# Lipschitz Optimization for Formal Verification of Homographies

arXiv: https://arxiv.org/abs/2605.23203 · HTML: https://mlsanity.com/paper/2605.23203/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 35 of its 57 references that are in the map.
- First posted on arXiv 22 May 2026.

## Builds on (most cited first, 30 of 35)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Universal adversarial perturbations](https://mlsanity.com/paper/1610.08401/) (2016)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [Algorithms for Verifying Deep Neural Networks](https://mlsanity.com/paper/1903.06758/) (2019)
- [Spatially Transformed Adversarial Examples](https://mlsanity.com/paper/1801.02612/) (2018)
- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [Vision-Centric BEV Perception: A Survey](https://mlsanity.com/paper/2208.02797/) (2022)
- [SoK: Certified Robustness for Deep Neural Networks](https://mlsanity.com/paper/2009.04131/) (2020)
- [General Cutting Planes for Bound-Propagation-Based Neural Network Verification](https://mlsanity.com/paper/2208.05740/) (2022)
- [Reachability Analysis of Deep Neural Networks with Provable Guarantees](https://mlsanity.com/paper/1805.02242/) (2018)
- [Marabou 2.0: A Versatile Formal Analyzer of Neural Networks](https://mlsanity.com/paper/2401.14461/) (2024)
- [The Second International Verification of Neural Networks Competition (VNN-COMP 2021): Summary and Results](https://mlsanity.com/paper/2109.00498/) (2021)
- [Neural Network Verification with Branch-and-Bound for General Nonlinearities](https://mlsanity.com/paper/2405.21063/) (2024)
- [Geometric robustness of deep networks: analysis and improvement](https://mlsanity.com/paper/1711.09115/) (2017)
- [Global optimization of Lipschitz functions](https://mlsanity.com/paper/1703.02628/) (2017)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [The Fourth International Verification of Neural Networks Competition (VNN-COMP 2023): Summary and Results](https://mlsanity.com/paper/2312.16760/) (2023)
- [The 6th International Verification of Neural Networks Competition (VNN-COMP 2025): Summary and Results](https://mlsanity.com/paper/2512.19007/) (2025)
- [Towards Compositional Adversarial Robustness: Generalizing Adversarial Training to Composite Semantic Perturbations](https://mlsanity.com/paper/2202.04235/) (2022)
- [Toward Certified Robustness Against Real-World Distribution Shifts](https://mlsanity.com/paper/2206.03669/) (2022)
- [Provable Defense Against Geometric Transformations](https://mlsanity.com/paper/2207.11177/) (2022)
- [LARD -- Landing Approach Runway Detection -- Dataset for Vision Based Landing](https://mlsanity.com/paper/2304.09938/) (2023)
- [Formal Verification of CNN-based Perception Systems](https://mlsanity.com/paper/1811.11373/) (2018)
- [ART-Point: Improving Rotation Robustness of Point Cloud Classifiers via Adversarial Rotation](https://mlsanity.com/paper/2203.03888/) (2022)
- [The Right (Angled) Perspective: Improving the Understanding of Road Scenes Using Boosted Inverse Perspective Mapping](https://mlsanity.com/paper/1812.00913/) (2018)
- [Towards Verifying the Geometric Robustness of Large-scale Neural Networks](https://mlsanity.com/paper/2301.12456/) (2023)
- [Verification of Geometric Robustness of Neural Networks via Piecewise Linear Approximation and Lipschitz Optimisation](https://mlsanity.com/paper/2408.13140/) (2024)

## Abstract (arXiv)

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 ℓp-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 .