The Geometry of Projective Reconstruction I: Matching Constraints and the Joint Image

Circulated in 1995. Accepted subject to revision to IJCV in 1995, but never completedThis paper studies the geometry of perspective projection into multiple images and the matching constraints that this induces between the images. The combined projections produce a 3D subspace of the space of combined image coordinates called the joint image. This is a complete projective replica of the 3D world defined entirely in terms of image coordinates, up to an arbitrary choice of certain scale factors. Projective reconstruction is a canonical process in the joint image requiring only the rescaling of image coordinates. The matching constraints tell whether a set of image points is the projection of a single world point. In 3D there are only three types of matching constraint: the fundamental matrix, Shashua's trilinear tensor, and a new quadrilinear 4 image tensor. All of these fit into a single geometric object, the joint image Grassmannian tensor. This encodes exactly the information needed for reconstruction: the location of the joint image in the space of combined image coordinates

Aspects of Geometric Algebra in Euclidean, Projective and Conformal Space

This report is meant to be a script of a tutorial on Clifford (or Geometric) algebra. It is therefore not complete in the description of the algebra and neither completely rigorous. The reader is also not likely to be able to perform arbitrary calculations with Clifford algebra after reading this script. The goal of this text is to give the reader a feeling for what Clifford algebra is about and how it may be used. It is attempted to convey the basic ideas behind the use of Clifford algebra in the description of geometry in Euclidean, projective and conformal space

Robust Multiple-View Geometry Estimation Based on GMM

Given three partially overlapping views of the scene from which a set of point or line correspondences have been extracted, 3D structure and camera motion parameters can be represented by the trifocal tensor, which is the key to many problems of computer vision on three views. Unlike in conventional typical methods, the residual value is the only rule to eliminate outliers with large value, we build a Gaussian mixture model assuming that the residuals corresponding to the inliers come from Gaussian distributions different from that of the residuals of outliers. Then Bayesian rule of minimal risk is employed to classify all the correspondences using the parameters computed from GMM. Experiments with both synthetic data and real images show that our method is more robust and precise than other typical methods because it can efficiently detect and delete the bad corresponding points, which include both bad locations and false matches

The joint image handbook

International audienceGiven multiple perspective photographs, point correspondences form the " joint image " , effectively a replica of three-dimensional space distributed across its two-dimensional projections. This set can be characterized by multilinear equations over image coordinates, such as epipolar and trifocal constraints. We revisit in this paper the geometric and algebraic properties of the joint image, and address fundamental questions such as how many and which multilinearities are necessary and/or sufficient to determine camera geometry and/or image correspondences. The new theoretical results in this paper answer these questions in a very general setting and, in turn, are intended to serve as a " handbook " reference about multilinearities for practitioners

Euclidean reconstruction and reprojection up to subgroups

The necessaryand sufficient conditionsfor being able to estimatescene structure, motion and camera calibration from a sequence of images are very rarely satisfied in practice. What exactly can be estimated in sequences of practical importance, when such conditions are not satisfied? In this paper we give a complete answer to this question. For every camera motion that fails to meet the conditions, we give explicit formulas for the ambiguities in the reconstructed scene, motion and calibration. Such a characterization is crucial both for designing robust estimation algorithms (that do not try to recover parameters that cannot be recovered), and for generating novel views of the scene by controlling the vantage point. To this end, we characterizeexplicitly all the vantage points that give rise to a valid Euclidean reprojection regardless of the ambiguity in the reconstruction. We also characterize vantage points that generate views that are altogether invariant to the ambiguity. All the results are presented using simple notation that involves no tensors nor complex projective geometry, and should be accessible with basic background in linear algebra. 1

Grassmann-Cayley algebra for modeling systems of cameras and the algebraic equations of the manifold of trifocal tensors

We show how to use the Grassmann-Cayley algebra to model systems of one, two and three cameras. We start with a brief introduction of the Grassmann-Cayley or double algebra and proceed to demonstrate its use for modeling systems of cameras. In the case of three cameras, we give a new interpretation of the trifocal tensors and study in detail some of the constraints that they satisfy. In particular we prove that simple subsets of those constraints characterize the trifocal tensors, in other words, we give the algebraic equations of the manifold of trifocal tensors

Du texte à la génération d'environnements virtuels 3D : application à la scénographie théâtrale

This thesis is part of a multidisciplinary project, the DRAMA project, which attempts to generate 3D virtual scenes from the descriptions which are obtained from theatrical text. This project aims to simplify, as soon as possible, the tasks of the end-users by providing simple, fast, and effective tools. Thus, the technique used in this study is focused on the declarative modeling of virtual environments that is based on three phases (description, generation and management of knowledge). The description phase allows the designer to describe the environment from a set of properties, interpreted as a set of constraints for a generation system which produces one or several virtual environments solutions. This project, new tagging methods have been proposed to detect essential for the creation of scene, including information on the placement of objects. In addition, users can also run queries in the text from these tags. Placement properties are translated into spatial constraints with the data originally stored in a knowledge base that uses XML. A technique adopting the method of metaheuristics is then used for solving constraints. The object physical properties (collision, gravity, friction) were also managed from a physics engine. At the end, the finals scenes solutions were be proposed to the user, using a 3D rendering engine.Cette thèse s'inscrit dans le cadre d'un projet pluridisciplinaire, le projet DRAMA, qui consiste à générer des scènes virtuelles 3D à partir des descriptions contenues dans les textes théâtraux. L'un des objectifs de ce projet consiste à simplifier au maximum la tâche des utilisateurs finaux en leur offrant un outil simple, rapide, et efficace. Ainsi, la technique adoptée dans cette étude est axée sur la modélisation déclarative d'environnements virtuels qui s'appuie sur trois phases (description, génération et prise de connaissances). La phase de description permet au concepteur de décrire l'environnement à partir d'un ensemble de propriétés, interprétées en un ensemble de contraintes destinées à un système de génération qui produit un ou plusieurs environnements virtuels solutions.Dans le cadre de ce projet DRAMA, des nouvelles méthodes de balisage ont été proposées afin de détecter les éléments essentiels pour la création d'une pièce théâtrale, notamment les informations sur les placements d'objets. Par ailleurs, les utilisateurs peuvent, aussi, lancer des requêtes au niveau du texte à partir de ces balises. Les propriétés sur les placements seront traduites en contraintes spatiales grâce aux données initialement stockées dans une base de connaissance qui utilise le langage XML. Une technique adoptant la méthode des métaheuristiques est ensuite utilisée pour la résolution des contraintes de placements obtenues précédemment. La gestion des propriétés physiques des objets (collision, gravité, friction) a été aussi gérée à partir d'un moteur physique. À la fin, les scènes solutions finales seront proposées à l'utilisateur, en utilisant un moteur de rendu 3D