Radon transform inversion using the shearlet representation

• Published in: Applied and computational harmonic analysis 2010-09, Vol.29 (2), p.232-250
• Main Authors: Colonna, Flavia, Easley, Glenn, Guo, Kanghui, Labate, Demetrio
• Format: Article
• Language: English
• Subjects: Directional wavelets; Estimating; Inverse problems; Inversions; Mathematical models; Radon; Radon transform shearlets; Reconstruction; Regularization; Transforms; Wavelets
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2009.10.005

The inversion of the Radon transform is a classical ill-posed inverse problem where some method of regularization must be applied in order to accurately recover the objects of interest from the observable data. A well-known consequence of the traditional regularization methods is that some important features to be recovered are lost, as evident in imaging applications where the regularized reconstructions are blurred versions of the original. In this paper, we show that the affine-like system of functions known as the shearlet system can be applied to obtain a highly effective reconstruction algorithm which provides near-optimal rate of convergence in estimating a large class of images from noisy Radon data. This is achieved by introducing a shearlet-based decomposition of the Radon operator and applying a thresholding scheme on the noisy shearlet transform coefficients. For a given noise level ε, the proposed shearlet shrinkage method can be tuned so that the estimator will attain the essentially optimal mean square error O ( log ( ε − 1 ) ε 4 / 5 ) , as ε → 0 . Several numerical demonstrations show that its performance improves upon similar competitive strategies based on wavelets and curvelets.