Factorization of Rational Curves in the Study Quadric and Revolute Linkages

Given a generic rational curve $C$ in the group of Euclidean displacements we construct a linkage such that the constrained motion of one of the links is exactly $C$. Our construction is based on the factorization of polynomials over dual quaternions. Low degree examples include the Bennett mechanisms and contain new types of overconstrained 6R-chains as sub-mechanisms.Comment: Changed arxiv abstract, corrected some type

Shape basis interpretation for monocular deformable 3D reconstruction

© 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.In this paper, we propose a novel interpretable shape model to encode object non-rigidity. We first use the initial frames of a monocular video to recover a rest shape, used later to compute a dissimilarity measure based on a distance matrix measurement. Spectral analysis is then applied to this matrix to obtain a reduced shape basis, that in contrast to existing approaches, can be physically interpreted. In turn, these pre-computed shape bases are used to linearly span the deformation of a wide variety of objects. We introduce the low-rank basis into a sequential approach to recover both camera motion and non-rigid shape from the monocular video, by simply optimizing the weights of the linear combination using bundle adjustment. Since the number of parameters to optimize per frame is relatively small, specially when physical priors are considered, our approach is fast and can potentially run in real time. Validation is done in a wide variety of real-world objects, undergoing both inextensible and extensible deformations. Our approach achieves remarkable robustness to artifacts such as noisy and missing measurements and shows an improved performance to competing methods.Peer ReviewedPostprint (author's final draft

GSLAM: Initialization-robust Monocular Visual SLAM via Global Structure-from-Motion

Many monocular visual SLAM algorithms are derived from incremental structure-from-motion (SfM) methods. This work proposes a novel monocular SLAM method which integrates recent advances made in global SfM. In particular, we present two main contributions to visual SLAM. First, we solve the visual odometry problem by a novel rank-1 matrix factorization technique which is more robust to the errors in map initialization. Second, we adopt a recent global SfM method for the pose-graph optimization, which leads to a multi-stage linear formulation and enables L1 optimization for better robustness to false loops. The combination of these two approaches generates more robust reconstruction and is significantly faster (4X) than recent state-of-the-art SLAM systems. We also present a new dataset recorded with ground truth camera motion in a Vicon motion capture room, and compare our method to prior systems on it and established benchmark datasets.Comment: 3DV 2017 Project Page: https://frobelbest.github.io/gsla

The Theory of Bonds: A New Method for the Analysis of Linkages

In this paper we introduce a new technique, based on dual quaternions, for the analysis of closed linkages with revolute joints: the theory of bonds. The bond structure comprises a lot of information on closed revolute chains with a one-parametric mobility. We demonstrate the usefulness of bond theory by giving a new and transparent proof for the well-known classification of overconstrained 5R linkages.Comment: more detailed explanations and additional reference

An Enhanced Structure-from-Motion Paradigm based on the Absolute Dual Quadric and Images of Circular Points

International audienceThis work aims at introducing a new unified Structure-from-Motion (SfM) paradigm in which images of circular point-pairs can be combined with images of natural points. An imaged circular point-pair encodes the 2D Euclidean structure of a world plane and can easily be derived from the image of a planar shape, especially those including circles. A classical SfM method generally runs two steps: first a projective factorization of all matched image points (into projective cameras and points) and second a camera self-calibration that updates the obtained world from projective to Euclidean. This work shows how to introduce images of circular points in these two SfM steps while its key contribution is to provide the theoretical foundations for combining “classical” linear self-calibration constraints with additional ones derived from such images. We show that the two proposed SfM steps clearly contribute to better results than the classical approach. We validate our contributions on synthetic and real images

The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201

Cusped light-like Wilson loops in gauge theories

We propose and discuss a new approach to the analysis of the correlation functions which contain light-like Wilson lines or loops, the latter being cusped in addition. The objects of interest are therefore the light-like Wilson null-polygons, the soft factors of the parton distribution and fragmentation functions, high-energy scattering amplitudes in the eikonal approximation, gravitational Wilson lines, etc. Our method is based on a generalization of the universal quantum dynamical principle by J. Schwinger and allows one to take care of extra singularities emerging due to light-like or semi-light-like cusps. We show that such Wilson loops obey a differential equation which connects the area variations and renormalization group behavior of those objects and discuss the possible relation between geometrical structure of the loop space and area evolution of the light-like cusped Wilson loops.Comment: Invited mini-review article to Physics of Particles and Nuclei. 16 pages, 9 eps figures; v2: references style changed, citations corrected and update