Interior tomography using 1D generalized total variation penalty and multiscale implementation by Bedrosian identity Interior tomography using 1D generalized total variation penalty and multiscale implementation by Bedrosian identity

In x-ray computed tomography, exact cone-beam reconstruction algorithms have been developed for various scanning trajectories. Volumes scanned by helical, saddle, and circle-plus-line trajectories can be also reconstructed exactly with these algorithms. Generally, the exact cone-beam reconstruction algorithms are divided into two types, such as filtered backprojection and backprojection filtration. Particularly, the backprojection filtration method has influenced to develop new interior tomography algorithms using the Hilbert transform relationship. One of them is to determine the region of interest uniquely when tiny subregions inside of the region of interest are known. The solution can be obtained by projections onto convex sets, but it is well-known that the convergence speed of the projections onto convex sets is slow. Hence, to overcome the limitation, we employs a parallel proximal algorithm to simultaneously consider multiple convex constraints rather than projecting on each of them sequentially as in the projections onto convex sets. Our simulation results show that the solution for the interior tomography problem can be accurately obtained using parallel proximal algorithm with a much smaller number of iterations than the projections onto convex sets. However, knowing subregions before reconstruction is impractical, so we propose a new method of interior tomography using a one-dimensional generalized total variation seminorm penalty. Unlike the existing interior tomography work with two-dimensional total variation, the proposed algorithm guarantees the exact recovery of twoand three-dimensional piecewise smooth images using a one-dimensional generalized total variation seminorm penalty. The one-dimensional seminorm penalty allow us to convert the original twoor three-dimensional problem into a set of one-dimensional problems, and it leads to a much faster reconstruction. The restrictions placed on each one-dimensional problem by our one-dimensional regularizer are much more relaxed than those imposed by the two-dimensional regularizer in previous works. For the optimization problem with total variation, we use fast gradient projection algorithm with a sublinear rate of convergence. To further accelerate the algorithm up to a level for clinical use, we propose 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. We show the efficacy of the proposed multiscale algorithm using circular fan-beam and helical cone-beam 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.