Distance Marching Algorithms

• Distance marching is a suite of algorithms that compute distance functions, geodesic maps, and shortest paths using numerical propagation techniques.
• It employs methods such as Fast Marching, FM-ASR, and FM-LBR, and applies to fields like robotics, medical imaging, and deep generative modeling.
• The approach balances theoretical guarantees, computational complexity, and practical applications with high scalability and precision.

Distance marching comprises a suite of algorithms and methodologies for computing distance functions, geodesic maps, shortest paths, and related characteristics over continuous or discrete domains. It is central to fields such as optimal control, Eikonal PDEs, level set methods, shape reconstruction, sub-Riemannian geometry, and generative modeling. The unifying principle is the numerical propagation of a "distance" (not always Euclidean) outward from a source or manifold by sequential or parallel updates governed by causality and upwind schemes. Contemporary formulations span Finsler and Riemannian metrics, anisotropic and asymmetric norms, high-dimensional and manifold-valued spaces, as well as hybrid implicit–explicit representations relevant for 3D geometric deep learning.

1. Mathematical Formulations of Distance Marching

At its core, distance marching addresses the computation of escape times or geodesic distances with respect to a prescribed local metric. In the classical form, the Eikonal equation

$$|\nabla T(x)| F(x) = 1\quad \text{with}\quad T|_{\partial\Omega} = 0$$

characterizes the arrival time from source or boundary $\partial\Omega$, with $F(x)$ a position-dependent (and potentially anisotropic) speed function (Yang et al., 2015). For Finsler or sub-Riemannian metrics, the direct generalization is an anisotropic Eikonal

$F^*_z\left(-\nabla T(z)\right) = 1$

with $F^*_z$ the dual norm to a possibly asymmetric $\mathcal{F}_z$ (Mirebeau, 2012). In level set approaches, the signed distance function $\phi$ satisfies $|\nabla\phi|^2 = 1$, with further constraints for higher regularity (Salac, 2011).

In generative modeling, the distance function $u_\theta:\mathbb R^D\to\mathbb R$ and the associated direction field $v_\theta:\mathbb R^D\to\mathbb R^D$ are learned to encode unsigned distance to a data manifold, forming the core of time-unconditional "Distance Marching" (Wang et al., 3 Feb 2026).

2. Algorithms and Discretization Methods

The principal computational scheme is the Fast Marching Method (FMM), a single-pass, upwind, heap-based approach. For isotropic Eikonal equations on Cartesian grids, FMM propagates the front by always updating the current minimal arrival time node and its upwind neighbors, using local finite-difference schemes and quadratic updates. In parallel settings, domain decomposition and narrow band strategies yield efficient distributed algorithms with near-linear scaling to $10^5$ cores, preserving $O(N\log N)$ complexity per pass (Yang et al., 2015).

For anisotropic or Finsler metrics, the Fast Marching using Anisotropic Stencil Refinement (FM-ASR) adapts local stencils by recursively refining triangle fans to achieve $F$-acuteness. The complexity of stencil construction scales poly-logarithmically with the metric's anisotropy ratio $\kappa$, resulting in average-case operation counts $O(N \ln^3\kappa)$ (Mirebeau, 2012). The FM-LBR variant addresses Riemannian or sub-Riemannian cases by using local lattice basis reduction to compute stencils aligned with the metric's principal directions (Sanguinetti et al., 2015).

In deep 3D shape synthesis, "distance marching" describes the iterative traversal along SDF zero-crossings in a tetrahedral mesh. Differentiable Marching Tetrahedra layers leverage this principle for high-resolution mesh extraction, supporting adaptive, memory-efficient refinement (Shen et al., 2021).

3. Applications Across Scientific Domains

Distance marching and its algorithmic variants underpin several domains:

• Motion planning and path finding: Optimal control problems, robotics navigation, and scene traversal exploit escape time computations under Finsler metrics (Mirebeau, 2012).
• Image segmentation and medical imaging: Fast marching is used for contour finding, vessel segmentation, and morphological filtering via geodesic and orientation-lifted distances (Sanguinetti et al., 2015).
• Level set methods and reinitialization: Accurate and smooth reinitialization of level set functions by augmented FMM maintains high-fidelity curvature fields necessary for interface propagation problems (Salac, 2011).
• 3D shape synthesis and reconstruction: Hybrid implicit-explicit frameworks employ distance marching to enable fine-detail isosurfacing and topology changes, outperforming uniform-grid baselines (Shen et al., 2021).
• Deep generative modeling: Distance Marching eliminates time-conditioning by learning a canonical distance map and marching directions, improving sample fidelity, reducing sampling steps, and enabling OOD detection through the learned distance field (Wang et al., 3 Feb 2026).

4. Theoretical Guarantees and Computational Complexity

Sequential FMM and its advanced generalizations provide strong causality and upwind propagation properties, crucial for uniqueness and accuracy. Theoretical complexity for basic FMM is $O(N \log N)$, where $N$ is the number of gridpoints. For FM-ASR, worst-case costs scale as $O(N \kappa \ln\kappa + N \ln N)$, with average-case $O(N \ln^3\kappa)$ over grid orientation (Mirebeau, 2012). Parallel FMM implementations achieve near-linear speedup and high efficiency by balancing band width ($\delta s$) and subdomain size (Yang et al., 2015).

Level set reinitialization via augmented FMM attains second-order accuracy for the signed distance and gradient fields and first-order accuracy for curvature, with overall $O(N \log N)$ complexity (Salac, 2011).

In generative modeling, loss reweighting and the dual field approach resolve multi-noise-level ambiguity and preserve convergence toward the data manifold (Wang et al., 3 Feb 2026). Quantitative benchmarks in high-dimensional generative domains show clear improvements in FID and sample complexity compared to time-conditioned and unconditional baselines.

5. Specialized Metrics: Finsler, Riemannian, and Sub-Riemannian Distance Marching

Anisotropy and geometric constraints require custom metric formulations:

• Finsler metrics: General, possibly asymmetric, convex and 1-homogeneous norms. FM-ASR handles strong anisotropy and directionality in 2D.
• Riemannian metrics: Symmetric, positive-definite local metric tensors; FM-LBR or standard FMM variants are applicable.
• Sub-Riemannian (SR) metrics: Constraints on allowable tangent vectors, as in the $SE(2)$ group for orientation-lifted path planning. Riemannian approximations enable the use of FMM via penalized directions, maintaining accurate SR geodesics and distance spheres (Sanguinetti et al., 2015).

For SR manifolds, geodesics can be validated against closed-form solutions, and the FM-LBR achieves first-order accuracy with dramatic improvements in computational cost over iterative PDE solvers.

6. Distance Marching in Modern Deep Generative and Geometric Learning

In high-resolution 3D shape synthesis, the DMTet framework applies distance marching through SDF field definition on deformable tetrahedral grids. The differentiable Marching Tetrahedra operation enables supervised mesh extraction and loss backpropagation. Adaptive grid refinement enumerates only those tetrahedra intersecting the zero-level set, supporting efficient, detail-preserving geometry reconstruction (Shen et al., 2021).

In generative modeling, the Distance Marching paradigm learns both the distance magnitude and directional fields, supporting sphere-tracing and gradient-descent sampling through field-driven updates. Losses that concentrate on closer targets in data-space remove denoising ambiguity and support advanced tasks such as adaptive early stopping and out-of-distribution detection (Wang et al., 3 Feb 2026).

7. Empirical Performance and Benchmark Results

Distance marching methods have been extensively benchmarked:

• FM-ASR exhibits speedups of one to four orders of magnitude versus ordered Gauss-Seidel methods in high-anisotropy regimes and maintains superior accuracy-complexity trade-offs (Mirebeau, 2012).
• FM-LBR for sub-Riemannian $SE(2)$ achieves first-order accuracy with respect to spatial grid size, with wall times orders of magnitude smaller than PDE-based iterative schemes, and enables real-time vessel segmentation (Sanguinetti et al., 2015).
• Augmented FMM for level set reinitialization achieves second-order $L_2$ accuracy for the distance and gradient fields, even for complex curve topologies (Salac, 2011).
• DMTet recovers fine details and thin structures in 3D shape benchmarks, outperforming ConvOnet and voxel-GAN on Chamfer, normal consistency, and visual fidelity metrics (Shen et al., 2021).
• Distance Marching generative models yield FID improvements of $9.5$–$24.7$\% across architectures and reach target FID with $60$\% of the sampling steps compared to flow matching, while providing robust OOD detection via the distance field (Wang et al., 3 Feb 2026).

• (Mirebeau, 2012) Mirebeau, Efficient Fast Marching with Finsler metrics
• (Yang et al., 2015) Zhao et al., A highly scalable massively parallel fast marching method for the Eikonal equation
• (Salac, 2011) Salac, The Augmented Fast Marching Method for Level Set Reinitialization
• (Sanguinetti et al., 2015) Sanguinetti et al., Sub-Riemannian Fast Marching in SE(2)
• (Shen et al., 2021) DMTet: Deep Marching Tetrahedra: a Hybrid Representation for High-Resolution 3D Shape Synthesis
• (Wang et al., 3 Feb 2026) Distance Marching for Generative Modeling