Differential Locally Injective Grid Deformation and Optimization

2026-01-08

arXiv.org

AI Summary

Uniform grids struggle to adapt to fine-scale geometric variations, and existing flip-free deformation methods suffer from limited computational efficiency. This work proposes a flip-free deformation framework based on a differential mesh representation, where spatially adaptive compression is achieved by optimizing differential weights and vertices are treated as differential units for optimization. The approach innovatively introduces vertex coloring to decouple the dense linear system into multiple independent vertex sets that can be optimized in parallel, significantly improving efficiency while preserving local injectivity. Combined with the Adam optimizer and UV-mesh refinement, the method yields smoother optimization landscapes and enhanced generalization across diverse tasks, including inverse rendering, iso-surface extraction, image compression, and mesh parameterization.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• grid deformation
• inversion-free
• adaptive resolution
• differential representation
• local injectivity

Innovation

• differential grid deformation
• inversion-free optimization
• vertex coloring
• local injectivity
• adaptive grid resolution