, X-ray transform

Publication Date

8-1-2013

Journal Title

Advances in Computational Mathematics

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