Patchwork: A Polyhedral Implicit Representation for Highly Detailed 3D Shapes

We introduce patchwork, a highly expressive shape representation that explicitly encodes 3D shapes as unions of polyhedra while admitting a simple closed-form zero-level-set equation. It is grounded in a rigorous mathematical framework that yields provable complexity bounds and guarantees approximation of arbitrary shapes to arbitrary precision. To fit patchwork to a given shape, we pair conventional gradient-based optimization with a novel combinatorial matching scheme. Experiments show that our approach achieves superior quality with 3–6$\times$ fewer parameters compared to existing alternatives on average, and scales to highly detailed artist-created assets. Patchwork further supports explicit polygonalization and conforming volume mesh extraction, making it a versatile, scalable, yet compact representation for downstream tasks, with potential for geometric learning.