PaperlayerView exact revision on arXiv

Source-linked AI summary

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:2609.00521v1cs.CVcs.LG

TL;DR

Soft-argmax fails when Hough coordinates represent undirected lines on a Möbius strip rather than a globally linear space. The paper uses a sign-invariant Veronese embedding to make averaging well defined, and reports seam-free recovery across resolvable lines. Evaluation remains limited to synthetic single lines, while real imagery and multi-line settings are untested.

  • Problem

    Soft-argmax can average geometrically adjacent undirected lines across the Hough seam into an incorrect result because the Möbius-strip label space is not linear.

  • Method

    VSmax applies soft-argmax to sign-invariant Veronese embeddings of line vectors, then recovers the line from the barycentre's leading eigenvector.

  • Results

    Seam EA-score rises from 0.69 with soft-argmax to 0.92 with VSmax, matching VSmax's interior score of 0.88.

  • Takeaways & Limitations

    Embedding the quotient label space makes differentiable line readout well defined and gives the L2 loss a projective-geometric interpretation.

  • Takeaways & Limitations

    Evaluation is restricted to single synthetic lines under additive noise; real imagery and multi-line settings remain untested.

Abstract

from arXiv · show

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 $(θ,ρ)$. 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 $H/\mathbb{Z}_2$: a Möbius 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 $\ell=(1+ρ^2)^{-1/2}(\cosθ,\sinθ,-ρ)^{\top}\in\mathbb{R}^3$ 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.

1 INTRODUCTION

Line recovery depends on a representation that preserves geometry in a space where linear operations are meaningful. The paper replaces seam-sensitive Hough-coordinate averaging with a sign-invariant Veronese representation and studies the resulting readout.

  • Neural line-recovery pipelines require parametrisations whose geometry remains meaningful under linear operations.
  • The Hough accumulator represents undirected lines as a flat rectangle, although boundary identification gives the space a Möbius-strip topology.
  • Soft-argmax averages cell coordinates across this seam, potentially producing a line unlike either geometrically adjacent input line.
  • The proposed VSmax maps sign-ambiguous unit homogeneous line vectors to outer products in a linear symmetric-matrix space.
  • VSmax extracts a barycentre in symmetric matrices and recovers the encoded line through the leading eigenvector.

2 RELATED WORK

Prior work contributes known operators, differentiable coordinate readouts, symmetry-aware processing, projection embeddings, and lifted geometric representations. The paper positions its contribution as combining these ideas for a differentiable, geometrically meaningful readout of undirected lines.

  • Known-operator learning inserts fixed differentiable operators such as the Hough transform, while some prior line systems retain non-differentiable peak extraction.
  • Differentiable coordinate readouts normalize heatmaps and return probability-weighted averages in the convex hull of a coordinate grid.
  • Symmetry-aware alternatives address Hough boundary structure through group-equivariant processing or angular unfolding, but unfolding duplicates every undirected line.
  • Projection-embedding methods represent sign-ambiguous vectors as symmetric outer-product matrices, enabling Euclidean computations while highlighting a mismatch with task-relevant distances.
  • Lifted rotation representations demonstrate regression in a higher-dimensional linear space followed by projection back, including quadratic-form recovery for sign-ambiguous quaternions.
  • The paper identifies the missing combination as a differentiable readout well posed on undirected-line space together with an objective measuring genuine line distance.

3 METHODS

The method replaces the seam-bearing Hough coordinates with a sign-invariant Veronese embedding, performs soft-argmax there, and decodes the result through a leading eigenvector.

  • 3.1 A SPACE-OPERATION MISMATCH: Undirected lines form a Möbius strip because identifying (θ,ρ) with (θ+π,−ρ) glues the angular boundaries with a ρ reflection.
  • 3.1 A SPACE-OPERATION MISMATCH: Soft-argmax averages cell coordinates, so mass spanning the seam can produce a line far from both geometrically adjacent lines.
  • 3.2 THE VERONESE EMBEDDING: Unit homogeneous line vectors reduce projective scaling to a sign ambiguity, yielding the quotient S2/{±1}=RP2.
  • 3.2 THE VERONESE EMBEDDING: The degree-two Veronese map represents each line by the sign-invariant outer product ℓℓ⊤, or its six-entry half-vectorisation in R6.
  • 3.2 THE VERONESE EMBEDDING: Diagonal weighting corrects the duplicated off-diagonal matrix entries, making weighted embedding L2 distance equal squared projective chordal distance.
  • 3.3 VERONESE SOFT-ARGMAX: VSmax applies the same softmax probability distribution as soft-argmax to cached Veronese embeddings, producing a seam-free barycentre in a linear space.
  • 3.4 LINE RECOVERY: Decoding inverts weighting and half-vectorisation, finds the closest rank-one symmetric matrix via Eckart–Young, and uses its leading eigenvector to recover Hough coordinates.
  • 3.4 LINE RECOVERY: Eigenvalue decomposition is differentiable under a strict leading-eigenvalue gap, while the proposed loss bypasses it by acting directly on the embedding.

4 EXPERIMENTS AND RESULTS

Experiments evaluate how readout, representation, noise, and architectural priors affect line recovery. The Veronese representation and VSmax improve seam behavior, robustness under noise, and generalisation beyond training support.

  • 4.1 OPERATOR VALIDATION: At σ = 0.6, VSmax raises seam EA-score to 0.92 versus 0.69 for standard soft-argmax, matching uniform recovery across the line space.The corresponding interior scores are 0.88 and 0.72; extreme offsets remain outlier regions because visible line segments are small and Hough responses weak.
  • 4.2 REPRESENTATION VALIDATION: Switching from polar to Veronese loss lifts seam EA-score by 28% for the MLP, 10% for standard soft-argmax, and 14% for VSmax.The MLP result closes the seam deficit to match the interior, while the broader comparison shows representation benefits across all pipelines.
  • 4.2 REPRESENTATION VALIDATION: VSmax reaches seam EA-score 0.92 versus 0.69 for soft-argmax under the Veronese loss, showing that representation and operator are complementary.The embedding loss constrains the output space, but only VSmax makes the averaging operation compatible with projective line geometry.
  • 4.2 REPRESENTATION VALIDATION: As test noise grows, Veronese loss degrades more gracefully than polar loss in the interior because it aligns with chordal line distance rather than distorted coordinate error.The polar loss can lead on clean data, but nonlinear dependence on ρ makes equal coordinate errors correspond to unequal geometric changes.
  • 4.3 ABLATION 1: WHY LEARN END-TO-END: The classical Hough readout is exact when clean but decays toward 0.5 with noise, while RHT collapses under mild noise and VSmax degrades gracefully.The end-to-end pipeline can learn to clean the accumulator before its differentiable readout; RHT’s recipe targets an unknown number of lines rather than single-line regression.
  • 4.4 ABLATION 2: WHY KNOWN OPERATORS: Across support settings g1–g4, the data-driven MLP collapses on held-out lines, whereas the known-operator pipeline remains almost uniform across the line space.The known-operator pipeline uses 37.6K parameters versus 11.3M for the CNN+MLP regressor, roughly 300× fewer.

5 CONCLUSIONS

The paper frames well-posed geometric readouts as requiring an embedding, a linear averaging space, and a closed-form inverse, while identifying limitations in evaluation scope and demonstrated applicability.

  • A valid geometric readout makes label-space identifications equalities before training, without trainable parameters or hyperparameters.The construction prevents chart seams from appearing when the three conditions hold.
  • The approach extends conceptually beyond lines to quotient label spaces such as axial orientations, signed quaternions, and discretely symmetric objects.The embedding determines both the averaging operator and the metric measured by its loss.
  • Evaluation is restricted to single synthetic lines under additive noise, leaving real imagery untested and multi-line settings unexplored.Real imagery would require a feature extractor introducing another uncertain variable.
  • The method is presented as a readout for known-operator line pipelines, while transfer to other projective label spaces remains un demonstrated.The paper suggests the same embed-average-recover recipe could apply elsewhere but does not demonstrate it.
  • The presented methods are not commercially available, and future availability is not guaranteed.

A.1 THE SQUARED L2 LOSS ON THE VERONESE EMBEDDING IS THE SQUARED CHORDAL DISTANCE

The appendix shows that the weighted squared L2 loss on Veronese vectors is exactly the squared chordal distance between the represented projective lines.

  • The training loss is the squared L2 distance between six-dimensional Veronese vectors formed by half-vectorizing outer-product matrices.
  • Weighting the off-diagonal Veronese components reconciles the six-entry vector norm with the nine-entry Frobenius matrix norm.Off-diagonal matrix entries otherwise contribute twice in the Frobenius norm.
  • Even without weighting, the vector and matrix losses share the same minimiser and remain monotone in the line angle.Thus the plain vector loss already trains in the correct ordering.
  • For unit representatives, the Frobenius distance between outer products equals the squared chordal distance between their projective points.The distance is invariant to sign changes and is therefore well defined for undirected lines.

A.2 M ¨OBIUS PADDING FOR THE SINGLE COVER

Single-cover Hough padding implements the Möbius seam by circularly wrapping orientation while reflecting the signed offset, preserving the continuing line values but not convolutional equivariance.

  • The single-cover accumulator indexes orientation over [0,π) and signed offset symmetrically, with offset reflection realizing ρ↦−ρ.An odd number of offset bins makes the centre bin exact.
  • Möbius padding maps virtual rows beyond either angular boundary to the opposite boundary with reflected offset indices.This is circular padding in orientation composed with a reflection in offset.
  • The padding splices exact heatmap continuations but cannot make translation-only plane convolutions equivariant to the orientation-reversing glide-reflection.A double cover replaces the glide-reflection with a pure translation, where ordinary circular padding is exact.

A.3 RHT: A MULTI-LINE DETECTOR APPLIED TO SINGLE-LINE REGRESSION

The RHT baseline uses a learned Gaussian-target heatmap and non-differentiable peak extraction; wider targets help on clean data, but all widths collapse under noise.

  • RHT regresses a Gaussian-target heatmap over the (θ,ρ) accumulator and recovers lines through a separate non-differentiable peak-extraction step.It was designed for an unknown number of lines and is evaluated here as a Hough-based reference.
  • On clean data, global EA-score rises from 0.79 at σt=0.5 to 0.89 at σt=3.0.The target width σt is swept because it is a free hyperparameter uncoupled from line error.
  • Under test noise, every target-heatmap width collapses, and no width recovers the RHT global EA-score.

A.4 TRAINING SETUP AND BASELINE CONFIGURATIONS

Table 2 reports the optimisation, batching, and baseline configurations used in the paper. The optimisation and batching blocks are identical across every reported run.

  • Table 2 documents the optimisation, batching, and baseline configurations for the reported experiments.
  • The optimisation block is identical across every run reported in the paper.
  • The batching block is identical across every run reported in the paper.
Loading 2609.00521v1…