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Camera Models and Fundamental Concepts Used in Geometric Compute...

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Camera Models and Fundamental Concepts Used in Geometric Computer Vision By Peter Sturm, Srikumar Ramalingam, Jean-Philippe Tardif, Simone Gasparini and Jo˜ao Barreto

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Foundations and Trends



Computer Graphics and Vision

Vol. 6, Nos. 1–2 (2010) 1–183

 2011 P. Sturm, S. Ramalingam, J.-P. Tardif,

S. Gasparini and J. Barreto

DOI: 10.1561/0600000023

Camera Models and Fundamental Concepts

Used in Geometric Computer Vision

By Peter Sturm, Srikumar Ramalingam,

Jean-Philippe Tardif, Simone Gasparini

and Jo˜ao Barreto

Contents

1 Introduction and Background Material 3

1.1 Introduction 3

1.2 Background Material 6

2 Technologies 8

2.1 Moving Cameras or Optical Elements 8

2.2 Fisheyes 14

2.3 Catadioptric Systems 15

2.4 Stereo and Multi-camera Systems 31

2.5 Others 33

3 Camera Models 36

3.1 Global Camera Models 41

3.2 Local Camera Models 66

3.3 Discrete Camera Models 72

3.4 Models for the Distribution of Camera Rays 75

3.5 Overview of Some Models 82

3.6 So Many Models . . . 84

4 Epipolar and Multi-view Geometry 90

4.1 The Calibrated Case 91

4.2 The Uncalibrated Case 92

4.3 Images of Lines and the Link between Plumb-line

Calibration and Self-calibration of Non-perspective

Cameras 100

5 Calibration Approaches 103

5.1 Calibration Using Calibration Grids 103

5.2 Using Images of Individual Geometric Primitives 110

5.3 Self-calibration 114

5.4 Special Approaches Dedicated to Catadioptric Systems 124

6 Structure-from-Motion 127

6.1 Pose Estimation 128

6.2 Motion Estimation 130

6.3 Triangulation 133

6.4 Bundle Adjustment 134

6.5 Three-Dimensional Scene Modeling 136

6.6 Distortion Correction and Rectiﬁcation 137

7 Concluding Remarks 143

Acknowledgements 145

References 146

Foundations and Trends



Computer Graphics and Vision

Vol. 6, Nos. 1–2 (2010) 1–183

 2011 P. Sturm, S. Ramalingam, J.-P. Tardif,

S. Gasparini and J. Barreto

DOI: 10.1561/0600000023

Camera Models and Fundamental Concepts

Used in Geometric Computer Vision

Peter Sturm

, Srikumar Ramalingam

Jean-Philippe Tardif

, Simone Gasparini

and Jo˜ao Barreto

INRIA Grenoble — Rhˆone-Alpes and Laboratoire Jean Kuntzmann,

Grenoble, Montbonnot, France, Peter.Sturm@inrialpes.fr

MERL, Cambridge, MA, USA, ramalingam@merl.com

NREC — Carnegie Mellon University, Pittsburgh, PA, USA,

tardifj@gmail.com

INRIA Grenoble — Rhˆone-Alpes and Laboratoire Jean Kuntzmann,

Grenoble, Montbonnot, France, Simone.Gasparini@inrialpes.fr

Coimbra University, Coimbra, Portugal, jpbar@deec.uc.pt

Abstract

This survey is mainly motivated by the increased availability and use

of panoramic image acquisition devices, in computer vision and various

of its applications. Diﬀerent technologies and diﬀerent computational

models thereof exist and algorithms and theoretical studies for geomet-

ric computer vision (“structure-from-motion”) are often re-developed

without highlighting common underlying principles. One of the goals

of this survey is to give an overview of image acquisition methods

used in computer vision and especially, of the vast number of cam-

era models that have been proposed and investigated over the years,

where we try to point out similarities between diﬀerent models. Results

on epipolar and multi-view geometry for diﬀerent camera models are

reviewed as well as various calibration and self-calibration approaches,

with an emphasis on non-perspective cameras. We ﬁnally describe what

we consider are fundamental building blocks for geometric computer

vision or structure-from-motion: epipolar geometry, pose and motion

estimation, 3D scene modeling, and bundle adjustment. The main goal

here is to highlight the main principles of these, which are independent

of speciﬁc camera models.

Introduction and Background Material

1.1 Introduction

Many diﬀerent image acquisition technologies have been investigated

in computer vision and other areas, many of them aiming at providing

a wide ﬁeld of view. The main technologies consist of catadioptric and

ﬁsheye cameras as well as acquisition systems with moving parts, e.g.,

moving cameras or optical elements. In this monograph, we try to give

an overview of the vast literature on these technologies and on com-

putational models for cameras. Whenever possible, we try to point out

links between diﬀerent models. Simply put, a computational model for

a camera, at least for its geometric part, tells how to project 3D entities

(points, lines, etc.) onto the image, and vice versa, how to back-project

from the image to 3D. Camera models may be classiﬁed according to

diﬀerent criteria, for example the assumption or not of a single view-

point or their algebraic nature and complexity. Also, recently several

approaches for calibrating and using “non-parametric” camera mod-

els have been proposed by various researchers, as opposed to classical,

parametric models.

In this survey, we propose a diﬀerent nomenclature as our main

criterion for grouping camera models. The main reason is that even

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