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Differentiable Voxelization of Surface Representations
Tobias Djuren, Ugo Finnendahl, Markus Worchel, Hendrik Meyer, Marc Alexa
TL;DR
Surface-based shape optimization needs derivatives linking surface parameters to volumetric energies. The paper derives winding-number gradients and an efficient voxel-grid method, demonstrating applications in intersection resolution, bandsaw manufacturability, and space tiling. These results support differentiable optimization across several volumetric shape tasks.
Problem
Surface representations are useful for modeling, but gradient-based optimization with volumetric energies requires derivatives of volume representations with respect to surface parameters.
Method
The paper derives gradients for winding numbers and computes them efficiently on regular voxel grids for triangle meshes using closed-form integration.
Results
The approach is demonstrated for resolving self-intersections, deforming meshes for three-direction bandsaw cutting, and creating shapes close to tiling 3D space.
Takeaways & Limitations
Differentiable voxelization transfers voxel-grid computational simplicity to explicit surface representations and supports applications such as constructive solid geometry and grid-based shape comparison.
Takeaways & Limitations
Self-intersections are resolved only approximately because voxel-integrated energies can miss intersections in half-empty voxels and negative winding numbers can hide intersections.
Abstract
from arXiv · showhide
Different shape representations facilitate different computations. Surface representations, in particular meshes, are often used for modeling, whereas volume representations are useful for spatial queries such as intersection or containment. Optimizing a surface representation based on a volumetric properties by gradient descent requires the derivatives of the volume relative to its bounding surface. We derive this gradient for winding numbers and show that it can be efficiently computed for volumetric values sampled on a regular grid (voxel representation) and surface parameters based on vertex sets (triangle meshes). This enables an efficient solution for a variety of optimization problems. We demonstrate the practical use of this approach at the examples of deforming meshes to resolve intersections, being manufacturable by cutting with a bandsaw from three directions, and creating shapes that are close to tiling 3D space.
1 Introduction
Solid shapes can be represented by surfaces or volumes, which support different computations. The paper derives efficient gradients connecting surface parameters to voxelized volume energies and demonstrates several optimization applications.
- Surface representations describe boundaries and include polygon meshes, parametric surfaces, and subdivision surfaces.
- Volume representations capture occupied space and support queries such as intersections and constructive solid geometry, with regular voxel grids offering efficient computation.
- Bandsaw-compatible modeling motivates optimizing a surface-controlled shape whose energy depends on its volumetric representation.
- The paper derives winding-number gradients from surface parameters to volumetric values and computes them efficiently for voxel grids and triangle meshes.
- The approach is evaluated through self-intersection resolution, bandsaw-cuttable deformation, and deformation toward space tiling.
2 Related Work
Prior work established voxelization, winding-number computation, and differentiable volumetric representations, but often relied on per-voxel integration or acceleration structures. This paper uses box-filtered winding numbers and regular-grid structure for efficient differentiable voxelization.
- Rasterization and Voxelization: Voxelization extends rasterization from pixels to voxels, while filtering reduces aliasing and supports boundary-based integration and differentiation.
- Rasterization and Voxelization: The primal voxelization uses a 3D scanline generalization that inherently handles overlapping shapes and is optimized for meshes.
- Volumetric Representations in Gradient-based Optimizations: Volumetric representations such as density and signed distance functions are widespread in computer vision and machine learning, where conversion between representations is common.
- Winding Numbers: Winding numbers support geometry-processing tasks including mesh booleans, constructive solid geometry, tetrahedral meshing, surface reconstruction, and mesh repair.
- Winding Numbers: Box-filtered winding numbers exploit regular voxel grids to avoid acceleration data structures, enabling highly efficient differentiable voxelization compared with Gaussian-smoothed approaches.
3 Differentiable Voxelization
The paper differentiates pre-filtered winding-number voxelizations with respect to surface parameters, including self-intersecting surfaces. For piecewise linear surfaces and box filters, it derives efficient voxelization and differentiation procedures for gradient-based optimization.
- 3 Differentiable Voxelization: Voxelization samples the winding-number field at regular-grid voxel centers, while pre-filtering produces values that vary continuously with the surface.For intersection-free surfaces, the winding number reduces to an inside/outside indicator; pre-filtering avoids derivatives that are zero almost everywhere or undefined at voxel centers.
- 3 Differentiable Voxelization: For gradient computation, the method evaluates smoothed winding numbers and their derivatives, then combines them with the energy’s derivatives with respect to the voxel grid.The voxel grid W supplies the objective E, whose optimization depends on surface parameters governing the surface.
- 3.3 Efficient Closed-Form Integration: Piecewise linear surfaces with box filters admit highly efficient voxelization and differentiation using closed-form integration and shadow contributions.The algorithm updates voxels below intersecting simplices; for triangle meshes, triangle subdivisions support integration across multiple voxels.
- 3 Differentiable Voxelization: Reynolds’ transport theorem converts the derivative into a boundary term because the winding-number field is piecewise constant and its interior derivative vanishes.The derivation partitions space into constant-winding-number regions and then simplifies their boundary integrals to one surface integral, including self-intersections.
- 3 Differentiable Voxelization: The resulting surface integral is independent of winding numbers and matches the result for intersection-free surfaces.This provides a general continuous derivative for winding numbers on possibly self-intersecting surfaces.
4 Evaluation and Applications
The evaluation shows that differentiable voxelization is computationally practical and supports applications including derivative-based self-intersection resolution and bandsaw-oriented shape deformation.
- 4.1 Evaluation: Runtime stays below one second for voxelization and gradient computation on grids up to 1024^3 voxels, even with meshes exceeding 100,000 faces.Timings were averaged over 30 repetitions on an Intel i9-12900K CPU.
- 4.1 Evaluation: The implementation is not fully optimized, while preliminary naive GPU experiments indicate speedups of 3.6× for voxelization and 14.5× for gradients.The reported GPU results use meshes with more than 100,000 faces on 1024^3 grids.
- 4.1 Evaluation: DVX produces expected numerical derivatives by sampling a smoothed winding-number field, unlike DOPH, whose derivatives are zero almost everywhere.DOPH uses a non-smoothed, piecewise-constant winding number; DVX derivatives match numerical results exactly.
- 4.2 Self-Intersection Resolving: Gradient descent with an ARAP-regularized self-intersection energy resolves intersections, although the voxel-based objective provides only an approximate resolution.The method penalizes voxels whose winding-number values exceed one and uses ARAP regularization to prioritize rigid deformations.
- 4.3 Shape Deformation for Bandsaw Cutting: The bandsaw application deforms input meshes to improve cutability while preserving their overall shape compared with naive silhouette-based cutting.The cut-out surface is represented as the intersection of extruded silhouettes from three orthogonal directions.
5 Discussion and Future Work
The method transfers voxel-grid computational simplicity to explicit surface representations, enabling difficult optimization problems and differentiable operations on triangular meshes. The authors identify further applications and extensions while noting that the current work focuses mainly on triangular meshes.
- Voxel grids simplify operations, but this work transfers that simplicity to common explicit surface representations.The authors specifically connect this transfer to triangular meshes and compare it with the prevalence of surface representations.
- Differentiable voxelization can support CSG, morphological operators, and grid-based machine-learning applications while differentiating back to triangular meshes.The discussion names intersections, differences, unions, morphological operators, and direct differentiable shape comparison in R3.
- The approach closes the mesh-to-voxel direction of a previously studied inverse loop, potentially enabling self-supervised optimization within the loop.The paper contrasts its direction with prior differentiable generation of surface meshes from voxel representations.
- The current work focuses mainly on triangular meshes, with extensions to other surface representations and faster smoothing filters left for future evaluation.The authors also identify parallelization as a possible route to reduce computation time.
A Detailed Results for the Bandsaw Cutting
The bandsaw-cutting evaluation reports how cut-volume differences and deformation distances are quantified for TOSCA categories. Relative improvement measures unnecessary material removed from the initial meshes.
- Table 1 reports average cut volume differences E_cut for each TOSCA category, with both values scaled by 10^3.
- Relative improvement is defined as 1 − E_cut(V̄)/E_cut(V), measuring unnecessary material removed through optimization.
- Two-sided Hausdorff distance H and mean surface distance M quantify mesh deformation relative to the bounding-box diagonal.
B Pseudocode
The pseudocode evaluates voxel occupancy and its derivatives by clipping simplices against voxel boundaries, integrating over clipped polytopes, and propagating adjoints to simplex vertices.
- Primal evaluation: The voxelization algorithm initializes an occupancy grid, computes voxel size and simplex bounds, and accumulates simplex contributions over covered voxels.Voxel grids span [−1,1]^d, and simplex vertices, outward normals, and the normal sign are used during evaluation.
- Primal evaluation: Clipping splits a barycentric-coordinate polytope by voxel-boundary hyperplanes into below and above parts for each axis.The procedure iterates over relevant voxel indices and retains the remaining polytope for subsequent dimensions.
- Primal evaluation: For each clipped polytope, projected area and average height provide the quantities used to accumulate voxel contributions.ProjectedArea integrates over the projected footprint, while AverageHeight integrates the final coordinate relative to surface measure.
- Derivatives: Forward- and backward-mode derivative algorithms reuse the primal clipping and coordinate computations to differentiate voxelization.Backward mode weights voxel adjoints and distributes them to simplex vertices through barycentric coordinates and surface area.
- Derivatives: The backward pass maps occupancy adjoints to vertex adjoints after computing clipped-polytope contributions.The pseudocode explicitly labels adjoint weighting and distribution to simplex vertices.