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Approximation of Set-Valued Functions with images sets in $\mathbb{R}^d$

Nira Dyn, David Levin

TL;DR

This work advances high-order approximation for set-valued functions $F:[0,1]\to K(\mathbb{R}^d)$ from finite samples, including the graph $Graph(F)$ in $\mathbb{R}^{d+1}$, by combining 1D MDL interpolation with cross-section reductions and a robust implicit-surface approach. It extends the $d=1$ theory to higher dimensions by (i) leveraging cross-sections to recast higher-dimensional problems as 1D problems, (ii) employing spline quasi-interpolation and signed-distance representations to achieve $O(h^s)$ convergence away from and near topology-change points (PCTs), and (iii) presenting a general, dimension-agnostic framework (and explicit $d=3$ and higher-$d$ procedures) for reconstructing both $F(t)$ and $Graph(F)$ with rigorous error bounds in the Hausdorff metric. The key contributions include the two-stage cross-section strategy, the implicit-function approach with signed-distance functions, and the extension to $d\ge 3$ with concrete algorithms and convergence guarantees. The methods enable high-fidelity reconstruction of evolving set-valued objects from scans or samples, with practical impact in fields requiring precise geometric data assimilation, computer graphics, and multidisciplinary simulations involving moving or uncertain shapes.

Abstract

Given a finite number of samples of a continuous set-valued function F, mapping an interval to non-empty compact subsets of $\mathbb{R}^d$, $F: [a,b] \to K(\mathbb{R}^d)$, we discuss the problem of computing good approximations of F. We also discuss algorithms for a direct high-order evaluation of the graph of $F$, namely, the set $Graph(F)=\{(t,y)\ | \ y\in F(t),\ t\in [a,b]\}\in K(\mathbb{R}^{d+1})$. A set-valued function can be continuous and yet have points where the topology of the image sets changes. The main challenge in set-valued function approximation is to derive high-order approximations near these points. In a previous paper, we presented with Q. Muzaffar, an algorithm for approximating set-valued functions with 1D sets ($d=1$) as images, achieving high approximation order near points of topology change. Here we build upon the results and algorithms in the $d=1$ case, first in more detail for the important case $d=2$, and later for approximating set-valued functions and their graphs in higher dimensions.

Approximation of Set-Valued Functions with images sets in $\mathbb{R}^d$

TL;DR

This work advances high-order approximation for set-valued functions from finite samples, including the graph in , by combining 1D MDL interpolation with cross-section reductions and a robust implicit-surface approach. It extends the theory to higher dimensions by (i) leveraging cross-sections to recast higher-dimensional problems as 1D problems, (ii) employing spline quasi-interpolation and signed-distance representations to achieve convergence away from and near topology-change points (PCTs), and (iii) presenting a general, dimension-agnostic framework (and explicit and higher- procedures) for reconstructing both and with rigorous error bounds in the Hausdorff metric. The key contributions include the two-stage cross-section strategy, the implicit-function approach with signed-distance functions, and the extension to with concrete algorithms and convergence guarantees. The methods enable high-fidelity reconstruction of evolving set-valued objects from scans or samples, with practical impact in fields requiring precise geometric data assimilation, computer graphics, and multidisciplinary simulations involving moving or uncertain shapes.

Abstract

Given a finite number of samples of a continuous set-valued function F, mapping an interval to non-empty compact subsets of , , we discuss the problem of computing good approximations of F. We also discuss algorithms for a direct high-order evaluation of the graph of , namely, the set . A set-valued function can be continuous and yet have points where the topology of the image sets changes. The main challenge in set-valued function approximation is to derive high-order approximations near these points. In a previous paper, we presented with Q. Muzaffar, an algorithm for approximating set-valued functions with 1D sets () as images, achieving high approximation order near points of topology change. Here we build upon the results and algorithms in the case, first in more detail for the important case , and later for approximating set-valued functions and their graphs in higher dimensions.
Paper Structure (16 sections, 7 theorems, 34 equations, 1 figure)

This paper contains 16 sections, 7 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Review of the results for the case $d=1$.
  4. Cross-sections
  5. The sampling density condition
  6. Spline quasi-interpolation operators
  7. Implicit approximation of smooth curves and surfaces
  8. Estimating the distance of a point from a surface
  9. From 2D cross-sections to 3D objects
  10. Approximating $F(t)$
  11. Approximating Graph(F) (the 3D object)
  12. The implicit function approach
  13. Building and analyzing the signed-distance function approximation
  14. Higher dimension procedures
  15. Computational algorithm for the case $d=3$
  16. ...and 1 more sections

Key Result

Proposition 2.5

Assume $\Gamma$ is a $C^4$ smooth curve in $[0,1]^2$ with minimal curvature radius $>R$, and such that its $R$-neighborhood is not self-intersecting and not intersecting the boundaries of $[0,1]^2$. Let $P$ be a square mesh of points in $[0,1]^2$, of mesh size $\delta<R/10$. Let $\Gamma$ subdivide $ Consider $Q_3$ to be the quasi-interpolation operator from vectors of values on $P$ to the space of

Figures (1)

  • Figure 2: A simple 3D object with cross-sections in two directions. The boundaries of the given cross-sections are in blue, and the boundaries of the cross-sections with constant $x_2$ are in red.

Theorems & Definitions (14)

  • Definition 2.2: Condition on the sampling density $SD(h)$
  • Definition 2.3: Condition $SD1(h)$
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Proposition 3.3
  • proof
  • Remark 3.5
  • ...and 4 more