CoShaRP: A Convex Program for Single-shot Tomographic Shape Sensing
This work is significant for medical imaging and industrial inspection, where single-shot tomography could reduce radiation exposure or speed up processes, by providing a method to reconstruct shapes from minimal data.
This paper addresses single-shot X-ray tomography, an extremely under-determined linear inverse problem, by transforming it into a non-linear problem of estimating roto-translations of known shapes. They propose CoShaRP, a convex program that recovers dictionary-coefficients, and demonstrate its ability to stably recover shapes from moderately noisy measurements.
We introduce single-shot X-ray tomography that aims to estimate the target image from a single cone-beam projection measurement. This linear inverse problem is extremely under-determined since the measurements are far fewer than the number of unknowns. Moreover, it is more challenging than conventional tomography where a sufficiently large number of projection angles forms the measurements, allowing for a simple inversion process. However, single-shot tomography becomes less severe if the target image is only composed of known shapes. Hence, the shape prior transforms a linear ill-posed image estimation problem to a non-linear problem of estimating the roto-translations of the shapes. In this paper, we circumvent the non-linearity by using a dictionary of possible roto-translations of the shapes. We propose a convex program CoShaRP to recover the dictionary-coefficients successfully. CoShaRP relies on simplex-type constraint and can be solved quickly using a primal-dual algorithm. The numerical experiments show that CoShaRP recovers shapes stably from moderately noisy measurements.