CoShaRP: A Convex Program for Single-shot Tomographic Shape Sensing

TL;DR

This paper presents CoShaRP, a convex optimization method for single-shot X-ray tomography that effectively estimates shapes from minimal measurements by leveraging shape priors and a dictionary of roto-translations.

Contribution

It introduces a convex program that transforms a non-linear shape estimation problem into a convex one using a dictionary approach, enabling fast and stable shape recovery.

Findings

Successfully recovers shapes from noisy measurements

Operates efficiently with a primal-dual algorithm

Performs well with moderately noisy data

Abstract

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.