Lipschitz Optimization for Formal Verification of Homographies

2026-05-21

AI Summary

Existing methods struggle to formally verify the robustness of vision neural networks under 3D camera motion, particularly homographic transformations. This work presents the first closed-form mapping from camera pose to pixel values, enabling formal verification of projective geometry without relying on simulation, surrogate networks, or explicit imaging models. By integrating Lipschitz optimization with piecewise continuity analysis, we derive tight linear bounds on perturbed pixels. Our approach achieves up to an 89% speedup and a 7% improvement in bound tightness compared to prior methods. Evaluation on the VNN-COMP benchmark reveals systematic vulnerabilities of models to projective perturbations, and a case study on runway classifiers demonstrates tangible safety risks in real-world applications.

Abstract

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 .

Problem

• formal verification
• homography
• camera motion
• robustness
• Lipschitz optimization

Innovation

• Lipschitz optimization
• formal verification
• homography
• camera motion robustness
• projective geometry