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.

This chapter gives an overview over recovery guarantees for total variation minimization in compressed sensing for different measurement scenarios. In addition to summarizing the results in the area, we illustrate why an approach that is common for synthesis sparse signals fails and different techniques are necessary. Lastly, we discuss a generalizations of recent results for Gaussian measurements to the subgaussian case.

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Designing computational experiments involving 1 minimization with linear constraints in a finite-dimensional, real-valued space for receiving a sparse solution with a precise number k of nonzero entries is, in general, difficult. Several conditions were introduced which guarantee that, for small k and for certain matrices, simply placing entries with desired characteristics on a randomly chosen support will produce vectors which can be recovered by 1 minimization.In this work, we consider the case of large k and propose both a methodology to quickly check whether a given vector is recoverable, and to construct vectors with the largest possible support. Moreover, we gain new insights in the recoverability in a non-asymptotic regime. The theoretical results are illustrated with computational experiments.