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Agent-Based Optimal Control for Image Processing

Alessio Oliviero, Simone Cacace, Giuseppe Visconti

TL;DR

The paper addresses image segmentation and colour quantization by recasting them as an optimal control problem for a fixed-grid multi-agent color-dynamics model. It first analyzes a quadratic color-gradient cost solved by a direct-adjoint looping (DAL) method, then develops a robust total-variation (ROF) based formulation solved via ADMM to preserve edges. The authors provide well-posedness results, describe numerical schemes, and demonstrate through CUDA-accelerated experiments that the TV-regularised approach yields sharper edges and more uniform clusters across both high-contrast and complex natural images. The work advances scalable, data-adaptive image processing by coupling agent-based dynamics with variational regularisation, offering practical potential for high-dimensional data and further extensions to colour images and stochastic settings.

Abstract

We investigate the use of multi-agent systems to solve classical image processing tasks, such as colour quantization and segmentation. We frame the task as an optimal control problem, where the objective is to steer the multi-agent dynamics to obtain colour clusters that segment the image. To do so, we balance the total variation of the colour field and fidelity to the original image. The solution is obtained resorting to primal-dual splitting and the method of multipliers. Numerical experiments, implemented in parallel with CUDA, demonstrate the efficacy of the approach and its potential for high-dimensional data.

Agent-Based Optimal Control for Image Processing

TL;DR

The paper addresses image segmentation and colour quantization by recasting them as an optimal control problem for a fixed-grid multi-agent color-dynamics model. It first analyzes a quadratic color-gradient cost solved by a direct-adjoint looping (DAL) method, then develops a robust total-variation (ROF) based formulation solved via ADMM to preserve edges. The authors provide well-posedness results, describe numerical schemes, and demonstrate through CUDA-accelerated experiments that the TV-regularised approach yields sharper edges and more uniform clusters across both high-contrast and complex natural images. The work advances scalable, data-adaptive image processing by coupling agent-based dynamics with variational regularisation, offering practical potential for high-dimensional data and further extensions to colour images and stochastic settings.

Abstract

We investigate the use of multi-agent systems to solve classical image processing tasks, such as colour quantization and segmentation. We frame the task as an optimal control problem, where the objective is to steer the multi-agent dynamics to obtain colour clusters that segment the image. To do so, we balance the total variation of the colour field and fidelity to the original image. The solution is obtained resorting to primal-dual splitting and the method of multipliers. Numerical experiments, implemented in parallel with CUDA, demonstrate the efficacy of the approach and its potential for high-dimensional data.
Paper Structure (12 sections, 2 theorems, 33 equations, 7 figures, 2 algorithms)

This paper contains 12 sections, 2 theorems, 33 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Keywords.
  2. MSC2020.
  3. Introduction
  4. Mathematical model
  5. The optimal control problem
  6. Numerical approximation
  7. Numerical tests
  8. Total variation minimisation
  9. Numerical approximation
  10. Numerical tests
  11. Conclusions and perspectives
  12. Derivation of the optimality system \ref{['eq:opt_system']}

Key Result

Theorem 3.1

Let $\alpha > 0$. Then the functional $J : H^1(\Omega) \to \mathbb{R}$ in Definition def:cost_functional has a unique global minimum.

Figures (7)

  • Figure 1: Magnetic resonance image of a brain tumour, from mri_scan. On the left side, the original picture; on the right, colour value $c_i \in [0,1]$ for every pixel $p_i,\ i=1,\ldots,N$.
  • Figure 2: Output of Algorithm \ref{['alg:dal']} for different values of the fidelity parameter $\alpha$, using the image in Figure \ref{['fig:mri']} as input. On each row, from left to right: output image, grey level for every pixel, time evolution of the semi-axes of the ellipsoid \ref{['eq:ellipsoid']}.
  • Figure 3: Portrait of baby Namou, S. Cacace's dog. On the left side, the original picture; on the right, colour value $c_i \in [0,1]$ for every pixel $p_i,\ i=1,\ldots,N$.
  • Figure 4: Output of Algorithm \ref{['alg:dal']} for different values of the fidelity parameter $\alpha$, using the image in Figure \ref{['fig:namou']} as input. On each row, from left to right: output image, grey level for every pixel, time evolution of the semi-axes of the ellipsoid \ref{['eq:ellipsoid']}.
  • Figure 5: Comparison of Algorithm \ref{['alg:dal']} and Algorithm \ref{['alg:admm']} on the test image in Figure \ref{['fig:mri']}. Output image on top, grey level for every pixel in the middle row, time evolution of the semi-axes of the ellipsoid \ref{['eq:ellipsoid']} on the bottom.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Remark 1: On the choice of $T$ and $E$
  • Remark 2: Convergence criteria for the DAL algorithm
  • Definition 4.1
  • Theorem 4.1
  • proof
  • Remark 3