cs.GRAug 16, 2026
SaveDifferentiable Voxelization of Surface Representations
Organizations: Technische Universität Berlin, Germany
Abstract
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.
Figures & tables
Figure 1. We present an efficient technique for differentiating voxel values with respect to the surface that generates the voxel grid. This enables the solution of various problems that can easily be solved on regular voxel grids, such as intersection detection, for surface representations like triangle meshes. The technique can be readily integrated into gradient-based optimization frameworks like PyTorch ( Paszke et al., 2019 ) . Moving the highlighted vertex (left) in the indicated direction changes the voxelization of all adjacent triangles (right). We discuss various applications, like a) the resolving of self-intersections, b) optimized cuts for manufacturing with bandsaws, and c) shape deformation to form repeatable tilings that fill the .
Figure 2. We can differentiate voxelizations, i.e., point-sampled winding numbers, for arbitrary surfaces. This example shows piecewise linear surfaces and circles in 2D. Our derivatives match the finite difference reference. This example uses Monte Carlo integration and a box filter of diameter 3.
Figure 3. The winding number field of a surface depicted in the left-most box is the sum of the four patch-wise contributions . Each defines for every point in space how often the patch (blue) is intersected in direction . As the ray intersects the patch at most once, is piecewise constant and either or depending on the normal relative to the ground plane.
Figure 4. Given a polygon, the integrated winding numbers in each voxel are updated whenever a simplex intersects a voxel above the voxel. The volume of the intersection of the signed shadow solid volume (green for positive and red for negative contribution) and a voxel (small boxes) can be computed as described in Sec. 3.3 and added to the value of each voxel, resulting in an update of all voxel values ‘below’ the simplex segment. Therefore, the integrated winding number of all voxels in the highlighted grid column can be computed as the sum of ‘shadow’ contributions of the five blue segments.
Figure 5. For high grid resolutions (up to voxels in each dimension), the runtime stays below a second even for meshes over faces for voxelization as well as the gradient computation. All timings are averaged over repetitions and were computed on an Intel i9-12900K CPU. The number of faces is shown in parentheses.
Figure 6. Luo et al. (2024) propose differentiable voxelization by automatically differentiating through a (non-smoothed) winding number ( DOPH ). Since the winding number is piecewise constant, the derivatives of their voxelization are zero if computed numerically with finite differences (FD), and their automatic derivatives are only spuriously non-zero very close to the surface. In contrast, our technique ( DVX ) is based on the smoothed winding number, which leads to expected numerical derivatives that match our results exactly.
Figure 7. Differentiable voxelization allows resolving self-intersections by an ARAP deformation of the input shape.
Figure 8. Left: Wooden figure that was created by cutting along handmade cut silhouettes ( Gauthier, 2019 ) (used with permission). Right: The cut out surface is defined as the intersection of extruded silhouettes.
Figure 9. The energy measures the volume difference between the deformed mesh and the deformed cut out shape . Larger vertex distances of the deformed mesh to are shown in red (left). is the discretized shape of where yellow voxels have high volume and green voxels have low volume. Voxels with volume lower than are hidden. is projected to the silhouette shapes and that are used to generate the cut shape . For visualization of the volume difference ( ), we highlight all voxels in red that are part of the shape but not in .
Figure 10. When optimized (bottom row), much more material that does not belong to the actual shape can be removed by cutting, when compared to the cut-out shape for meshes without optimization (top row). All voxels of that do not belong to are highlighted in red. The color of the optimized triangle meshes shows the difference to the initial meshes. Gray voxels are part of the voxelization of the triangle meshes and red voxels are additional voxels that are not removed by cutting. The relative improvements denote how much unnecessary material of the initial meshes can be removed with the optimization ( ). Minimal deformations, such as straightening the fingers of the hand or adjusting the tail and ears of Tweety and Bunny, significantly improve the shapes’ cutability.
Figure 11. We test the ‘cutability’ of optimized cut shapes by cutting various optimized shapes using a bandsaw (left top and bottom) and a polystyrene foam cutter (bottom right). The references for the cut shapes are at the top right.
Figure 12. Differentiable voxelization can be used to optimize tilings in to periodically fill the space. Left: Tiling procedure for a single shape instance. Right: Each row belongs to one optimized packing. The packed meshes (first column) can be repeated along all three axis to fill the space. The packed meshes are composed of one or more deformed meshes that are all similar (within a class of animals). When voxelized and repeated in all dimensions, the packing fills (third column). The last column shows the interior of the space filling voxels by hiding some instances (for the space filling for multiple animal types with some hidden instances, all Tweety birds are shaded in reds and all Spots are shaded in blues).