Helly-Type Theorems in Property Testing

Helly’s theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If [Formula: see text] is a set of [Formula: see text] points in [Formula: see text], we say that [Formula: see text] is [Formula: see text]-clusterable if it can be partitioned into [Formula: see text] clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object [Formula: see text]. In this paper, as an application of Helly’s theorem, by taking a constant size sample from [Formula: see text], we present a testing algorithm for [Formula: see text]-clustering, i.e., to distinguish between the following two cases: when [Formula: see text] is [Formula: see text]-clusterable, and when it is [Formula: see text]-far from being [Formula: see text]-clusterable. A set [Formula: see text] is [Formula: see text]-far [Formula: see text] from being [Formula: see text]-clusterable if at least [Formula: see text] points need to be removed from [Formula: see text] in order to make it [Formula: see text]-clusterable. We solve this problem when [Formula: see text], and [Formula: see text] is a symmetric convex object. For [Formula: see text], we solve a weaker version of this problem. Finally, as an application of our testing result, in the case of clustering with outliers, we show that with high probability one can find the approximate clusters by querying only a constant size sample.

On a Problem of Danzer

Let C be a bounded convex object in ℝd, and let P be a set of n points lying outside C. Further, let cp, cq be two integers with 1 ⩽ cq ⩽ cp ⩽ n - ⌊d/2⌋, such that every cp + ⌊d/2⌋ points of P contain a subset of size cq + ⌊d/2⌋ whose convex hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex hulls are disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time.In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner (p, q) numbers for balls in ℝd. For example, it follows from our theorem that when p > q = (1+β)⋅d/2 for β > 0, then any set of balls satisfying the (p, q)-property can be hit by O((1+β)2d2p1+1/β logp) points. This is the first improvement over a nearly 60 year-old exponential bound of roughly O(2d).Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p, q) for convex sets in ℝd for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.

Multiple robots motion control to transport an object

This paper presents transporting algorithm for multiple robots to transport a concave or convex object. The object transporting includes three processes: calculating proper points process; approaching proper points process; and transporting an object process. Using fuzzy sliding mode control algorithm, we design a kinematic velocity controller. We also propose a dynamic torque controller by adaptive sliding mode control algorithm. Finally, simulations and experiment show good performance of proposed methods.

3D Modeling for Environmental Informatics

3D modeling plays an important role in the field of computer vision and image processing. It provides a convenient tool set for many environmental informatics tasks, such as taxonomy and species identification. This chapter discusses a novel way of building the 3D models of objects from their varying 2D views. The appearance of a 3D object depends on both the viewing directions and illumination conditions. What is the set of images of an object under all viewing directions? In this chapter, a novel image representation is proposed, which transforms any n-pixel image of a 3D object to a vector in a 2n-dimensional pose space. In such a pose space, it is proven that the transformed images of a 3D object under all viewing directions form a parametric manifold in a 6-dimensional linear subspace. With in-depth rotations along a single axis in particular, this manifold is an ellipse. Furthermore, it is shown that this parametric pose manifold of a convex object can be estimated from a few images in different poses and used to predict object's appearances under unseen viewing directions. These results immediately suggest a number of approaches to object recognition, scene detection, and 3D modeling, applicable to environmental informatics. Experiments on both synthetic data and real images were reported, which demonstrates the validity of the proposed representation.

3D Modeling for Environmental Informatics

3D modeling plays an important role in the field of computer vision and image processing. It provides a convenient tool set for many environmental informatics tasks, such as taxonomy and species identification. This chapter discusses a novel way of building the 3D models of objects from their varying 2D views. The appearance of a 3D object depends on both the viewing directions and illumination conditions. What is the set of images of an object under all viewing directions? In this chapter, a novel image representation is proposed, which transforms any n-pixel image of a 3D object to a vector in a 2n-dimensional pose space. In such a pose space, it is proven that the transformed images of a 3D object under all viewing directions form a parametric manifold in a 6-dimensional linear subspace. With in-depth rotations along a single axis in particular, this manifold is an ellipse. Furthermore, it is shown that this parametric pose manifold of a convex object can be estimated from a few images in different poses and used to predict object's appearances under unseen viewing directions. These results immediately suggest a number of approaches to object recognition, scene detection, and 3D modeling, applicable to environmental informatics. Experiments on both synthetic data and real images were reported, which demonstrates the validity of the proposed representation.

Interactive and On-Line Learning System for Assembly Task Motion Planning

This paper presents a novel interactive motion planning system for assembly/disassembly operations. Our system consists of three layers: interaction layer, learning layer and motion library layer. In interaction layer, user’s manipulation in difficult scenario is liberated by relaxing collision constraints. The resulting path is retracted and connected by random retraction method and BiRRT algorithm. A motion path which successfully passed through the narrow passage or information of geometrical interference in failed case is returned to user. In learning layer, motion primitives corresponding to prior similar scenario are selected by scenario comparison which is based on medial axis, and then transformed to generate new motions. Significant improvement for motion planning of non-convex object in challenging scenarios with narrow passages is obtained by interactive process. The introduction of learning mechanism can reduce global planning time and obtain experiential knowledge.