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

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, Fréchet-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].

Figures (5)

• Figure 1: Theoretical models of a nuclear weapon as presented in fetter1990.
• Figure 2: A schematic setup with which the measurements are made according to the physical cryptographic method kemp2016physical. During the procedure, an Electron beam is directed through a Bremsstrahlung radiator, generating a continuous X-ray beam. This X-ray beam subsequently transmits through the treaty accountable item (TAI). As it does so, a portion of the radiation is absorbed, leading to the formation of isotope-specific absorption lines. To ensure the data's confidentiality, resonance fluorescence photons are produced using a specialized Encrypting foil. Any low-energy photons that might carry sensitive information are then filtered out with a Tungsten low-energy filter. The remaining radiation is captured using a single-pixel HPGe detector, producing a single value. This value can be compared across different objects for consistency and verification.
• Figure 3: On the left, a source $r$, which emits X-rays of intensity $I_{0}$, is shown. In this cone, the black box, containing an unknown object $f$, is placed. While the X-rays pass through the black box, a part of their intensity is absorbed. The remaining intensity $I_{1}$ is measured with a single-pixel detector $D$
• Figure 4: Results of the reconstruction of objects from Single Pixel X-Ray Transform measurement values. The left side always shows a cross-section through the corresponding object, while the right side shows the related discretized reconstruction.
• Figure 5: Analysing the dependency of the SSIM value on the noise and the regularization parameter

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• proof
• Lemma 1
• proof
• Definition 4