Grids are a general representation for capturing regularly-spaced information, but since they are uniform in space, they cannot dynamically allocate resolution to regions with varying levels of detail. There has been some exploration of indirect grid adaptivity by replacing uniform grids with tetrahedral meshes or locally subdivided grids, as inversion-free deformation of grids is difficult. This work develops an inversion-free grid deformation method that optimizes differential weight to adaptively compress space. The method is the first to optimize grid vertices as differential elements using vertex-colorings, decomposing a dense input linear system into many independent sets of vertices which can be optimized concurrently. This method is then also extended to optimize UV meshes with convex boundaries. Experimentally, this differential representation leads to a smoother optimization manifold than updating extrinsic vertex coordinates. By optimizing each sets of vertices in a coloring separately, local injectivity checks are straightforward since the valid region for each vertex is fixed. This enables the use of optimizers such as Adam, as each vertex can be optimized independently of other vertices. We demonstrate the generality and efficacy of this approach through applications in isosurface extraction for inverse rendering, image compaction, and mesh parameterization.

翻译：网格是捕获规则间距信息的通用表示方法，但由于其在空间上具有均匀性，无法动态地将分辨率分配给具有不同细节层次的区域。由于网格的无翻转变形较为困难，已有研究通过用四面体网格或局部细分网格替代均匀网格来间接实现网格自适应性。本研究提出了一种无翻转网格变形方法，通过优化差分权重来自适应压缩空间。该方法首次利用顶点着色将网格顶点作为差分元素进行优化，将稠密的输入线性系统分解为多个可并行优化的独立顶点集合。此方法进一步扩展至具有凸边界的UV网格优化。实验表明，这种差分表示相比更新顶点外显坐标能产生更平滑的优化流形。通过分别优化着色中的各顶点集合，局部单射性检验变得直接——因为每个顶点的有效区域是固定的。这使得Adam等优化器得以应用，每个顶点可独立于其他顶点进行优化。我们通过等值面提取（用于逆向渲染）、图像压缩和网格参数化等应用，证明了该方法的通用性与有效性。