Soft-Argmax for the Projective Plane via the Veronese Embedding

Abstract

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 $(\theta,\rho)$. Differentiable pipelines extract coordinates via \emph{soft-argmax}, a probability-weighted average that is only meaningful in a globally linear space. However, $(\theta,\rho)$ and $(\theta+\pi,-\rho)$ describe the same undirected line, so $H$ double-covers the space of undirected lines $H/\mathbb{Z}_2$: a M\"obius 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 lines apart. Thus we need a $\mathbb{Z}_2$-invariant embedding of lines into a linear space, on which soft-argmax is well-defined. We achieve this by parametrising lines via unit-norm homogeneous vectors $\ell=(1+\rho^2)^{-1/2}(\cos\theta,\sin\theta,-\rho)^{\top}\in\mathbb{R}^3$ and applying the Veronese map $v_2(\ell)=\ell\ell^{\top}$ that satisfies $v_2(\ell)=v_2(-\ell)$. This descends continuously to an embedding of the quotient $H/\mathbb{Z}_2$ into the linear space $\mathrm{Sym}^2(\mathbb{R}^3)$, where the antipodal ambiguity vanishes. Line extraction becomes a barycentre in $\mathrm{Sym}^2(\mathbb{R}^3)$, projected back via its leading eigenvector. We validate our \emph{Veronese soft-argmax} in a Hough transform-based network across all resolvable lines, confirming uniform and seam-free recovery. We further derive that the $L_2$-loss on isometrically weighted Veronese embeddings equals the squared chordal distance between lines in projective space, enabling a geometrically precise training objective.