# American Institute of Mathematical Sciences

May  2015, 9(2): 431-446. doi: 10.3934/ipi.2015.9.431

## Empirical average-case relation between undersampling and sparsity in X-ray CT

 1 Department of Applied Mathematics and Computer Science, Technical University of Denmark, Richard Petersens Plads, Building 324, 2800 Kgs. Lyngby, Denmark, Denmark 2 Department of Radiology, University of Chicago, 5841 South Maryland Avenue, Chicago, IL 60637, United States, United States

Received  July 2014 Revised  January 2015 Published  March 2015

In X-ray computed tomography (CT) it is generally acknowledged that reconstruction methods exploiting image sparsity allow reconstruction from a significantly reduced number of projections. The use of such reconstruction methods is inspired by recent progress in compressed sensing (CS). However, the CS framework provides neither guarantees of accurate CT reconstruction, nor any relation between sparsity and a sufficient number of measurements for recovery, i.e., perfect reconstruction from noise-free data. We consider reconstruction through 1-norm minimization, as proposed in CS, from data obtained using a standard CT fan-beam sampling pattern. In empirical simulation studies we establish quantitatively a relation between the image sparsity and the sufficient number of measurements for recovery within image classes motivated by tomographic applications. We show empirically that the specific relation depends on the image class and in many cases exhibits a sharp phase transition as seen in CS, i.e., same-sparsity images require the same number of projections for recovery. Finally we demonstrate that the relation holds independently of image size and is robust to small amounts of additive Gaussian white noise.
Citation: Jakob S. Jørgensen, Emil Y. Sidky, Per Christian Hansen, Xiaochuan Pan. Empirical average-case relation between undersampling and sparsity in X-ray CT. Inverse Problems & Imaging, 2015, 9 (2) : 431-446. doi: 10.3934/ipi.2015.9.431
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