[Paper Review] On the Accuracy of Anisotropic Fast Marching

Jean‐Marie Mirebeau|arXiv (Cornell University)|May 23, 2012

Robotic Mechanisms and Dynamics7 references3 citations

TL;DR

This paper analyzes the accuracy of anisotropic fast marching, a variant of the fast marching algorithm for solving the generalized eikonal equation under large anisotropies. Using lattice basis reduction, the method maintains low computational complexity and, in the case of a constant Riemannian metric in dimensions 1–4, achieves numerical accuracy that is independent of the anisotropy ratio in an average sense over all directions.

ABSTRACT

The fast marching algorithm, and its variants, solves numerically the generalized eikonal equation associated to an underlying riemannian metric. A major challenge for these algorithms is the non-isotropy of the riemannian metric. Applications of the eikonal equation to image processing often involve pronounced anisotropies, which motivated the design of new algorithms. A recently introduced variant of the fast marching algorithm addresses the problem of large anisotropies using an algebraic tool named lattice basis reduction. The numerical complexity of this algorithm is insensitive to anisotropy, under weak assumptions. We establish in this paper, in the simplified setting of a constant riemannian metric, that the accuracy of this algorithm is also extremely robust to anisotropy : in an average sense, it is independent of the anisotropy ratio. We also extend this algorithm to higher dimension.

Motivation & Objective

• To analyze the numerical accuracy of a lattice basis reduction-based fast marching variant designed for high anisotropy in Riemannian metrics.
• To investigate whether the algorithm's accuracy degrades as the anisotropy ratio κ(M) increases, particularly in the context of image processing applications.
• To extend the algorithm to four-dimensional domains, motivated by applications in medical imaging and data analysis.
• To provide theoretical justification for the observed numerical robustness in prior experiments, especially under extreme anisotropy.
• To establish that the error in the algorithm remains bounded and independent of κ(M) when averaged over all spatial directions, under constant metric assumptions.

Proposed method

• Employs lattice basis reduction to reconfigure the computational stencil in the fast marching algorithm, enabling robustness to high anisotropy.
• Uses M-reduced bases (u₁, ..., u₄) in R^d to define a structured, non-orthogonal grid that adapts to the local metric M.
• Applies a geometric acuteness property derived from the M-reduced basis: for any integer combination z of basis vectors excluding uᵢ, 2|⟨z, uᵢ⟩ₘ| ≤ ‖z‖ₘ².
• Defines a triangulation T of R^d ∖ {0} using simplices formed from integer combinations of the M-reduced basis vectors.
• Constructs a finite difference scheme on the triangulation that approximates the generalized eikonal equation: ‖∇D(z)‖ₘ⁻¹ = 1.
• Analyzes the error in the numerical solution by examining the scalar product between vertices of simplices in the triangulation, proving non-negativity of angles to ensure stability and accuracy.

Experimental results

Research questions

• RQ1 Does the numerical error of the lattice basis reduction-based fast marching algorithm increase with the anisotropy ratio κ(M) in the case of a constant Riemannian metric?
• RQ2 Is the algorithm's accuracy robust to extreme anisotropy, particularly in an average sense over all spatial directions?
• RQ3 Can the method be extended to four-dimensional domains while preserving its accuracy and low computational complexity?
• RQ4 What geometric and algebraic properties of the M-reduced basis ensure the stability and boundedness of the numerical error?
• RQ5 How does the error in this algorithm compare to alternative methods in the presence of high anisotropy, especially in terms of both magnitude and directional dependence?

Key findings

• The numerical error of the algorithm is independent of the anisotropy ratio κ(M) when averaged over all space directions, even for large values such as κ(M) = 100.
• The error bound grows only in the worst-case scenario, but in the average case, it remains uniformly bounded regardless of anisotropy magnitude.
• The method maintains computational complexity O(N ln N + N ln κ(M)) under the condition that N ≫ κ(M), which holds in typical applications.
• The algorithm is successfully extended to four-dimensional domains, demonstrating its generalizability beyond 2D and 3D.
• Numerical experiments confirm a 7-fold error reduction and 200-fold speedup compared to an alternative method on a 1200×1200 grid with κ(M) = 100.
• The acuteness property of the triangulation—ensured by the M-reduced basis—guarantees non-negative angles between simplex vertices, which underpins the error stability.