Deformable Polygonal Flow Matching with Informed Priors and Hierarchical Graph Constraints

This paper presents a novel method, called Deformable Polygonal Flow Matching (DPFM), for the generation of polygonal arrangements such as jigsaw puzzles and floor plans. DPFM is a Flow Matching framework that enables the generation process to deform, rotate, and translate polygons while decoupling these transformation, allowing to toggle them individually. Able to combine the spatial reasoning capabilities of arrangement models with the flexibility of position-based models, it covers a wide range of applications within a unified formulation, from noiseless puzzle solving using rigid alignments to unconstrained floor plan generation. We represent data using a hierarchical graph composed of a topological subgraph encoding connectivity information and semantics (such as room types for floor plans), and a geometrical subgraph encoding the 1D polygonal loop of each shape. DPFM also leverages Flow Matching's arbitrary prior distributions for geometric constraints by designing priors with domain knowledge. Rather than starting the generation process from uninformed distributions, the generation is constrained through the informed priors at the initialization stage. The qualitative and quantitative evaluations of our method, ran on the RPLAN and jigsaw puzzle datasets, demonstrate strong performance. DPFM outperforms taskspecific methods, becoming the new state-of-the-art for 2D arrangement generation. Our results show that DPFM is able to solve novel tasks, such as puzzle denoising, where pieces are reconstructed from noisy versions and arranged into a valid puzzle in parallel.