MODELING THE OPERATION OF AN IMAGE RECOGNITION SYSTEM FOR DYED MATERIALS

Abstract

This paper presents a mathematically model of a surface structure recognition system for painted materials based on a variational filtering framework. The problem is formulated as an optimal estimation task derived from the minimization of an energy functional that combines a data fidelity term with gradient-type regularization.
A discrete formulation is constructed using the finite difference method, leading to a symmetric positive definite linear algebraic system. The model is extended to the full RGB case through a block operator representation that accounts for inter-channel correlations via a coupling matrix. It is proven that the positive definiteness of the inter-channel interaction matrix guarantees the preservation of the symmetric positive definite structure of the global system.
A spectral analysis of the resulting operator is performed. Bounds for eigenvalues and estimates of the condition number are derived, demonstrating the influence of discretization parameters and regularization coefficients on stability and numerical robustness. The existence and uniqueness of the solution are established for both scalar and vector formulations.
The conjugate gradient method is analyzed as an efficient numerical solver for the block system. Convergence estimates are obtained in terms of the condition number, confirming the computational efficiency of the proposed approach for high-resolution images.
The developed model provides a theoretically justified and interpretable alternative to purely data-driven techniques in surface quality assessment. It enables stable, well-conditioned, and computationally tractable processing of multichannel images in industrial inspection tasks.