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

TL;DR

This work addresses reconstructing a bivariate function from edge-integral data on triangular meshes by introducing a density-weighted, locally quadratic histopolation framework. It develops two two-parameter families of generalized truncated normal edge densities, derives unisolvency and explicit basis functions, and provides an algorithm for adaptive parameter selection that enhances robustness. The framework extends to any edge density that defines a valid probability density on edges, demonstrated through a general theory of edge functionals and orthogonal polynomials. Numerical experiments show clear accuracy gains over classical linear histopolation, with potential extensions to three-dimensional settings and adaptive imaging applications.

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.

Figures (6)

• Figure 1: Normalized probability density functions $\widetilde{k}_{\sigma,\mu}$ on $[-1,1]$. Left: effect of the scale parameter, with fixed $\mu=2$ and different values of $\sigma$. Right: effect of the shape parameter, with fixed $\sigma=0.50$ and varying values of $\mu$, illustrating the increased sharpness and concentration around $t=0$. These plots emphasize the flexibility provided by the two parameters.
• Figure 2: Normalized probability density functions $\widetilde{g}_{\sigma,\mu}$ on $[-1,1]$. Left: effect of the scale parameter, with fixed $\mu=2$ and different values of $\sigma$. Right: effect of the shape parameter, with fixed $\sigma=0.50$ and varying values of $\mu$, illustrating the increased sharpness and concentration around $t=0$. These plots emphasize the flexibility provided by the two parameters.
• Figure 3: Successive regular Friedrichs--Keller triangulations $\mathcal{T}_n$, with $n=20$ (top left), $n=30$ (top right), $n=40$ (bottom left), and $n=50$ (bottom right).
• Figure 4: Semi-log plot of the $L^1$ approximation error for $f_1$ (left) and $f_2$ (right) obtained with the classical histopolation method $\mathcal{CH}$ (blue) and the weighted quadratic enriched method $\mathcal{H}^{\mathrm{enr}}_{1,\sigma^{\star},\mu^{\star}}$ (red), as the number of triangles in the Friedrichs--Keller triangulations increases.
• Figure 5: Semi-log plot of the $L^1$ approximation error for $f_3$ (left) and $f_4$ (right) obtained with the classical histopolation method $\mathcal{CH}$ (blue) and the weighted quadratic enriched method $\mathcal{H}^{\mathrm{enr}}_{1,\sigma^{\star},\mu^{\star}}$ (red), as the number of triangles in the Friedrichs--Keller triangulations increases.

Theorems & Definitions (50)

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