Generalized Probability Density Approach to Histopolation Schemes of Arbitrary Order

Abstract: In this paper, we investigate the reconstruction of a bivariate function from weighted edge integrals on a triangular mesh, a problem of central importance in tomography, computer vision, and numerical approximation. Our approach is based on local histopolation methods defined through unisolvent triples, where the edge weights are induced by probability densities. We present a general strategy that applies to arbitrary polynomial order~$k$, in which edge moments are taken against orthogonal polynomials associated with the chosen densities. This yields a systematic framework for weighted reconstructions of any degree, with theoretical guarantees of unisolvency and fully explicit basis functions. As a concrete and flexible instance, we introduce a two-parameter family of Jacobi-type distributions on $[-1,1]$, together with its symmetric Gegenbauer subclass, and show how these densities generate new quadratic reconstruction operators that generalize the standard linear histopolation scheme while preserving its simplicity and locality. We employ an adaptive parameter selection algorithm for Jacobi densities, which automatically tunes the distribution parameters to minimize the global reconstruction error. This strategy enhances robustness and adaptivity across different function classes and mesh resolutions. The effectiveness of the proposed operators is demonstrated through extensive numerical experiments, which confirm their superior accuracy in approximating both smooth and highly oscillatory functions. Finally, the framework is sufficiently general to accommodate any admissible edge density, thus providing a flexible and broadly applicable tool for weighted function reconstruction.