Mathematical Modeling of Smooth Curve from Sampled Data via the Discrete Biharmonic Equation

Abstract

Curve reconstruction is the process of estimating a smooth function or curve that fits a given set of data points, either exactly (interpolation) or approximately (fitting). Classical approaches, including global polynomial interpolation, splines, Hermite interpolation, and radial basis function fitting, face challenges when data are sparse, irregularly distributed, or noisy. In this paper, we propose a curve reconstruction method based on the discrete form of the biharmonic equation. The method formulates reconstruction as a constrained quadratic optimization problem, incorporating both equality and inequality constraints and producing globally C1 smooth curves. The approach is physically interpretable, penalizing excessive bending, as in the case of a thin elastic beam, and can be extended to higher-dimensional surface reconstruction. Performance is evaluated through numerical experiments on known functions and synthetic data with various distributions and constraints, including small perturbation tests to assess stability and robustness. The results demonstrate that the proposed method reproduces the data, enforces the prescribed bounds, and remains stable under irregular sampling and noise.