Inverting Laguerre tessellations: Recovering tessellations from the volumes and centroids of their cells using optimal transport

TL;DR

The paper addresses recovering a Laguerre tessellation from cell volumes $v_i$ and centroids $b_i$, and fitting a Laguerre diagram to target volume/centroid data, by leveraging semi-discrete optimal transport and convex optimization. It proves that the tessellation is uniquely determined by $(v_i,b_i)$ and provides a constructive convex method to recover it, with a generalisation to anisotropic Laguerre tessellations. For the fitting problem, the authors show that, under suitable data assumptions, local minimisers of the centroid error correspond to solutions of a convex reformulation, and they validate the approach with numerical experiments including EBSD data. The work demonstrates both a rigorous theoretical framework and practical algorithms, highlighting applications in materials science and offering a publicly available Python implementation.

Abstract

In this paper we study an inverse problem in convex geometry, inspired by a problem in materials science. Firstly, we consider the question of whether a Laguerre tessellation (a partition by convex polytopes) can be recovered from only the volumes and centroids of its cells. We show that this problem has a unique solution and give a constructive way of computing it using optimal transport theory and convex optimisation. Secondly, we consider the problem of fitting a Laguerre tessellation to synthetic volume and centroid data. Given some target volumes and centroids, we seek a Laguerre tessellation such that the difference between the volumes and centroids of its cells and the target volumes and centroids is minimised. For an appropriate objective function and suitable data, we prove that local minimisers of this problem can be constructed using convex optimisation. We also illustrate our results numerically. There is great interest in the computational materials science community in fitting Laguerre tessellations to electron backscatter diffraction (EBSD) and x-ray diffraction images of polycrystalline materials. As an application of our results we fit a 2D Laguerre tessellation to an EBSD image of steel.

Figures (9)

• Figure 1: Recovering a Laguerre diagram $\{ \mathrm{Lag}_i(\mathbf{X}_0,\mathbf{w}_0) \}_{i=1}^n$ from the areas ${\bf{v}}$ and centroids $\mathbf{B}$ of its cells; see Section \ref{['sec: numerics - diagram exists']}. Left: The Laguerre tessellation $\{ L_i(\mathbf{X}_{\textrm{init}};{\bf{v}})\}_{i=1}^n$ with cells of areas ${\bf{v}}$ (up to a $0.1\%$ error) generated from a random collection of seeds $\mathbf{X}_{\textrm{init}}$. Right: Numerical approximation of the unique Laguerre tessellation with cells of areas ${\bf{v}}$ and centroids $\mathbf{B}$. This figure shows the Laguerre tessellation corresponding to an approximate maximiser of the constrained optimisation problem \ref{['eq:maxHcon']}, computed by scipy.optimize.minimize using the initial guess $\mathbf{X}_{\textrm{init}}$ and $229$ iterations. The red dots are the centroids of the computed Laguerre cells. The blue dots (which are almost indistinguishable from the red dots) are the target centroids $\mathbf{B}$. The cells have areas ${\bf{v}}$ up to a $0.1\%$ error. We have not plotted the true Laguerre diagram $\{ \mathrm{Lag}_i(\mathbf{X}_0,\mathbf{w}_0) \}_{i=1}^n$ since it is indistinguishable from the computed Laguerre diagram.
• Figure 2: Recovering a Laguerre diagram -- convergence of the algorithm from Section \ref{['sec: numerics - diagram exists']}. Left: The value of the objective function $H$ converges to its maximum value of $0$. Right: The nonlinear least squares error $f$ also converges to $0$.
• Figure 3: Fitting a Laguerre diagram to synthetic data $({\bf{v}},\mathbf{B}^\varepsilon)$ with $\varepsilon=0.001$ (small perturbation); see Section \ref{['subsec: small']}. Left: The initial guess $\{ L_i(\mathbf{X}_{\textrm{init}};{\bf{v}})\}_{i=1}^n$ for the maximiser of \ref{['eq:maxHcon']}, where $\mathbf{X}_{\mathrm{init}}=\mathbf{B}^\varepsilon$. Right: An approximate maximiser of the constrained optimisation problem \ref{['eq:maxHcon']}, computed by scipy.optimize.minimize using the initial guess $\mathbf{X}_{\textrm{init}}$ and $402$ iterations. The centroids of the Laguerre cells (red dots) are almost indistinguishable from the target centroids (blue dots).
• Figure 4: Fitting a Laguerre diagram to synthetic data (small perturbation) - convergence of the algorithm from Section \ref{['subsec: small']}. Left: Convergence of the objective function $H$. It converges to a positive value because there does not exist a Laguerre diagram with cells of areas and centroids $({\bf{v}},\mathbf{B}^\varepsilon)$. Right: Convergence of the least squares error $f$. The $y$-axis shows the ratio of the value of $f$ at each iteration to its initial value $f(\mathbf{X}_{\mathrm{init}})$.
• Figure 5: Fitting a Laguerre diagram to synthetic data (see Section \ref{['sec:fitting']}) - minimum distance between the seeds for $\varepsilon=0.001$ (left) and $\varepsilon=0.05$ (right). The minimum distance between the seeds is normalised by the minimum allowed distance $\delta=10^{-3}$. The dotted line corresponds to when at least one of the constraints $c_{ij} \ge 0$ is active. Left (small perturbation): At the end of the algorithm none of the constraints are active. The minimum distance between the seeds is at least $30 \delta$. Right (larger perturbation): At the end of the algorithm at least one of the constraints is active. This suggests that if we solved \ref{['eq:maxHcon']} without the constraints $c_{ij} \ge 0$, then some of the seeds would collide.

Theorems & Definitions (29)

• Lemma 2.1: See Meyron2019, Proposition 6
• Theorem 2.2: See MerigotThibertOT, Theorem 40 & Proposition 37
• Proposition 4.1: Derivative of $W_2^2$; cf. MerigotSantambrogioSarrazin, Proposition 1
• Theorem 4.2: Properties of $H$
• proof
• Remark 4.3: Consequences of Theorem \ref{['thm:properties H']} for Goal \ref{['goal: fitting']}
• Definition 4.4: Compatible diagrams
• Theorem 4.5: Uniqueness of compatible diagrams
• proof
• Theorem 4.6: Goal \ref{['goal: recovering']} is a convex optimisation problem