Soft-Argmax for the Projective Plane via the Veronese Embedding

Visual front-ends for robotic localization typically rely on point-based features such as Oriented FAST and Rotated BRIEF (ORB), which frequently fail in structured environments dominated by strong linear structures or textureless surfaces. While line-based Simultaneous Localization and Mapping (SLAM) systems mitigate this by utilizing line segments, conventional line extraction and description algorithms are computationally prohibitive for real-time edge robotics. To address this fundamental bottleneck, we propose HOME (Hough-space One-dimensional Matching of Extrema), an ultra-lightweight, training-free feature matching framework. HOME transforms images into Hough space, mapping global linear structures to stable local extrema, which serve as keypoints, thereby reformulating complex line matching into highly efficient one-dimensional point matching. The proposed 1D radial descriptor mathematically guarantees rotational and translational invariance without the overhead of explicit orientation estimation. As a proof of concept to validate the matching accuracy and efficiency of HOME, this paper focuses on homography estimation. Extensive evaluations demonstrate that HOME achieves robust registration in challenging scenarios where point-based methods fail, operating at a much faster speed than existing line-based methods. Extending this robust matching engine to full 3D pose estimation remains a highly promising future direction.

The adoption of vision neural networks in regulated industries requires formal robustness guarantees, especially in safety-critical domains such as healthcare, autonomous vehicles, and aerospace. However, current approaches are confined to incomplete statistical verification or robustness to $\ell_p$-norm and affine transforms, which cover only a narrow subset of perturbations to the image formation process. In particular, robustness to camera motion remains an open problem despite being key to deploy many vision applications. We present a formal verification approach that targets robustness against 3D motion perturbations of the capturing camera. We first establish a closed-form mapping from camera pose to pixel values. By analyzing the continuity properties of the resulting homographies, we show that recent work on Lipschitz optimization and piecewise continuity can be extended to derive tight linear bounds on perturbed pixel values. Our approach applies to scenes with predominantly planar structure, such as ground planes in augmented reality, road markings and traffic signs in autonomous driving, or planar workspaces in robotic manipulation. This enables the first formal verification of projective geometry transforms, without complex simulation, surrogate networks, or explicit image-formation models. We validate our implementation and show up to 89% speedup and 7% tighter bounds over prior work. We then evaluate our method on the VNN-COMP benchmark and reveal systematic weaknesses to projective perturbations. Finally, we demonstrate a real-world case study on a safety-critical runway classifier, highlighting practical vulnerabilities to camera motion, and addressing a key challenge in the certification of learned models. Data and code are publicly available at https://github.com/jeangud/homography-verification .

Image registration is essential in applications such as electronic image stabilization. Scale-Invariant Feature Transform (SIFT), a widely used local keypoint detector and descriptor, typically provides accurate registration; however, it often fails in scenes with strong linear structures (e.g., shutters), where local features become ambiguous. We propose Hough-SIFT, a robust registration method that performs SIFT descriptor matching in Hough space. In this domain, linear structures form distinctive peaks that restore descriptor discriminability. Experiments demonstrate that Hough-SIFT is robust in linear scenes where SIFT frequently fails, while maintaining accuracy comparable to SIFT in normal scenes.