26 research outputs found
Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-ray CT
Iterative image reconstruction (IIR) with sparsity-exploiting methods, such
as total variation (TV) minimization, investigated in compressive sensing (CS)
claim potentially large reductions in sampling requirements. Quantifying this
claim for computed tomography (CT) is non-trivial, because both full sampling
in the discrete-to-discrete imaging model and the reduction in sampling
admitted by sparsity-exploiting methods are ill-defined. The present article
proposes definitions of full sampling by introducing four sufficient-sampling
conditions (SSCs). The SSCs are based on the condition number of the system
matrix of a linear imaging model and address invertibility and stability. In
the example application of breast CT, the SSCs are used as reference points of
full sampling for quantifying the undersampling admitted by reconstruction
through TV-minimization. In numerical simulations, factors affecting admissible
undersampling are studied. Differences between few-view and few-detector bin
reconstruction as well as a relation between object sparsity and admitted
undersampling are quantified.Comment: Revised version that was submitted to IEEE Transactions on Medical
Imaging on 8/16/201
Empirical average-case relation between undersampling and sparsity in X-ray CT
Abstract. 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 recon-struction methods is motivated by recent progress in compressed sensing (CS). However, the CS framework provides neither guarantees of accurate CT re-construction, 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 em-pirical 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 ex-hibits a sharp phase transition as seen in CS, i.e. same-sparsity image require the same number of projections for recovery. Finally we demonstrate that th
How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray CT
We introduce phase-diagram analysis, a standard tool in compressed sensing,
to the X-ray CT community as a systematic method for determining how few
projections suffice for accurate sparsity-regularized reconstruction. In
compressed sensing a phase diagram is a convenient way to study and express
certain theoretical relations between sparsity and sufficient sampling. We
adapt phase-diagram analysis for empirical use in X-ray CT for which the same
theoretical results do not hold. We demonstrate in three case studies the
potential of phase-diagram analysis for providing quantitative answers to
questions of undersampling: First we demonstrate that there are cases where
X-ray CT empirically performs comparable with an optimal compressed sensing
strategy, namely taking measurements with Gaussian sensing matrices. Second, we
show that, in contrast to what might have been anticipated, taking randomized
CT measurements does not lead to improved performance compared to standard
structured sampling patterns. Finally, we show preliminary results of how well
phase-diagram analysis can predict the sufficient number of projections for
accurately reconstructing a large-scale image of a given sparsity by means of
total-variation regularization.Comment: 24 pages, 13 figure
Testable uniqueness conditions for empirical assessment of undersampling levels in total variation-regularized X-ray CT
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.Comment: 18 pages, 7 figures, submitte