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 piecewise smooth data. This approach 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 applies a thresholding scheme to the 3D shearlet transform coefficients of the noisy data. For a given noise level $\epsilon$, the threshold can be set so that the estimator attains an essentially optimal mean square error rate $O(\epsilon^{2/3})$, as \epsilon \to 0. This approach outperforms standard SVD estimation methods as well as methods based on the Wavelet-Vaguelettes decomposition.