发表机构
Siemens Healthineers AG; Artificial Intelligence Lab, Otto-von-Guericke-University（西门子医疗健康公司; 奥托·冯·格里克大学人工智能实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对霍夫空间二重覆盖导致的soft-argmax无法处理无向直线的问题，提出Veronese嵌入的soft-argmax方法，实现直线的无缝恢复，且其损失函数对应投影空间直线的平方弦距离，可用于精确训练。
AI 中文摘要
从地平线检测到X射线成像中的纤维结构，许多视觉任务通过霍夫空间$H=S^1\times\mathbb{R}$（方向-偏移对$(\theta,\rho)$的定义域）中的峰值检测来恢复直线。可微流程通过\emph{soft-argmax}提取坐标，这是一种概率加权平均，仅在全局线性空间中有意义。然而，$(\theta,\rho)$和$(\theta+\pi,-\rho)$描述同一条无向直线，因此$H$是无向直线空间$H/\mathbb{Z}_2$的二重覆盖：通过在$\mathbb{Z}_2$作用下识别每对得到的莫比乌斯带。Soft-argmax在覆盖空间$H$上操作，但由于$H/\mathbb{Z}_2$不具备线性结构，它会将几何上相邻的直线割裂。因此，我们需要将直线嵌入线性空间的$\mathbb{Z}_2$不变嵌入，使soft-argmax在该空间上有良好定义。我们通过将直线参数化为单位范数齐次向量$\ell=(1+\rho^2)^{-1/2}(\cos\theta,\sin\theta,-\rho)^\top\in\mathbb{R}^3$并应用满足$v_2(\ell)=v_2(-\ell)$的Veronese映射$v_2(\ell)=\ell\ell^\top$来实现这一点。这连续下降为商空间$H/\mathbb{Z}_2$到线性空间$\mathrm{Sym}^2(\mathbb{R}^3)$的嵌入，其中对极模糊消失。直线提取变为$\mathrm{Sym}^2(\mathbb{R}^3)$中的重心，通过其主导特征向量投影返回。我们在基于霍夫变换的网络中针对所有可分辨直线验证了我们的\emph{Veronese soft-argmax}，确认其能实现均匀且无缝的直线恢复。我们进一步推导得出，等距加权Veronese嵌入上的$L_2$损失等于投影空间中直线之间的平方弦距离，从而实现了几何上精确的训练目标。
英文摘要
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.
