Camera Calibration and Stereo via a Single Image of a Spherical Mirror

TL;DR

This work tackles camera calibration and catadioptric stereo using a single image of a spherical mirror. It derives a geometric framework where the sphere’s image contour appears as an ellipse described by the conic $v^T C v=0$, and recovers the camera projection matrix $P$ and sphere center $B$ from either the camera’s reflection or point correspondences, employing a radius-1 unit. The key contribution is a closed-form solution that combines conic geometry with sphere-based constraints to estimate $P$ and $B$ from a single view, and it is validated on synthetic and real data with sub-3% accuracy in several cases. The approach enables practical, real-time catadioptric stereo using a mirrored sphere and offers potential extensions to 3D point localization and omnidirectional imaging centered on the sphere, including scenarios where a spherical mirror is incidental in the scene.

Abstract

This paper presents a novel technique for camera calibration using a single view that incorporates a spherical mirror. Leveraging the distinct characteristics of the sphere's contour visible in the image and its reflections, we showcase the effectiveness of our method in achieving precise calibration. Furthermore, the reflection from the mirrored surface provides additional information about the surrounding scene beyond the image frame. Our method paves the way for the development of simple catadioptric stereo systems. We explore the challenges and opportunities associated with employing a single mirrored sphere, highlighting the potential applications of this setup in practical scenarios. The paper delves into the intricacies of the geometry and calibration procedures involved in catadioptric stereo utilizing a spherical mirror. Experimental results, encompassing both synthetic and real-world data, are presented to illustrate the feasibility and accuracy of our approach.

Figures (9)

• Figure 1: Spherical mirror in scene
• Figure 2: The ellipse which is the projected sphere's contour in red
• Figure 3: The rays from the camera to the mirror $0\rightarrow H$, from the mirror to the camera $H\rightarrow 0$ and the normal at the mirror coincide.
• Figure 5: A 2d cross section of a pair of correspondence points
• Figure 7: Finding the sphere's center from a single pair of corresponding points and the major axis of the ellipse. The green line connects the corresponding points, while the red line represents the major axis of the ellipse.