McGrids: Monte Carlo-Driven Adaptive Grids for Iso-Surface Extraction

TL;DR

This paper tackles the high computational burden of iso-surface extraction from neural implicit fields by introducing McGrids, an adaptive tetrahedral grid generated through an iterative Monte Carlo sampling process. A density function $d(x) = κ /( γ + |f(x) - α| )$, weighted by curvature, guides point sampling and, together with Centroidal Voronoi Tessellation, yields a progressively refined grid around the iso surface. After refinement, the surface is extracted with marching tetrahedra, achieving high fidelity with far fewer field queries and lower memory usage than dense grid baselines. The approach is validated on analytical SDFs and learned implicit fields, and is demonstrated to integrate into differentiable multiview reconstruction pipelines, offering a plug-and-play tool for efficient high-detail surface meshes.

Abstract

Iso-surface extraction from an implicit field is a fundamental process in various applications of computer vision and graphics. When dealing with geometric shapes with complicated geometric details, many existing algorithms suffer from high computational costs and memory usage. This paper proposes McGrids, a novel approach to improve the efficiency of iso-surface extraction. The key idea is to construct adaptive grids for iso-surface extraction rather than using a simple uniform grid as prior art does. Specifically, we formulate the problem of constructing adaptive grids as a probability sampling problem, which is then solved by Monte Carlo process. We demonstrate McGrids' capability with extensive experiments from both analytical SDFs computed from surface meshes and learned implicit fields from real multiview images. The experiment results show that our McGrids can significantly reduce the number of implicit field queries, resulting in significant memory reduction, while producing high-quality meshes with rich geometric details.

Figures (11)

• Figure 1: McGrids can efficiently extract a highly adaptive and accurate mesh from complex implicit fields. Left: Mesh extracted using Marching Cubes with a voxel grid resolution of 1024 (RAM usage: $>$150GB, #Queries: 1,073,741,824). Middle: Mesh extracted by McGrids (RAM: $<$ 10 GB, #Queries: 62,012,058). Right: Ground truth mesh. Top row: Zoom in for better visualization.
• Figure 2: A simplified 2D illustration of McGrids that builds an adaptive grid around the object's iso-surface. With given implicit function (SDF), we define a probability density function (PDF), i.e. Eq. \ref{['eq:density_function']}, to guide the generation of grid points. We initialize approximation with a uniform distribution, then iteratively sample new grid points according to \ref{['eq:density_function']}. The positions and the implicit values of new samples are used to refine the PDF estimation. Once the iterations converge, the tetrahedron grids with all the sample point positions are used to extract the final mesh using marching tetrahedron.
• Figure 3: Underlying data structure of McGrids and the process of inserting a new point into McGrids. From left to right: (a) Sample points; (b) Delaunay triangulation and dual Voronoi diagram; (c) Insert a new point to McGrids; (d) CVT relaxation that moves the newly inserted point to its Voronoi centroid; (e) Updated McGrids.
• Figure 4: 2D Illustration of the McGrids sampling process. For the current McGrids, we calculate discrete probabilities for each Voronoi region, and normalize and aggregate them to form a discrete cumulative distribution. By randomly sampling a number (0.684), an indexed Voronoi region (#4) is selected. This chosen region is then tessellated into multiple tetrahedra. After that, we calculate the volumes of these tetrahedra and form another discrete cumulative distribution for sampling. Using the selected tetrahedron (#2), we sample a new point and insert it back into the McGrids.
• Figure 5: Visual comparisons of the extracted meshes. We compare McGrids with Marching Cubes marchingcubes1987, Octree method Kazhdan2007octree, and Neural Dual Contouring chen2022MDC with a resolution of $256^3$. For a fair comparison, we run McGrids with a terminating threshold of $1e^{-4}$. We can see that McGrids extracts more detailed and accurate meshes from complex implicit fields.