Estimating the Reach of a Manifold via its Convexity Defect Function

The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard (Ann. Statist. 47(1), 177–204 (2019)), an estimator for the reach is given. A uniform expected loss bound over a $${\mathscr {C}}^k$$ C k model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost achieves these rates in the $${\mathscr {C}}^3$$ C 3 and $${\mathscr {C}}^4$$ C 4 cases, with a gap given by a logarithmic factor.

Affine Image Registration Based on Geometric Inference

Image registration is widely used in applications for mapping one image to another. As it is often formulated as a point matching problem, in this paper, a novel method, called the Geometric Inference (GI) algorithm, is proposed for feature point based image registration. Firstly, according to affine distance invariant, the global geometric relationship between collinear correspondences is deduced and used for collinear point matching. Secondly, utilizing affine area invariant, geometric relationship between noncollinear correspondences is inferred and used for noncollinear point matching. Finally, the best affine transformation can be discovered from the correspondences composed of the collinear and noncollinear corresponding point pairs. Experiments on synthesized and real data demonstrate that GI is well-adapt to image registration as it is fast and robust to missing points, outliers, and noise.