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    We developed a novel parametric regression method for the Grassmann manifold, enabling intrinsic curve fitting. This approach simplifies complex vision tasks like shape and crowd analysis.

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    Area of Science:

    • Computer Vision
    • Differential Geometry
    • Machine Learning

    Background:

    • Parametric regression in Euclidean space relies on energy minimization.
    • Generalizing regression to non-Euclidean spaces like the Grassmann manifold is challenging.
    • Existing methods often require tailored solutions for different problems.

    Purpose of the Study:

    • To develop a unified framework for intrinsic parametric regression on the Grassmann manifold.
    • To generalize energy minimization principles from Euclidean space to Riemannian manifolds.
    • To provide a simple, extensible, and easy-to-implement solution for parametric regression.

    Main Methods:

    • Generalizing energy minimization for linear least-squares using an optimal-control perspective.
    • Specializing the method to the Grassmann manifold for intrinsic regression.
    • Extending the basic geodesic model to time-warped variants and cubic splines.

    Main Results:

    • A novel, unified framework for parametric regression on the Grassmann manifold.
    • The method successfully extends to time-warped models and cubic splines.
    • Demonstrated utility across diverse computer vision tasks including shape regression, traffic speed estimation, and crowd counting.

    Conclusions:

    • The proposed Grassmann manifold regression framework offers a versatile and efficient solution.
    • It enables solving multiple vision problems within a single, adaptable pipeline.
    • This approach facilitates intrinsic analysis and modeling on complex data manifolds.