Multiscale Interior Tomography Using 1D Generalized Total Variation

We propose a method for accurate and fast reconstruction of the interior of a 2D or 3D tomographic image from its incomplete local Radon transform. Unlike the existing interior tomography work with 2D total variation, the proposed algorithm guarantees exact recovery using a 1D generalized total variation semi-norm for regularization. The restrictions placed on an image by our 1D regularizer are much more relaxed than those imposed by the 2D regularizer in previous works. Furthermore, to further accelerate the algorithm up to a level of clinical use, we propose a multi-resolution reconstruction method by exploiting the Bedrosian theorem for the Hilbert transform. More specifically, as the high frequency part of the image can be quickly recovered using Hilbert transform thanks to the Bedrosian equality, we show that computationally expensive iterative reconstruction can be applied only for the low resolution images in downsampled domain, which significantly reduces the computational burden. We demonstrate the efficacy of the algorithm using circular fan-beam and helical cone-beam data.