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    This study introduces a framework to analyze numerical conditioning in minimal problems for multiple view geometry. It explains why 5-point and 7-point algorithms can fail, even without outliers, due to intrinsic instabilities.

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    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.