Reconstruction of sparse-view tomography via preconditioned Radon sensing matrix

Abstract

Computed Tomography (CT) is one of the signicant research areas in the eld of medical image analysis. As X-rays used in CT image reconstruction are harmful to the human body, it is necessary to reduce the X-ray dosage while also maintaining good quality of CT images. Since medical images have a natural sparsity, one can directly employ compressive sensing (CS) techniques to reconstruct the CT images. In CS, sensing matrices having low coherence (a measure providing correlation among columns) provide better image reconstruction. However, the sensing matrix constructed through the incomplete angular set of Radon projections typically possesses large coherence. In this paper, we attempt to reduce the coherence of the sensing matrix via a square and invertible preconditioner possessing a small condition number, which is obtained through a convex optimization technique. The stated properties of our preconditioner imply that it can be used eectively even in noisy cases. We demonstrate empirically that the preconditioned sensing matrix yields better signal recovery than the original sensing matrix.

Nullspace Property for Optimality of Minimum Frame Angle Under Invertible Linear Operators

“…In particular, in contrast to existing results, we show and demonstrate numerically that it is not always possible to obtain a strictly incoherent and equivalent frame. In light of the present contribution, given a frame, one can verify if there is any scope for obtaining its coherence improvement as low coherence has a bearing on the quality of the underlying signal to be reconstructed and/or number of measurements to be used in such applications as tomographic reconstruction [16].…”

“…Consequently, the performances of sparse recovery algorithms can be different. Several methods exist in the literature [1], [4], [9], [5], [16] for finding an equivalent frame with optimal coherence.…”

“…The results reported, for example, in [1], [4], [9] proposed ways of finding the incoherent frames and their relevance to applications. It was shown in [16] that the incoherence obtained via preconditioning results in significant improvement in tomographic image reconstruction.…”

Sparsity-driven deterministic sampling strategy for coded aperture x-ray computed tomography

“…In recent years, image reconstruction methods that optimize the sparsity of the CT images in a transformed domain have become popular. As a result, techniques from compressed sensing (CS) as applied to CT image reconstruction have attracted the attention of the research community [10,11,17,19,22,25]. Notwithstanding the potential of CS-based ideas, one, needs to be careful while applying CS techniques to CT image reconstruction because the subsampled Radon sensing matrices can become rank-deficient and ill-conditioned [10,11,17,25].…”

TGV-regularized inversion of the Radon transform for photoacoustic tomography

“…Second, with respect to Radon inversion, the use of sophisticated image reconstruction methods allows a reduction of the amount of projection directions without introducing image distortion and stripe artifacts [22]. And third, appropriate regularization functionals have already been successfully employed for few-view computed tomography, for example, the total variation (TV) [23,24]. Nevertheless, other regularization strategies exist, such as Bayesian inversion or inversion based on deep learning [25].…”