Differential Locally Injective Grid Deformation and Optimization

TL;DR

This paper tackles the inefficiency of uniform grids in representing spatial detail by introducing an inversion-free grid deformation framework. It represents each vertex as a differential element via convex combinations of its neighbors and uses graph coloring to decouple optimization into independent, parallel color-classes, enabling the use of optimizers like Adam. Barrier energies are employed to prevent inversions, and the method is demonstrated across 2D grids, UV parameterization, image compaction, and differentiable isosurface extraction, showing smoother optimization and preserved injectivity compared to direct vertex updates. The approach preserves regular topology while achieving adaptive, high-detail representations, with practical impact on inverse rendering, texture mapping, and geometry processing.

Abstract

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.

Figures (15)

• Figure 1: Previous work flexicubes deforms each vertex along half of each grid cell. This work optimizes vertices within the kernel of its neighbors. Valid positions in the current configuration are shown in blue.
• Figure 2: The unconstrained convex combination of a polygon may lead to vertices exiting the polygon's interior, shown in red. When vertices are inside the polygon, some regions lead to non-injectivity, shown in orange. The region which has no inversion is the kernel of the polygon, shown in green. To ensure vertices stay in the kernel, explicit checks for inversion are required and optimization is reverted if elements flipped.
• Figure 3: To optimize vertices in a uniform grid, alternating steps are taken (Eq. \ref{['eq:grid-alt']}), fixing all odd vertices, and optimizing even vertices. Then, all even vertices are fixed, and odd vertices are optimized. The checkerboard region indicates regions vertices can be optimized within initially.
• Figure 4: To optimize a differential mesh representation, first vertices are colored, then all vertices of one color are optimized concurrently while other colors are fixed. From this it is easy to verify if a vertex is not in its one-ring neighborhood. Different shading indicates the one-ring for different vertices, showing the optimizable regions for each.
• Figure 5: Example of direct optimization of position values per grid vertex, with alternating optimization on even and odd vertices, and representing vertices as convex combinations of their neighbors. There is no visible bias introduced, and vertices optimize smoothly. The zoomed region is indicated by the red box.