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Accuracy Improvements for Convolutional and Differential Distance Function Approximations

Alexander Belyaev, Pierre-Alain Fayolle

TL;DR

This work tackles accurate estimation of the distance to the boundary $\mathrm{dist}(x,\partial\Omega)$ for $x\in\Omega$ by linking convolution-based distance transforms with Laplace asymptotics and integrating them into the heat-method framework. It introduces two Taylor-based extrapolation schemes for flat domains and a gradient-normalization step to boost accuracy, alongside a blending strategy that combines LogConv and SoftMin estimators with data-driven weights. The differential-approximation approach yields two practical proxies, $-\frac{v'_{\lambda}}{v}$ and its second-order correction, which, when normalized, outperform the conventional heat method in planar settings. The results indicate substantial accuracy gains with potential extensions to graphs, surfaces, and more general metric settings, while acknowledging the need for further theoretical guarantees and broader applicability.

Abstract

Given a bounded domain, we deal with the problem of estimating the distance function from the internal points of the domain to the boundary of the domain. Convolutional and differential distance estimation schemes are considered and, for both the schemes, accuracy improvements are proposed and evaluated. Asymptotics of Laplace integrals and Taylor series extrapolations are used to achieve the improvements.

Accuracy Improvements for Convolutional and Differential Distance Function Approximations

TL;DR

This work tackles accurate estimation of the distance to the boundary for by linking convolution-based distance transforms with Laplace asymptotics and integrating them into the heat-method framework. It introduces two Taylor-based extrapolation schemes for flat domains and a gradient-normalization step to boost accuracy, alongside a blending strategy that combines LogConv and SoftMin estimators with data-driven weights. The differential-approximation approach yields two practical proxies, and its second-order correction, which, when normalized, outperform the conventional heat method in planar settings. The results indicate substantial accuracy gains with potential extensions to graphs, surfaces, and more general metric settings, while acknowledging the need for further theoretical guarantees and broader applicability.

Abstract

Given a bounded domain, we deal with the problem of estimating the distance function from the internal points of the domain to the boundary of the domain. Convolutional and differential distance estimation schemes are considered and, for both the schemes, accuracy improvements are proposed and evaluated. Asymptotics of Laplace integrals and Taylor series extrapolations are used to achieve the improvements.

Paper Structure

This paper contains 6 sections, 25 equations, 3 figures.

Table of Contents

  1. Introduction and contribution
  2. Laplace's method
  3. Convolutional distance function estimation
  4. Differential distance function estimation
  5. Numerical experiments
  6. Discussion, conclusion, and future work

Figures (3)

  • Figure 1: (a) A 2-D shape (binary image) and its 1-D slice which are used to evaluate the proposed combination of the LogConv and SoftMin approximations of the distance to the boundary function. (b) Graphs of the restrictions of the exact distance function, the two convolution-based approximations proposed in Karam-etal_spl19 and corresponding to the LogConv (\ref{['eq:logConv']}) and SoftMin (\ref{['eq:softMin']}) formulas, and the combination of the approximations with weights given by (\ref{['eq:weights']}) onto the slice. (c) Graphs of the approximation errors. (d) Approximation errors for the SoftMin (top) and proposed approximation (bottom) are visualized for the whole shape.
  • Figure 2: Left: 1-D slices of the distance function approximations (\ref{['eq:heatDistApprox']}), (\ref{['eq:dist1']}), and (\ref{['eq:dist2']}). Middle: dependencies of the $L^2$ distance function approximation errors for (\ref{['eq:heatDistApprox']}), (\ref{['eq:dist1']}), and (\ref{['eq:dist2']}) as functions of parameter $t\equiv1/\lambda^2$. Right: dependencies of the $L^\infty$ (maxnorm) distance function approximation errors for (\ref{['eq:heatDistApprox']}), (\ref{['eq:dist1']}), and (\ref{['eq:dist2']}) as functions of parameter $t$. In all these experiments, the normalized version of (\ref{['eq:dist2']}) shows the best performance.
  • Figure 3: Distance function error maps for several binary shapes (binary images). Top row: the heat method Crane-Weischedel-Wardetzky_tog13 with $t=1$ is used. Middle row: second-order Taylor-based extrapolation (\ref{['eq:dist2']}) with $t\equiv1/\lambda^2=5$ is used. Bottom row: for each shape, the magnitude of the distance function gradient is visualized: high gradient magnitudes correspond to the skeleton of the shape; it looks plausible that the regions near the skeleton branch points contribute most to the distance estimation error.