Research
Soft-Argmax for the Projective Plane via the Veronese Embedding
arXiv:2609.00521v1 Announce Type: new Abstract: From horizon detection to fibre structures in X-ray imaging, many vision tasks recover lines via peak detection in Hough space H=S^1imesR, the domain of
arXiv:2609.00521v1 Announce Type: new Abstract: From horizon detection to fibre structures in X-ray imaging, many vision tasks recover lines via peak detection in Hough space H=S^1imesR, the domain of orientation-offset pairs (heta,rho). Differentiable pipelines extract coordinates via soft-argmax, a probability-weighted average that is only meaningful in a globally linear space. However, (heta,rho) and (heta+pi,-rho) describe the same undirected line, so H double-covers the space of undirected lines H/Z_2: a Mobius strip, obtained by identifying each pair under Z_2 action. Soft-argmax operates on the cover H, but since H/Z_2 admits no linear structure, it tears geometrically adjacent lines apart. Thus we need a 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}(osheta,sinheta,-rho)^{op}inR^3 and applying the Veronese map v_2(ell)=ellell^{op} that satisfies v_2(ell)=v_2(-ell). This descends continuously to an embedding of the quotient H/Z_2 into the linear space Sym^2(R^3), where the antipodal ambiguity vanishes. Line extraction becomes a barycentre in Sym^2(R^3), 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 L_2-loss on isometrically weighted Veronese embeddings equals the squared chordal distance between lines in projective space, enabling a geometrically precise training objective.
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Source: arXiv cs.CV | 2026-09-02