Image based visual servoing using algebraic curves applied to shape alignment

Visual servoing schemes generally employ various image features (points, lines, moments etc.) in their control formulation. This paper presents a novel method for using boundary information in visual servoing. Object boundaries are modeled by algebraic equations and decomposed as a unique sum of product of lines. We propose that these lines can be used to extract useful features for visual servoing purposes. In this paper, intersection of these lines are used as point features in visual servoing. Simulations are performed with a 6 DOF Puma 560 robot using Matlab Robotics Toolbox for the alignment of a free-form object. Also, experiments are realized with a 2 DOF SCARA direct drive robot. Both simulation and experimental results are quite promising and show potential of our new method

Detecting Similarity of Rational Plane Curves

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees.Comment: 22 page

Dual-to-kernel learning with ideals

In this paper, we propose a theory which unifies kernel learning and symbolic algebraic methods. We show that both worlds are inherently dual to each other, and we use this duality to combine the structure-awareness of algebraic methods with the efficiency and generality of kernels. The main idea lies in relating polynomial rings to feature space, and ideals to manifolds, then exploiting this generative-discriminative duality on kernel matrices. We illustrate this by proposing two algorithms, IPCA and AVICA, for simultaneous manifold and feature learning, and test their accuracy on synthetic and real world data.Comment: 15 pages, 1 figur

Determining Critical Points of Handwritten Mathematical Symbols Represented as Parametric Curves

We consider the problem of computing critical points of plane curves represented in a finite orthogonal polynomial basis. This is motivated by an approach to the recognition of hand-written mathematical symbols in which the initial data is in such an orthogonal basis and it is desired to avoid ill-conditioned basis conversions. Our main contribution is to assemble the relevant mathematical tools to perform all the necessary operations in the orthogonal polynomial basis. These include implicitization, differentiation, root finding and resultant computation

Neural 3D Morphable Models: Spiral Convolutional Networks for 3D Shape Representation Learning and Generation

Generative models for 3D geometric data arise in many important applications in 3D computer vision and graphics. In this paper, we focus on 3D deformable shapes that share a common topological structure, such as human faces and bodies. Morphable Models and their variants, despite their linear formulation, have been widely used for shape representation, while most of the recently proposed nonlinear approaches resort to intermediate representations, such as 3D voxel grids or 2D views. In this work, we introduce a novel graph convolutional operator, acting directly on the 3D mesh, that explicitly models the inductive bias of the fixed underlying graph. This is achieved by enforcing consistent local orderings of the vertices of the graph, through the spiral operator, thus breaking the permutation invariance property that is adopted by all the prior work on Graph Neural Networks. Our operator comes by construction with desirable properties (anisotropic, topology-aware, lightweight, easy-to-optimise), and by using it as a building block for traditional deep generative architectures, we demonstrate state-of-the-art results on a variety of 3D shape datasets compared to the linear Morphable Model and other graph convolutional operators.Comment: to appear at ICCV 201

Reverse engineering of CAD models via clustering and approximate implicitization

In applications like computer aided design, geometric models are often represented numerically as polynomial splines or NURBS, even when they originate from primitive geometry. For purposes such as redesign and isogeometric analysis, it is of interest to extract information about the underlying geometry through reverse engineering. In this work we develop a novel method to determine these primitive shapes by combining clustering analysis with approximate implicitization. The proposed method is automatic and can recover algebraic hypersurfaces of any degree in any dimension. In exact arithmetic, the algorithm returns exact results. All the required parameters, such as the implicit degree of the patches and the number of clusters of the model, are inferred using numerical approaches in order to obtain an algorithm that requires as little manual input as possible. The effectiveness, efficiency and robustness of the method are shown both in a theoretical analysis and in numerical examples implemented in Python

Symmetry Detection of Rational Space Curves from their Curvature and Torsion

We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly faster, simpler, and more general than an earlier method addressing a similar problem. To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage.Comment: 25 page