让网格模型通过梯度优化实现体积化操作，提升建模与制造效率。
Differentiable Voxelization of Surface Representations
- 基于网格顶点的表面参数，推导出体素体积对表面的可微分梯度。
- 支持在规则网格上高效计算梯度，实现交集消除、三向切割制造等优化。
- 适用于几何优化、工业制造与空间密铺设计，适合计算机图形学研究者。
不同形状表示方式适用于不同计算任务。表面表示（如网格）常用于建模，而体表示则便于空间查询（如相交或包含判断）。通过梯度下降优化表面表示以满足体素性质，需计算体积对边界表面的导数。本文推导了绕数（winding numbers）对应的梯度，并证明其可在规则网格采样（体素表示）与基于顶点的表面参数（三角网格）下高效计算。该方法可有效求解多种优化问题，例如通过变形网格消除几何交叠、生成可从三个方向用带锯切割制造的模型，以及构造接近三维空间密铺的形状。
原文摘要 · Abstract (English)
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.
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