Image Restoration via Integration of Optimal Control Techniques and the Hamilton-Jacobi-Bellman Equation

TL;DR

This paper introduces a new image restoration method that combines optimal control theory with the Hamilton-Jacobi-Bellman equation, effectively improving image quality through a mathematically grounded approach.

Contribution

It presents a novel framework integrating optimal control and HJB equations for image restoration, reducing computational complexity under radial symmetry assumptions.

Findings

Significant improvement in image quality metrics like PSNR and SSIM

Effective solution of HJB via shooting method under radial symmetry

Validation of the theoretical model through numerical experiments

Abstract

In this paper, we propose a novel image restoration framework that integrates optimal control techniques with the Hamilton-Jacobi-Bellman (HJB) equation. Motivated by models from production planning, our method restores degraded images by balancing an intervention cost against a state-dependent penalty that quantifies the loss of critical image information. Under the assumption of radial symmetry, the HJB equation is reduced to an ordinary differential equation and solved via a shooting method, from which the optimal feedback control is derived. Numerical experiments, supported by extensive parameter tuning and quality metrics such as PSNR and SSIM, demonstrate that the proposed framework achieves significant improvement in image quality. The results not only validate the theoretical model but also suggest promising directions for future research in adaptive and hybrid image restoration techniques.