AI RESEARCHarXiv9d ago

Soft-Argmax for the Projective Plane via the Veronese Embedding

Benjamin El-Zein · Dominik Eckert · Paul Zech · Christopher Syben · Bernhard Geiger · Steffen Kappler · Sebastian Stober

arXiv:2609.00521v1Computer VisionMachine Learning

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 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 / Z_{2}

: a Möbius 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

ℓ = (1 + ρ^{2})^{- 1/2} (cos θ, sin θ, - ρ)^{⊤} \in R^{3}

and applying the Veronese map

v_{2} (ℓ) = ℓ ℓ^{⊤}

that satisfies

v_{2} (ℓ) = v_{2} (- ℓ)

. This descends continuously to an embedding of the quotient

H / Z_{2}

into the linear space

S y m^{2} (R^{3})

, where the antipodal ambiguity vanishes. Line extraction becomes a barycentre in

S y m^{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.

Discussion · 0

Original paper on arXiv