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

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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(ε 1/2) and O(ε 1/3), respectively.

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Authors and Affiliations

  1. Department of Mathematics, Missouri State University, Springfield, MO, 65804, USA

    Kanghui Guo

  2. Department of Mathematics, University of Houston, Houston, TX, 77204, USA

    Demetrio Labate

Authors
  1. Kanghui Guo
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  2. Demetrio Labate
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Correspondence to Demetrio Labate.

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Communicated by: Yang Wang.

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Guo, K., Labate, D. Optimal recovery of 3D X-ray tomographic data via shearlet decomposition. Adv Comput Math 39, 227–255 (2013). https://doi.org/10.1007/s10444-012-9276-x

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

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