Testable uniqueness conditions for empirical assessment of undersampling levels in total variation-regularized x-ray CT

TL;DR

This paper develops computational methods to verify solution uniqueness in total variation-regularized CT, enabling more reliable and efficient assessment of the number of measurements needed for sparse image recovery, revealing a phase transition similar to compressed sensing.

Contribution

It introduces new algorithms for testing solution uniqueness in CT, bridging a gap in recoverability assessment and empirically establishing a phase transition in measurement sufficiency.

Findings

Uniqueness tests are often faster than reconstruction.

A sharp phase transition in recoverability was observed.

Empirical relation between sparsity and measurements was established.

Abstract

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, e.g., 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.