Soft-Argmax for the Projective Plane via the Veronese Embedding
From horizon detection to fibre structures in X-ray imaging, many vision tasks recover lines via peak detection in Hough space $H=S^1\times\mathbb{R}$, the domain of orientation-offset pairs $(θ,ρ)$. Differentiable pipelines extract coordinates via \emph{soft-argmax}, a probability-weighted average that is only meaningful in a globally linear space. However, $(θ,ρ)$ and $(θ+π,-ρ)$ describe the same undirected line, so $H$ double-covers the space of undirected lines $H/\mathbb{Z}_2$: a Möbius strip, obtained by identifying each pair under $\mathbb{Z}_2$ action. Soft-argmax operates on the cover $H$, but since $H/\mathbb{Z}_2$ admits no linear structure, it tears geometrically adjacent line...
| Authors | Benjamin El-Zein, Dominik Eckert, Paul Zech et al. |
|---|---|
| Primary category | cs.CV |
| Published | 1 September 2026 |
| Last updated at source | 1 September 2026 |
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