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

This is the 2000th blog entry with a Compressive Sensing tag. Woohoo ! In the Map Makers, one can see that the sharp phase transitions can be used in many ways. One of the ways is to figure ut if a specifc mesurement essemble can be used efficiently. Another is to figure out if a specific technology can be improved. That discussion was somehow started for CT by Xiaochuan Pan, Emil Sidky and Michael Vannier in Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction? and later [2]. A while back, I used a rule of thumb using sparsity-only to look into the issue of CT with an assumption on the measurement matrix [1]. Today, we have a very well designed and deep study of the subject with the intent described by the authors that "the goal was to simply document phase transition behavior for CT measurements.". This is awesome ! Without further ado: Testable uniqueness conditions for empirical assessment of undersampling levels in total variation-regularized x-ray CT by Jakob S. Jørgensen, Christian Kruschel, Dirk A. Lorenz

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.

One of the author, Dirk Lorenz, also blogs at Regularize.

[1] Why You Should Care About A Compressive Sensing Approach to CT
[2] Toward optimal X-ray ﬂux utilization in breast CT by Jakob H. Jørgensen, Per Christian Hansen, Emil Sidky, Ingrid S. Reiser, and Xiaochuan Pan.