Reconstructions of Single Pixel X-Ray Transforms with Applications in Nuclear-Disarmament Verification

TL;DR

This paper analyzes the mathematical properties of the Single Pixel X-Ray Transform used in nuclear verification, and introduces a reconstruction method from noisy measurements, focusing on rotational symmetric objects.

Contribution

It provides a detailed mathematical analysis of the Single Pixel X-Ray Transform and proposes a novel reconstruction algorithm using Douglas-Rachford splitting and total variation denoising.

Findings

The Single Pixel X-Ray Transform is non-linear, continuous, Fréchet-differentiable, and convex.

A reconstruction method based on finite noisy measurements is developed.

Implementation demonstrates effectiveness for rotational symmetric objects.

Abstract

In nuclear arms control and disarmament processes, it is crucial to determine whether an object is a nuclear weapon or not without revealing sensitive information about it. At the MIT: Laboratory for Nuclear Security and Policy, such a nuclear verification method was developed, showcasing a transmission-based approach [1]. This method's essential part rests on a mathematical operation, the Single-Pixel X-Ray Transform: a cone of X-rays transmits an object and the remaining intensity is measured with a single-pixel detector. This transformation and the recovery of objects from dimensionless single-pixel measurements more generally has only been analyzed to a limited extent. In this work, we investigate some of the Single Pixel X-Ray Transform's mathematical properties. More specifically, we show that the Single Pixel X-ray transform is non-linear, continuous, Fr\'echet-differentiable and convex. We also introduce a method of reconstructing an object based only on a finite number of dimensionless, noisy Single Pixel X-Ray Transform measurement values. This method is based on Douglas-Rachford splitting and uses total variation denoising. We present an implementation for this method, focusing on rotational symmetric objects, as they allow the use of a one-dimensional direct total variation denoising algorithm [2].