Estimating the Reach of a Manifold

Various problems in manifold estimation make use of a quantity called the reach, denoted by $\tau\_M$, which is a measure of the regularity of the manifold. This paper is the first investigation into the problem of how to estimate the reach. First, we study the geometry of the reach through an approximation perspective. We derive new geometric results on the reach for submanifolds without boundary. An estimator $\hat{\tau}$ of $\tau\_{M}$ is proposed in a framework where tangent spaces are known, and bounds assessing its efficiency are derived. In the case of i.i.d. random point cloud $\mathbb{X}\_{n}$, $\hat{\tau}(\mathbb{X}\_{n})$ is showed to achieve uniform expected loss bounds over a $\mathcal{C}^3$-like model. Finally, we obtain upper and lower bounds on the minimax rate for estimating the reach

Model-based estimation of off-highway road geometry using single-axis LADAR and inertial sensing

This paper applies some previously studied extended Kalman filter techniques for planar road geometry estimation to the domain of autonomous navigation of off-highway vehicles. In this work, a clothoid model of the road geometry is constructed and estimated recursively based on road features extracted from single-axis LADAR range measurements. We present a method for feature extraction of the road centerline in the image plane, and describe its application to recursive estimation of the road geometry. We analyze the performance of our method against simulated motion of varied road geometries and against closed-loop detection, tracking and following of desert roads. Our method accomodates full 6 DOF motion of the vehicle as it navigates, constructs consistent estimates of the road geometry with respect to a fixed global reference frame, and requires an estimate of the sensor pose for each range measurement

Tracking control with adaption of kites

A novel tracking paradigm for flying geometric trajectories using tethered kites is presented. It is shown how the differential-geometric notion of turning angle can be used as a one-dimensional representation of the kite trajectory, and how this leads to a single-input single-output (SISO) tracking problem. Based on this principle a Lyapunov-based nonlinear adaptive controller is developed that only needs control derivatives of the kite aerodynamic model. The resulting controller is validated using simulations with a point-mass kite model.Comment: 20 pages, 12 figure

3D Geometric Analysis of Tubular Objects based on Surface Normal Accumulation

This paper proposes a simple and efficient method for the reconstruction and extraction of geometric parameters from 3D tubular objects. Our method constructs an image that accumulates surface normal information, then peaks within this image are located by tracking. Finally, the positions of these are optimized to lie precisely on the tubular shape centerline. This method is very versatile, and is able to process various input data types like full or partial mesh acquired from 3D laser scans, 3D height map or discrete volumetric images. The proposed algorithm is simple to implement, contains few parameters and can be computed in linear time with respect to the number of surface faces. Since the extracted tube centerline is accurate, we are able to decompose the tube into rectilinear parts and torus-like parts. This is done with a new linear time 3D torus detection algorithm, which follows the same principle of a previous work on 2D arc circle recognition. Detailed experiments show the versatility, accuracy and robustness of our new method.Comment: in 18th International Conference on Image Analysis and Processing, Sep 2015, Genova, Italy. 201

Principal arc analysis on direct product manifolds

We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS370 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Inference Under Convex Cone Alternatives for Correlated Data

In this research, inferential theory for hypothesis testing under general convex cone alternatives for correlated data is developed. While there exists extensive theory for hypothesis testing under smooth cone alternatives with independent observations, extension to correlated data under general convex cone alternatives remains an open problem. This long-pending problem is addressed by (1) establishing that a "generalized quasi-score" statistic is asymptotically equivalent to the squared length of the projection of the standard Gaussian vector onto the convex cone and (2) showing that the asymptotic null distribution of the test statistic is a weighted chi-squared distribution, where the weights are "mixed volumes" of the convex cone and its polar cone. Explicit expressions for these weights are derived using the volume-of-tube formula around a convex manifold in the unit sphere. Furthermore, an asymptotic lower bound is constructed for the power of the generalized quasi-score test under a sequence of local alternatives in the convex cone. Applications to testing under order restricted alternatives for correlated data are illustrated.Comment: 31 page