Relaxation-based dynamical Ising machines for discrete tomography

TL;DR

This paper demonstrates that a relaxation-based dynamical Ising machine, driven by the V2 model, can exactly solve discrete tomography problems by converging to binary images consistent with tomographic data, showcasing a novel application of dynamical systems.

Contribution

The paper introduces a new approach using the V2 dynamical Ising model to solve discrete tomography problems exactly, highlighting the model's unique non-local transition capabilities.

Findings

High success probability in reconstructing images from limited rays

Convergence time depends weakly on image size for certain cases

Exact solutions achieved through dynamical features of the V2 model

Abstract

Dynamical Ising machines are continuous dynamical systems that evolve from a generic initial state to a state strongly related to the ground state of the classical Ising model. We show that such a machine driven by the V${}_2$ dynamical model can solve exactly discrete tomography problems about reconstructing a binary image from the pixel sums along a discrete set of rays. In contrast to usual applications of Ising machines, targeting approximate solutions to optimization problems, the randomly initialized V${}_2$ model converges with high probability ($P_{\mathrm{succ}} \approx 1$) to an image precisely satisfying the tomographic data. For the problems with at most two rays intersecting at each pixel, the V${}_2$ model converges in internal machine time that depends only weakly on the image size. Our consideration is an example of how specific dynamical systems can produce exact solutions to highly non-trivial data processing tasks. Crucially, this solving capability arises from the dynamical features of the V${}_2$ model itself, in particular its equations of motion that enable non-local transitions of the discrete component of the relaxed spin beyond Hamming-neighborhood constraints, rather than from merely recasting the tomography problem in spin form.