We study recoverability in fan-beam computed tomography (CT) with sparsity and total variation priors: how many underdetermined linear measurements suffice for recovering images of given sparsity? Results from compressed sensing (CS) establish such conditions for example for random measurements, but not for CT. Recoverability is typically tested by checking whether a computed solution recovers the original. This approach cannot guarantee solution uniqueness and the recoverability decision therefore depends on the optimization algorithm. We propose new computational methods to test recoverability by verifying solution uniqueness conditions. Using both reconstruction and uniqueness testing, we empirically study the number of CT measurements sufficient for recovery on new classes of sparse test images. We demonstrate an average-case relation between sparsity and sufficient sampling and observe a sharp phase transition as known from CS, but never established for CT. In addition to assessing recoverability more reliably, we show that uniqueness tests are often the faster option.
Bibliographical noteThis is an Open Access article distributed under the terms of the Creative Commons Attribution License http://creativecommons. org/licenses/by/3.0/, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The moral rights of the named author(s) have been asserted.
- computed tomography
- total variation
- sparse regularization
- uniqueness conditions
Jørgensen, J. S., Kruschel, C., & Lorenz, D. A. (2015). Testable uniqueness conditions for empirical assessment of undersampling levels in total variation-regularized X-ray CT. Inverse Problems in Science and Engineering, 23(8), 1283–1305. https://doi.org/10.1080/17415977.2014.986724