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

Dragos-Patru Covei

TL;DR

The paper addresses image restoration under noise by formulating it as a stochastic optimal-control problem driven by the Hamilton–Jacobi–Bellman (HJB) framework. By assuming radial symmetry, the HJB is reduced to a nonlinear ODE in a single radius variable, solved via a shooting method to obtain a unique, decreasing value function $V(y)$ and the corresponding optimal feedback $p^*(y)$. The authors implement this approach with Python, extracting the optimal control from $\nabla V$ and simulating restorative diffusion, reporting significant improvements in PSNR and SSIM on degraded images. The work provides a principled, adaptive denoising mechanism that balances noise suppression with detail preservation, and it points to promising extensions such as multi-scale schemes and hybrid integrations with data-driven methods for real-time applicability.

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.

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

TL;DR

The paper addresses image restoration under noise by formulating it as a stochastic optimal-control problem driven by the Hamilton–Jacobi–Bellman (HJB) framework. By assuming radial symmetry, the HJB is reduced to a nonlinear ODE in a single radius variable, solved via a shooting method to obtain a unique, decreasing value function and the corresponding optimal feedback . The authors implement this approach with Python, extracting the optimal control from and simulating restorative diffusion, reporting significant improvements in PSNR and SSIM on degraded images. The work provides a principled, adaptive denoising mechanism that balances noise suppression with detail preservation, and it points to promising extensions such as multi-scale schemes and hybrid integrations with data-driven methods for real-time applicability.

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.
Paper Structure (19 sections, 2 theorems, 56 equations)

This paper contains 19 sections, 2 theorems, 56 equations.

Table of Contents

  1. Mathematical Formulation of the Problem
  2. Radial Reduction and Main Result
  3. Optimality and Verification
  4. Structural Properties: Monotonicity and Uniqueness
  5. Numerical Simulation and Procedure
  6. Model Parameters and Notation
  7. Image Cost Function
  8. Reduction to a Radial ODE
  9. Optimal Feedback and Simulation of the Dynamics
  10. A Python Code for mathematical implementation
  11. Interpretation of the Parameter $\alpha$
  12. Interpretation and Applications in Image Restoration
  13. A Concrete Example of the Application of the Theory in Image Restoration
  14. The Image Restoration Model as an Optimal Control Problem
  15. Numerical Implementation
  16. ...and 4 more sections

Key Result

Theorem 2.1

Assume that the cost function $h:[0,R]\rightarrow \lbrack 0,\infty )$ is continuous, that $\alpha \in (1,2]$, $\sigma >0$, and that $g\in \mathbb{R}$ is a suitable parameter. Then the boundary value problem eq:radialODE with has a unique classical solution satisfying $u^{\prime }(r)<0$ for all $r\in (0,R]$. Consequently, the radially symmetric function is the unique classical solution of the corre

Theorems & Definitions (4)

  • Theorem 2.1: Existence and Uniqueness of the Radial Solution
  • proof
  • Proposition 3.1
  • proof : Proof Sketch