Darboux cyclides and webs from circles

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the classical approach to these surfaces based on the spherical model of 3D Moebius geometry, we provide computational tools for the identification of circle families on a given cyclide and for the direct design of those. In particular, we show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure

Signature Sequence of Intersection Curve of Two Quadrics for Exact Morphological Classification

We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in PR3, 3D real projective space; here, the term morphology is used in a broad sense to mean the shape, topological, and algebraic properties of a QSIC, including singularity, reducibility, the number of connected components, and the degree of each irreducible component, etc. There are in total 35 different QSIC morphologies with non-degenerate quadric pencils. For each of these 35 QSIC morphologies, through a detailed study of the eigenvalue curve and the index function jump we establish a characterizing algebraic condition expressed in terms of the Segre characteristics and the signature sequence of a quadric pencil. We show how to compute a signature sequence with rational arithmetic so as to determine the morphology of the intersection curve of any two given quadrics. Two immediate applications of our results are the robust topological classification of QSIC in computing B-rep surface representation in solid modeling and the derivation of algebraic conditions for collision detection of quadric primitives

Feature correspondences From Multiple Views of Coplanar Ellipses

International audienceWe address the problem of feature correspondences in images of coplanar ellipses with objective to benefit of robust ellipse fitting algorithm. The main difficulty is the lack of projective invariant points immediately available. Therefore, our key idea is to construct virtual line and point features using the property of tangent invariance under perspective projection. The proposed method requires first a robust detection of ellipse edge points to fit a parametric model on each ellipse. The feature lines are then obtained by computing the 4 bitangents to each couple of ellipses. The points are derived by considering the tangent points and the intersection points between bitangents. Results of experimental studies are presented to demonstrate the reliability and robustness of the feature extraction process. Subpixel accuracy is easily achieved. A real application to camera self-calibration is also described

Stereo image processing system for robot vision

More and more applications (path planning, collision avoidance methods) require 3D description of the surround world. This paper describes a stereo vision system that uses 2D (grayscale or color) images to extract simple 2D geometric entities (points, lines) applying a low-level feature detector. The features are matched across views with a graph matching algorithm. During the projective reconstruction the 3D description of the scene is recovered. The developed system uses uncalibrated cameras, therefore only projective 3D structure can be detected defined up to a collineation. Using the Euclidean information about a known set of predefined objects stored in database and the results of the recognition algorithm, the description can be updated to a metric one

Vision-Based Object Recognition and 3-D Pose Estimation Using Conic Features

This thesis deals with monocular vision-based object recognition and 3-D pose estimation based on conic features. Conic features including circles and ellipses are frequently observed in many man-made objects in real word as well as have the merit of robustness potentially in feature extraction in vision-based applications. Although the 3-D pose estimation problem of conic features in 3-D space has been studied well since 1990, the previous work has not provided a unique solution completely for full 3-D pose parameters (i.e., 3-orientations and 3-positions) due to complexity from high nonlinearity of a general conic. This thesis, therefore, renews conic features in a new perspective on geometric invariants in both 3-D space and 2-D projective space, incorporating other geometric features with conics. First, as the most essential step in dealing with conics, this thesis shows that the pose parameters of a circular feature in 3-D space can be derived analytically from incorporating a coplanar point. A procedure of pose parameter recovery is described in detail, and its performance is evaluated and discussed in view of pose estimation errors and sensitivity. Second, it is also revealed that the pose of an elliptic feature can be resolved when two coplanar points are incorporated on the basis of the polarity of two points for a conic in 2-D projective space. This thesis proposes a series of algorithms to determine the 3-D pose parameters uniquely, and evaluates the proposed method through a measure of estimation performance and sensitivity depending on point locations. Third, a pair of two conics is dealt with, which is regarded as an extension of the idea of the incorporation scheme to another conic feature from point features. Under the polarity concept, this thesis proves that the problem involving a pair of two conics can be formulated with the problem of one ellipse with two points so that its solution is derived in the same form as in the ellipse case. In order to treat two or more conic objects as well as to deal with an object recognition problem, the rest of thesis concentrates on the theoretical foundation of multiple object recognition. First, some effective modeling approaches are described. A general object model is specially designed to model multiple objects for object recognition and pose recovery in view of spatial geometry. In particular, this thesis defines a pairwise conic model that can describes the geometrical relation between two conics invariantly in 2-D projective space, which consists of a pairwise conic (PC), a pairwise conic invariant (PCI), and a pairwise conic pole (PCP). Based on the two kinds of models, an object learning and recognition system is proposed as a general framework for multiple object recognition. Considering simplicity and flexibility in object learning stage, this thesis introduces a semi-automatic learning scheme to construct the multiple object model from a model image at once. To utilize geometric relations among multiple objects effectively in object recognition, this thesis specifies some feature functions based on the pairwise conic model, and then describes an object recognition method in a fashion of linear-chain conditional random field (CRF). In particular, as a post refinement step of the recognition, a geometric alignment procedure is also proposed in algorithmic details to improve recognition performance against noisy conditions. Last, the multiple object recognition method is evaluated intensively through two practical applications that deal with a place recognition and an elevator button recognition problem for service robots. A series of experiment results supports the effectiveness of the proposed method, maintaining reliable performance against noisy conditions in the presence of perspective distortion and partial object occlusions.Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Research objective and expected contribution . . . . . . . . . . . . . . . . . . 6 1.4 Organization of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 3-D Pose Estimation of a Circular Feature 10 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.4 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Preliminaries: an elliptic cone in 3-D space and its homogeneous representation in 2-D projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.1 Homogeneous representation . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.2 Principal planes of a cone versus diagonalization of a conic matrix Q . 16 2.3 3-D interpretation of a circular feature for 3-D pose estimation . . . . . . . . 19 2.3.1 3-D orientation estimation . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 3-D position estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.3 Composition of homogeneous transformation and discrimination for the unique solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Experiment results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.1 A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.2 Evaluation of pose estimation performance . . . . . . . . . . . . . . . 29 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3 3-D Pose Estimation of an Elliptic Feature 35 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Interpretation of an elliptic feature with coplanar points in 2-D projective space 38 3.2.1 The minimal number of points for pose estimation . . . . . . . . . . . 39 3.2.2 Analysis of possible constraints for relative positions of two points to an ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.3 Feature selection scheme for stable homography estimation . . . . . . 43 3.3 3-D pose estimation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.1 Extraction of triangular features from an elliptic object . . . . . . . . 47 3.3.2 Homography decomposition . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.3 Composition of homogeneous transformation matrix with unique solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4 Experiment results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.2 Evaluation of the proposed method . . . . . . . . . . . . . . . . . . . . 54 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 3-D Pose Estimation of a Pair of Conic Features 61 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 3-D pose estimation of a conic feature incorporated with line features . . . . 61 4.3 3-D pose estimation of a conic feature incorporated with another conic feature 63 4.3.1 Some examples of self-polar triangle and invariants . . . . . . . . . . . 65 4.3.2 3-D pose estimation of a pair of coplanar conics . . . . . . . . . . . . . 67 4.3.3 Examples of 3-D pose estimation of a conic feature incorporated with another conic feature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5 Multiple Object Recognition Based on Pairwise Conic Model 77 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2 Learning of geometric relation of multiple objects . . . . . . . . . . . . . . . . 78 5.3 Pairwise conic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3.1 De_nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4 Multiple object recognition based on pairwise conic model and conditional random _elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.4.1 Graphical model for multiple object recognition . . . . . . . . . . . . . 86 5.4.2 Linear-chain conditional random _eld . . . . . . . . . . . . . . . . . . 87 5.4.3 Determination of low-level feature functions for multiple object recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4.4 Range selection trick for e_ciently computing the costs of low-level feature functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.5 Evaluation of observation sequence . . . . . . . . . . . . . . . . . . . . 93 5.4.6 Object recognition based on hierarchical CRF . . . . . . . . . . . . . . 95 5.5 Geometric alignment algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6 Application to Place Recognition for Service Robots 105 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.1.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.2 Feature extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2.1 Detection of 2-D geometric shapes . . . . . . . . . . . . . . . . . . . . 107 6.2.2 Examples of shape feature extraction . . . . . . . . . . . . . . . . . . . 109 6.3 Object modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.3.1 A place model that describes multiple landmark objects . . . . . . . . 112 6.3.2 Pairwise conic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.3.3 Incorporation of non-conic features with a pairwise conic model . . . . 114 6.4 Place learning and recognition system . . . . . . . . . . . . . . . . . . . . . . 121 6.4.1 HCRF-based recognition . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.5 Experiment results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.5.2 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7 Application to Elevator Button Recognition 136 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.1.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.1.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.2 Object modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2.1 Geometric model for multiple button objects . . . . . . . . . . . . . . 140 7.2.2 Pairwise conic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.3 Learning and recognition system . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.3.1 Button object learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.3.2 CRF-based recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.4 Experiment results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.4.2 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8 Concluding remarks 159 8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 References 161 Summary (in Korean) 16

Angular variation as a monocular cue for spatial percepcion

Monocular cues are spatial sensory inputs which are picked up exclusively from one eye. They are in majority static features that provide depth information and are extensively used in graphic art to create realistic representations of a scene. Since the spatial information contained in these cues is picked up from the retinal image, the existence of a link between it and the theory of direct perception can be conveniently assumed. According to this theory, spatial information of an environment is directly contained in the optic array. Thus, this assumption makes possible the modeling of visual perception processes through computational approaches. In this thesis, angular variation is considered as a monocular cue, and the concept of direct perception is adopted by a computer vision approach that considers it as a suitable principle from which innovative techniques to calculate spatial information can be developed. The expected spatial information to be obtained from this monocular cue is the position and orientation of an object with respect to the observer, which in computer vision is a well known field of research called 2D-3D pose estimation. In this thesis, the attempt to establish the angular variation as a monocular cue and thus the achievement of a computational approach to direct perception is carried out by the development of a set of pose estimation methods. Parting from conventional strategies to solve the pose estimation problem, a first approach imposes constraint equations to relate object and image features. In this sense, two algorithms based on a simple line rotation motion analysis were developed. These algorithms successfully provide pose information; however, they depend strongly on scene data conditions. To overcome this limitation, a second approach inspired in the biological processes performed by the human visual system was developed. It is based in the proper content of the image and defines a computational approach to direct perception. The set of developed algorithms analyzes the visual properties provided by angular variations. The aim is to gather valuable data from which spatial information can be obtained and used to emulate a visual perception process by establishing a 2D-3D metric relation. Since it is considered fundamental in the visual-motor coordination and consequently essential to interact with the environment, a significant cognitive effect is produced by the application of the developed computational approach in environments mediated by technology. In this work, this cognitive effect is demonstrated by an experimental study where a number of participants were asked to complete an action-perception task. The main purpose of the study was to analyze the visual guided behavior in teleoperation and the cognitive effect caused by the addition of 3D information. The results presented a significant influence of the 3D aid in the skill improvement, which showed an enhancement of the sense of presence.Las señales monoculares son entradas sensoriales capturadas exclusivamente por un solo ojo que ayudan a la percepción de distancia o espacio. Son en su mayoría características estáticas que proveen información de profundidad y son muy utilizadas en arte gráfico para crear apariencias reales de una escena. Dado que la información espacial contenida en dichas señales son extraídas de la retina, la existencia de una relación entre esta extracción de información y la teoría de percepción directa puede ser convenientemente asumida. De acuerdo a esta teoría, la información espacial de todo le que vemos está directamente contenido en el arreglo óptico. Por lo tanto, esta suposición hace posible el modelado de procesos de percepción visual a través de enfoques computacionales. En esta tesis doctoral, la variación angular es considerada como una señal monocular, y el concepto de percepción directa adoptado por un enfoque basado en algoritmos de visión por computador que lo consideran un principio apropiado para el desarrollo de nuevas técnicas de cálculo de información espacial. La información espacial esperada a obtener de esta señal monocular es la posición y orientación de un objeto con respecto al observador, lo cual en visión por computador es un conocido campo de investigación llamado estimación de la pose 2D-3D. En esta tesis doctoral, establecer la variación angular como señal monocular y conseguir un modelo matemático que describa la percepción directa, se lleva a cabo mediante el desarrollo de un grupo de métodos de estimación de la pose. Partiendo de estrategias convencionales, un primer enfoque implanta restricciones geométricas en ecuaciones para relacionar características del objeto y la imagen. En este caso, dos algoritmos basados en el análisis de movimientos de rotación de una línea recta fueron desarrollados. Estos algoritmos exitosamente proveen información de la pose. Sin embargo, dependen fuertemente de condiciones de la escena. Para superar esta limitación, un segundo enfoque inspirado en los procesos biológicos ejecutados por el sistema visual humano fue desarrollado. Está basado en el propio contenido de la imagen y define un enfoque computacional a la percepción directa. El grupo de algoritmos desarrollados analiza las propiedades visuales suministradas por variaciones angulares. El propósito principal es el de reunir datos de importancia con los cuales la información espacial pueda ser obtenida y utilizada para emular procesos de percepción visual mediante el establecimiento de relaciones métricas 2D- 3D. Debido a que dicha relación es considerada fundamental en la coordinación visuomotora y consecuentemente esencial para interactuar con lo que nos rodea, un efecto cognitivo significativo puede ser producido por la aplicación de métodos de L estimación de pose en entornos mediados tecnológicamente. En esta tesis doctoral, este efecto cognitivo ha sido demostrado por un estudio experimental en el cual un número de participantes fueron invitados a ejecutar una tarea de acción-percepción. El propósito principal de este estudio fue el análisis de la conducta guiada visualmente en teleoperación y el efecto cognitivo causado por la inclusión de información 3D. Los resultados han presentado una influencia notable de la ayuda 3D en la mejora de la habilidad, así como un aumento de la sensación de presencia