ADAPTIVE IMAGE APPROXIMATION MODEL BASED ON RIGONOMETRIC SPLINES IN COMPUTER VISION TASKS

Abstract

This paper proposes an adaptive image approximation model based on trigonometric splines for computer vision tasks. The model integrates methods of trigonometric approximation, digital signal processing, and computer vision, enabling accurate representation of image intensity profiles and object contours. Images are interpreted as discrete signals containing sharp intensity transitions corresponding to object boundaries.

Special attention is paid to the analysis of the Gibbs phenomenon, which manifests itself as oscillatory artifacts near discontinuities when spectral approximation methods are applied. Such artifacts distort contour geometry and reduce the accuracy of edge detection, segmentation, and image reconstruction algorithms. The proposed approach employs trigonometric splines with parameterized smoothing, allowing controlled adjustment of approximation smoothness and reduction of Gibbs-related oscillations.

A comparative analysis of trigonometric polynomial approximation and trigonometric spline approximation is performed. Numerical experiments demonstrate that spline-based approximation provides improved stability near discontinuities and significantly reduces reconstruction errors compared to classical Fourier-based methods. The influence of spline parameters and smoothing factors on approximation accuracy is investigated. In particular, the use of linear summation methods and Rogozinsky-type smoothing factors is shown to be effective in suppressing oscillatory artifacts.

The proposed model can be interpreted as an adaptive image reconstruction framework whose parameters are tuned according to the structural properties of image signals. The obtained results confirm the feasibility of using trigonometric splines as a mathematical basis for adaptive approximation models in computer vision tasks, including edge detection, contour approximation, and image reconstruction.