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.

翻译：本文研究从三角形网格边上的加权积分重构二元函数的问题，该问题在层析成像、计算机视觉和数值逼近中具有核心重要性。我们的方法依赖于通过唯一可解三元组定义的局部直方插值方法，其中边权重由适当的概率密度函数诱导。特别地，我们引入了两个新的双参数广义截断正态分布族，它们扩展了经典的指数型分布律，并在捕捉目标函数局部特征方面提供了额外的灵活性。这些分布催生了新的二次重构算子，这些算子推广了标准的线性直方插值方案，同时保留了其简洁性和局部性。我们建立了其理论基础，证明了唯一可解性并推导了显式基函数，并通过大量数值实验展示了其提升的精度。此外，我们设计了一种用于最优选择分布参数的算法，确保重构的鲁棒性和自适应性。最后，我们证明所提出的框架可自然推广至任意限制在边上能定义有效概率密度的二元函数，从而凸显了其普适性和广泛的应用潜力。