A General Probability Density Framework for Local Histopolation and Weighted Function Reconstruction from Mesh Line Integrals

TL;DR

This paper introduces a flexible probabilistic framework for reconstructing bivariate functions from mesh line integrals, utilizing new distribution families to improve accuracy and adaptivity in tomography and related fields.

Contribution

It develops a novel local histopolation approach using generalized truncated normal distributions, extending classical methods with improved theoretical foundations and practical algorithms.

Findings

New quadratic reconstruction operators derived from distribution families.

Theoretical proof of unisolvency and explicit basis functions.

Numerical tests show enhanced accuracy and robustness.

Abstract

In this paper, we study the reconstruction of a bivariate function from weighted integrals along the edges of a triangular mesh, a problem of central importance in tomography, computer vision, and numerical approximation. Our approach relies on local histopolation methods defined through unisolvent triples, where the edge weights are induced by suitable probability densities. In particular, we introduce two new two-parameter families of generalized truncated normal distributions, which extend classical exponential-type laws and provide additional flexibility in capturing local features of the target function. These distributions give rise to new quadratic reconstruction operators that generalize the standard linear histopolation scheme, while retaining its simplicity and locality. We establish their theoretical foundations, proving unisolvency and deriving explicit basis functions, and we demonstrate their improved accuracy through extensive numerical tests. Moreover, we design an algorithm for the optimal selection of the distribution parameters, ensuring robustness and adaptivity of the reconstruction. Finally, we show that the proposed framework naturally extends to any bivariate function whose restriction to the edges defines a valid probability density, thus highlighting its generality and broad applicability.