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=S1×ℝ, the domain of orientation-offset pairs (θ,ρ). Differentiable pipelines extract coordinates via 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/mathbbZ2: a M"obius strip, obtained by identifying each pair under mathbbZ2 action. Soft-argmax operates on the cover H, but since H/mathbbZ2 admits no linear structure, it tears geometrically adjacent lines apart. Thus we need a mathbbZ2-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)top∈ℝ3 and applying the Veronese map v2(ℓ)=ℓℓtop that satisfies v2(ℓ)=v2(-ℓ). This descends continuously to an embedding of the quotient H/mathbbZ2 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 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.