Initial state reconstruction on graphs

TL;DR

This paper introduces a spectral graph-based regularization framework for the unstable backward diffusion equation on graphs, providing stability, convergence results, and numerical experiments to improve initial state reconstruction in noisy environments.

Contribution

It proposes a novel spectral graph regularization method for backward diffusion, addressing instability issues and enabling stable initial state reconstruction.

Findings

The method achieves stable reconstructions in noisy settings.

Convergence of the proposed solution is theoretically established.

Numerical experiments demonstrate effectiveness of the approach.

Abstract

The presence of noise is an intrinsic problem in acquisition processes for digital images. One way to enhance images is to combine the forward and backward diffusion equations. However, the latter problem is well known to be exponentially unstable with respect to any small perturbations on the final data. In this scenario, the final data can be regarded as a blurred image obtained from the forward process, and that image can be pixelated as a network. Therefore, we study in this work a regularization framework for the backward diffusion equation on graphs. Our aim is to construct a spectral graph-based solution based upon a cut-off projection. Stability and convergence results are provided together with some numerical experiments.