Condition Numbers in Multiview Geometry, Instability in Relative Pose Estimation, and RANSAC

Area of Science:

• Computer Vision
• Computational Geometry
• Numerical Analysis

Background:

• Relative pose estimation is crucial for 3D reconstruction.
• Standard algorithms like 5-point and 7-point RANSAC can fail unexpectedly.
• These failures can occur even with sufficient inlier data.

Purpose of the Study:

• To develop a general framework for analyzing numerical conditioning in minimal problems.
• To understand the intrinsic instabilities of 5- and 7-point minimal problems.
• To provide computational tests for assessing problem conditioning.

Main Methods:

• Utilizing tools from computational algebra and Riemannian geometry.
• Analyzing the numerical conditioning of minimal problems in multiple view geometry.
• Characterizing instabilities based on scene geometry and image data.

Main Results:

• Identified intrinsic instabilities in 5- and 7-point minimal problems.
• Developed methods to assess the condition number before solving.
• Demonstrated that RANSAC implicitly selects well-conditioned data.

Conclusions:

• The framework provides insights into the failure of pose estimation algorithms.
• Numerical conditioning is a critical factor in the success of minimal problem solvers.
• RANSAC's practical success is partly due to its selection of well-conditioned data.