Generalized Probability Density Approach to Histopolation Schemes of Arbitrary Order

TL;DR

The paper addresses reconstructing a bivariate function from edge-integral data on triangular meshes by introducing a generalized histopolation framework that weights edge data with probability densities and supports arbitrary polynomial order $k$. It establishes unisolvency for enriched triples $\mathcal{H}_k^\omega$, derives explicit basis functions, and defines the reconstruction operator $\Pi_k^\omega$, with a focus on Jacobi-type densities including the Gegenbauer subclass. An adaptive, mesh-level parameter selection strategy is developed to tune density parameters $(\alpha,\beta)$ for accuracy, demonstrated through numerical experiments. Results show that enriched histopolation operators based on Jacobi/Gegenbauer densities outperform classical linear histopolation in both smooth and oscillatory scenarios, signaling strong practical impact for tomography and related edge-data reconstruction tasks.

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.

Figures (4)

• Figure 1: Normalized Jacobi probability density functions $\omega_{\alpha,\beta}$ on $[-1,1]$. Left: effect of the parameter $\beta$, with fixed $\alpha = 2$ and different values of $\beta$. Right: effect of the parameter $\alpha$, with fixed $\beta = 2$ and varying values of $\alpha$, illustrating how the weight near the endpoints $t=\pm 1$ can be shifted and modulated. These plots highlight the flexibility of Jacobi densities in shaping local features through the choice of $\alpha$ and $\beta$.
• Figure 2: Normalized Gegenbauer probability density functions $\omega_{\gamma}$ on $[-1,1]$ for different values of $\gamma$. As $\gamma$ increases, the density becomes more concentrated around $t=0$ and vanishes more rapidly near the endpoints $t=\pm 1$. This illustrates how the single parameter $\gamma$ controls the balance between central concentration and endpoint weight.
• Figure 3: Examples of regular Friedrichs--Keller triangulations $\mathcal{T}_n$, for $n=20$ (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 $f_1$ (top left), $f_2$ (top right), $f_3$ (middle left), $f_4$ (middle right), $f_5$ (bottom left), and $f_6$ (bottom right). Comparison between the classical histopolation method $\mathcal{CH}$ (blue) and the enriched Jacobi-based method $\mathcal{H}^{\alpha^\star,\beta^\star}_2$ (red), as the number of triangles in the Friedrichs--Keller triangulations increases.

Theorems & Definitions (17)

• Remark 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• Remark 2.9
• Theorem 2.10