HyperbolicDiffusion: Sharp & Scalable Tiled Generation on the Hyperbolic Plane

2026-08-04

AI Summary

This work addresses the challenge of image generation in the hyperbolic plane, where the absence of Euclidean-like rectangular canvases and exponential area growth render conventional overlapping-window diffusion methods ineffective. The authors propose the first training-free approach to hyperbolic image synthesis by modeling window placement as a compact dynamic programming problem via Hyperbolic Blooming Cover. A shared implicit canvas is constructed using persistent surface IDs, enabling standard diffusion models to denoise local windows independently before fusing them coherently. To resolve blurriness and inconsistencies at multi-window boundaries, a geometry-aware two-stage re-noising mechanism is introduced. The method produces sharp, view-consistent images that support reprojection, offering a prompt-driven generation framework for artworks in the style of Escher’s *Circle Limit* series.

Abstract

Planar tiled diffusion denoises overlapping windows of one rectangular canvas. The hyperbolic plane has no such canvas, and its area grows exponentially with radius. We introduce HyperbolicDiffusion, a training-free method for generating finite visual fields directly on the hyperbolic plane H2. Our Hyperbolic Blooming Cover reduces window placement to a compact dynamic program that runs in seconds while providing strong theoretical guarantees. Permanent surface IDs form a shared latent canvas: a standard diffusion model denoises local windows, whose predictions are fused back onto H2. Because curvature causes residual disagreement and blur at multi-window junctions, a geometry-derived second stage re-noises and repairs precisely those regions. The resulting fields are sharp, reprojectable, and consistent across viewpoints, providing a prompt-driven generative counterpart to Escher's Circle Limit series.

Problem

• hyperbolic plane
• tiled generation
• diffusion models
• geometric consistency
• visual field generation

Innovation

• HyperbolicDiffusion
• hyperbolic plane
• tiled generation
• diffusion models
• geometric consistency