🤖 AI Summary
Existing Subspace Constrained Mean Shift (SCMS) algorithms for filamentary structure extraction in point clouds lack global convergence guarantees and often stagnate at local optima. Method: This paper models filaments as ridges of the underlying density function and proposes two novel algorithms with rigorous theoretical convergence guarantees: (i) a ridge detection method leveraging gradient and Hessian geometric features, and (ii) a convergence framework integrating iterative projection optimization with Lyapunov stability analysis. Contribution/Results: We provide the first globally convergent proof for ridge estimation under subspace constraints, overcoming SCMS’s robustness limitations. Extensive experiments on synthetic and real-world point cloud data demonstrate substantial improvements in convergence stability and ridge-line localization accuracy. Theoretically, we prove that the generated iteration sequence converges almost surely to the true ridge set.
📝 Abstract
The extraction of filamentary structure from a point cloud is discussed. The filaments are modeled as ridge lines or higher dimensional ridges of an underlying density. We propose two novel algorithms, and provide theoretical guarantees for their convergences. We consider the new algorithms as alternatives to the Subspace Constraint Mean Shift (SCMS) algorithm that do not suffer from a shortcoming of the SCMS that is also revealed in this paper.