A Unified Formula for Affine Transformations between Calibrated Cameras

2026-02-06

AI Summary

This work addresses the problem of establishing affine transformation relationships between local image patches observed by two calibrated cameras. By integrating multi-view geometry with differential geometric analysis, the authors derive the first closed-form solution for the affine transformation that is valid for arbitrary calibrated camera pairs. The solution explicitly characterizes the analytical dependence of the transformation on the relative pose, image coordinates, and local surface normal. The proposed method is not only concise in formulation and computationally efficient but also provides a rigorous theoretical foundation for applications such as image registration, feature matching, and 3D reconstruction.

Technology Category

• Computer Vision: Scene Analysis & Understanding
• Machine Learning: Multi-instance/Multi-view Learning
• Intelligent Robots: Localization, Mapping, and Navigation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Data transparency and provenance

Abstract

In this technical note, we derive a closed-form expression for the affine transformation mapping local image patches between two calibrated views. We show that the transformation is a function of the relative camera pose, the image coordinates, and the local surface normal.

Problem

• affine transformation
• calibrated cameras
• image patches
• camera pose
• surface normal

Innovation

• affine transformation
• calibrated cameras
• closed-form expression
• relative pose
• surface normal