Differential Locally Injective Grid Deformation and Optimization

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.