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

Title: Abstract: I will discuss 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. This decomposition yields a highly effective reconstruction algorithm providing 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 based on a thresholding scheme applied to the 3D shearlet transform coefficients of the noisy data which, for a given noise level , can be tuned so that the estimator attains an essentially optimal mean square error rate O( ), as → 0. This is the first result to achieve this type of error estimate, outperforming methods based on Wavelet-Vaguelettes decomposition and SVD. University of Houston E-mail address: [email protected]