Interior Tomography Using 1D Generalized Total Variation. Part II: Multiscale Implementation

Abstract

Abstract. To address the classic interior tomography problem where projections at each view extend only to the shadow of a circular region completely interior to the subject being scanned, previously we showed that the exact recovery of two-and three-dimensional piecewise smooth images is guaranteed using a one-dimensional generalized total variation seminorm penalty which allows a much faster reconstruction. To further accelerate the algorithm up to a level for clinical use, this paper proposes a novel multiscale reconstruction method by exploiting the Bedrosian identity of the Hilbert transform. More specifically, we show that the high frequency parts of the one-dimensional signals can be quickly recovered analytically with the Hilbert transform because of the Bedrosian identity. This implies that computationally expensive iterative reconstruction need only be applied to low resolution images in the downsampled domain, which significantly reduces the computational burden. Moreover, even for incomplete trajectories such as circular cone-beam geometry, we demonstrate that the proposed multiscale interior tomography approach can be combined with a novel spectral blending method in order to mitigate cone-beam artifacts from missing frequency regions. We show the efficacy of the proposed multiscale algorithm using circular fan-beam, helical cone-beam data, and circular conebeam geometry. With a graphics processing unit implementation, we demonstrate that the speed of the algorithm can be significantly accelerated up to the level for clinical use for various acquisition geometries.

On Few-View Tomography and Staircase Artifacts

“…It is known that if the object is not piecewise-constant, some higher order methods are effective [14]–[17]. This paper argues that higher order methods may not be necessary.…”

“…When the TV-norm is minimized, it is believed that piecewise constant regions are encouraged. Based on this consideration, methods using higher-order derivatives were suggested to battle the staircase artifacts or for non-piecewise-constant objects [14]–[17]. …”

A weighted rebinned backprojection‐filtration algorithm from partially beam‐blocked data for a single‐scan cone‐beam CT with hybrid type scatter correction

“…The orthogonal butterfly filter approach has been proposed as a way to reduce missing frequency problems through the sum of the data sets of the valid frequency domain along the scan trajectory of [, ] . In particular, simulation results were reported to reduce streak artifacts in BPF algorithm . In our study, we applied orthogonal butterfly filtering for each of the two PI‐line data sets in the direction perpendicular to 360 degree in both simulation and experiment studies.…”

“…For image reconstruction, we developed a weighted rebinned BPF (wrBPF) algorithm. Also, we applied a butterfly filtering method which was previously reported to be an efficient technique for reducing cone‐beam artifacts and streak artifacts . We have conducted a simulation and an experimental study to validate and evaluate the utility of the proposed method.…”

A practical local tomography reconstruction algorithm based on a known sub-region

“…In (Yang et al, 2010), (Yu & Wang, 2009), (Lee et al, 2015), Total Variation method is used to reconstruct the ROI. In (Niinimäki et al, 2007), the function is assumed to be sparse in the wavelet domain, and a multi-resolution scheme reduces the number of unknown by keeping only fine-scale wavelet coefficients inside the ROI.…”