From horizon detection to fibre structures in X-ray imaging, many vision tasks recover lines via peak detection in Hough space
H=S1×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/Z2: a Möbius strip, obtained by identifying each pair under
Z2 action. Soft-argmax operates on the cover
H, but since
H/Z2 admits no linear structure, it tears geometrically adjacent lines apart. Thus we need a
Z2-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
ℓ=(1+ρ2)−1/2(cosθ,sinθ,−ρ)⊤∈R3 and applying the Veronese map
v2(ℓ)=ℓℓ⊤ that satisfies
v2(ℓ)=v2(−ℓ). This descends continuously to an embedding of the quotient
H/Z2 into the linear space
Sym2(R3), where the antipodal ambiguity vanishes. Line extraction becomes a barycentre in
Sym2(R3), 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
L2-loss on isometrically weighted Veronese embeddings equals the squared chordal distance between lines in projective space, enabling a geometrically precise training objective.