Interior tomography using 1D generalized total variation. Part II: Multiscale implementation

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.