机器人视觉中的多视图几何学及其应用

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"Multiple View Geometry in Computer Vision（Second Edition）是由Richard Hartley和Andrew Zisserman合著的一本书，主要探讨了计算机视觉中的多视角几何理论及其在机器人视觉感知与控制中的应用。书中利用多视角几何来从高维图像信息中提取低维度的几何信息，以解决机器人系统的感知和控制问题。同时，还介绍了如何在模型不确定性和多重约束条件下，运用Lyapunov方法设计稳定控制律。" 这本书是计算机视觉领域的经典之作，第二版对初版进行了更新和扩展，涵盖了多视角几何学的最新进展。多视角几何是计算机视觉中的核心概念，它涉及到通过不同视角拍摄的图像来重建三维场景几何结构的问题。该书深入浅出地讲解了如何从多个视图的图像中提取关键的几何信息，例如对应点匹配、基础矩阵和本质矩阵的计算、三维重建以及摄像机姿态估计等。移动机器人和SLAM（Simultaneous Localization And Mapping，即同时定位与建图）是书中讨论的重要应用领域。多视角几何在SLAM中发挥着至关重要的作用，帮助机器人在未知环境中定位自身并构建环境地图。通过理解和应用多视角几何，机器人可以更准确地理解其周围环境，实现自主导航和交互。书中的另一亮点是引入了Lyapunov方法，这是一种稳定性分析和控制器设计的技术，常用于动态系统。在机器人控制中，Lyapunov方法可以帮助设计出即使在模型不确定性和存在多种约束条件下的稳定控制策略。这种方法确保了机器人的运动控制能够在面对环境变化时保持稳定，这对于实现精确的机器人操作和避障策略至关重要。 "Multiple View Geometry in Computer Vision（Second Edition）"不仅提供了多视角几何的基础理论，还将其与实际的机器人系统相结合，为研究者和工程师提供了宝贵的理论工具和实践经验。这本书对于深入理解计算机视觉、移动机器人技术以及SLAM算法的开发具有极高的参考价值。

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xiv Preface

that are required at each stage of a computation. This avoids computing parameters

that have no effect on the ﬁnal result, and results in simpler algorithms. It is also

worth correcting a possible misconception. In the uncalibrated framework, entities (for

instance point positions in 3-space) are often recovered to within a precisely deﬁned

ambiguity. This ambiguity does not mean that the points are poorly estimated.

More practically, it is often not possible to calibrate cameras once-and-for-all; for

instance where cameras are moved (on a mobile vehicle) or internal parameters are

changed (a surveillance camera with zoom). Furthermore, calibration information is

simply not available in some circumstances. Imagine computing the motion of a cam-

era from a video sequence, or building a virtual reality model from archive ﬁlm footage

where both motion and internal calibration information are unknown.

The achievements in multiple view geometry have been possible because of develop-

ments in our theoretical understanding, but also because of improvements in estimating

mathematical objects from images. The ﬁrst improvement has been an attention to the

error that should be minimized in over-determined systems – whether it be algebraic,

geometric or statistical. The second improvement has been the use of robust estimation

algorithms (such as RANSAC), so that the estimate is unaffected by “outliers” in the

data. Also these techniques have generated powerful search and matching algorithms.

Many of the problems of reconstruction have now reached a level where we may

claim that they are solved. Such problems include:

(i) Estimation of the multifocal tensors from image point correspondences, par-

ticularly the fundamental matrix and trifocal tensors (the quadrifocal tensor

having not received so much attention).

(ii) Extraction of the camera matrices from these tensors, and subsequent projective

reconstruction from two, three and four views.

Other signiﬁcant successes have been achieved, though there may be more to learn

about these problems. Examples include:

(i) Application of bundle adjustment to solve more general reconstruction prob-

lems.

(ii) Metric (Euclidean) reconstruction given minimal assumptions on the camera

matrices.

(iii) Automatic detection of correspondences in image sequences, and elimination

of outliers and false matches using the multifocal tensor relationships.

Roadplan. The book is divided into six parts and there are seven short appendices.

Each part introduces a new geometric relation: the homography for background, the

camera matrix for single view, the fundamental matrix for two views, the trifocal tensor

for three views, and the quadrifocal tensor for four views. In each case there is a

chapter describing the relation, its properties and applications, and a companion chapter

describing algorithms for its estimation from image measurements. The estimation

algorithms described range from cheap, simple, approaches through to the optimal

algorithms which are currently believed to be the best available.

Preface xv

Part 0: Background. This part is more tutorial than the others. It introduces the

central ideas in the projective geometry of 2-space and 3-space (for example

ideal points, and the absolute conic); how this geometry may be represented,

manipulated, and estimated; and how the geometry relates to various objectives

in computer vision such as rectifying images of planes to remove perspective

distortion.

Part 1: Single view geometry. Here the various cameras that model the perspective

projection from 3-space to an image are deﬁned and their anatomy explored.

Their estimation using traditional techniques of calibration objects is described,

as well as camera calibration from vanishing points and vanishing lines.

Part 2: Two view geometry. This part describes the epipolar geometry of two

cameras, projective reconstruction from image point correspondences, methods

of resolving the projective ambiguity, optimal triangulation, transfer between

views via planes.

Part 3: Three view geometry. Here the trifocal geometry of three cameras is de-

scribed, including transfer of a point correspondence from two views to a third,

and similarly transfer for a line correspondence; computation of the geometry

from point and line correspondences, retrieval of the camera matrices.

Part 4: N-views. This part has two purposes. First, it extends three view geometry

to four views (a minor extension) and describes estimation methods applica-

ble to N-views, such as the factorization algorithm of Tomasi and Kanade for

computing structure and motion simultaneously from multiple images. Sec-

ond, it covers themes that have been touched on in earlier chapters, but can

be understood more fully and uniformly by emphasising their commonality.

Examples include deriving multi-linear view constraints on correspondences,

auto-calibration, and ambiguous solutions.

Appendices. These describe further background material on tensors, statistics, pa-

rameter estimation, linear and matrix algebra, iterative estimation, the solution

of sparse matrix systems, and special projective transformations.

Acknowledgements. We have beneﬁted enormously from ideas and discussions with

our colleagues: Paul Beardsley, Stefan Carlsson, Olivier Faugeras, Andrew Fitzgibbon,

Jitendra Malik, Steve Maybank, Amnon Shashua, Phil Torr, Bill Triggs.

If there are only a countable number of errors in this book then it is due to Antonio

Criminisi, David Liebowitz and Frederik Schaffalitzky who have with great energy and

devotion read most of it, and made numerous suggestions for improvements. Similarly

both Peter Sturm and Bill Triggs have suggested many improvements to various chap-

ters. We are grateful to other colleagues who have read individual chapters: David

Capel, Lourdes de Agapito Vicente, Bob Kaucic, Steve Maybank, Peter Tu.

We are particularly grateful to those who have provided multiple ﬁgures: Paul Beard-

sley, Antonio Criminisi, Andrew Fitzgibbon, David Liebowitz, and Larry Shapiro; and

for individual ﬁgures from: Martin Armstrong, David Capel, Lourdes de Agapito Vi-

cente, Eric Hayman, Phil Pritchett, Luc Robert, Cordelia Schmid, and others who are

explicitly acknowledged in ﬁgure captions.

xvi Preface

At Cambridge University Press we thank David Tranah for his constant source of

advice and patience, and Michael Behrend for excellent copy editing.

A small number of minor errors have been corrected in the reprinted editions, and

we thank the following readers for pointing these out: Luis Baumela, Niclas Borlin,

Mike Brooks, Jun ho. Choi, Wojciech Chojnacki, Carlo Colombo, Nicolas Dano, An-

drew Fitzgibbon, Bogdan Georgescu, Fredrik Kahl, Bob Kaucic, Jae-Hak Kim, Han-

sung Lee, Dennis Maier, Karsten Muelhmann, David Nister, Andreas Olsson, St

ephane

Paris, Frederik Schaffalitzky, Bill Severson, Pedro Lopez de Teruel Alcolea, Bernard

Thiesse, Ken Thornton, Magdalena Urbanek, Gergely Vass, Eugene Vendrovsky, Sui

Wei, and Tom

sWerner.

The second edition. This new paperback edition has been expanded to include some

of the developments since the original version of July 2000. For example, the book

now covers the discovery of a closed form factorization solution in the projective case

when a plane is visible in the scene, and the extension of afﬁne factorization to non-

rigid scenes. We have also extended the discussion of single view geometry (chapter 8)

and three view geometry (chapter 15), and added an appendix on parameter estimation.

In preparing this second edition we are very grateful to colleagues who have made

suggestion for improvements and additions. These include Marc Pollefeys, Bill Triggs

and in particular Tom

s Werner who provided excellent and comprehensive comments.

We also thank Antonio Criminisi, Andrew Fitzgibbon, Rob Fergus, David Liebowitz,

and particularly Josef

Sivic, for proof reading and very helpful comments on parts of

the new material. As always we are grateful to David Tranah of CUP.

The ﬁgures appearing in this book can be downloaded from

http://www.robots.ox.ac.uk/∼vgg/hzbook.html

This site also includes Matlab code for several of the algorithms, and lists the errata of

earlier printings.

I am never forget the day my ﬁrst book is published. Every chapter I stole from somewhere else. Index

I copy from old Vladivostok telephone directory. This book, this book was sensational!

Excerpts from “Nikolai Ivanovich Lobachevsky” by Tom Lehrer.

Introduction – a Tour of Multiple View Geometry

This chapter is an introduction to the principal ideas covered in this book. It gives an

informal treatment of these topics. Precise, unambiguous deﬁnitions, careful algebra,

and the description of well honed estimation algorithms is postponed until chapter 2

and the following chapters in the book. Throughout this introduction we will generally

not give speciﬁc forward pointers to these later chapters. The material referred to can

be located by use of the index or table of contents.

1.1 Introduction – the ubiquitous projective geometry

We are all familiar with projective transformations.When we look at a picture, we see

squares that are not squares, or circles that are not circles. The transformation that

maps these planar objects onto the picture is an example of a projective transformation.

So what properties of geometry are preserved by projective transformations? Cer-

tainly, shape is not, since a circle may appear as an ellipse. Neither are lengths since

two perpendicular radii of a circle are stretched by different amounts by the projective

transformation. Angles, distance, ratios of distances – none of these are preserved,

and it may appear that very little geometry is preserved by a projective transformation.

However, a property that is preserved is that of straightness. It turns out that this is

the most general requirement on the mapping, and we may deﬁne a projective trans-

formation of a plane as any mapping of the points on the plane that preserves straight

lines.

To see why we will require projective geometry we start from the familiar Euclidean

geometry. This is the geometry that describes angles and shapes of objects. Euclidean

geometry is troublesome in one major respect – we need to keep making an exception

to reason about some of the basic concepts of the geometry – such as intersection of

lines. Two lines (we are thinking here of 2-dimensional geometry) almost always meet

in a point, but there are some pairs of lines that do not do so – those that we call parallel.

A common linguistic device for getting around this is to say that parallel lines meet “at

inﬁnity”. However this is not altogether convincing, and conﬂicts with another dictum,

that inﬁnity does not exist, and is only a convenient ﬁction. We can get around this by

2 1 Introduction – a Tour of Multiple View Geometry

enhancing the Euclidean plane by the addition of these points at inﬁnity where parallel

lines meet, and resolving the difﬁculty with inﬁnity by calling them “ideal points.”

By adding these points at inﬁnity, the familiar Euclidean space is transformed into a

new type of geometric object, projective space. This is a very useful way of thinking,

since we are familiar with the properties of Euclidean space, involving concepts such as

distances, angles, points, lines and incidence. There is nothing very mysterious about

projective space – it is just an extension of Euclidean space in which two lines always

meet in a point, though sometimes at mysterious points at inﬁnity.

Coordinates. A point in Euclidean 2-space is represented by an ordered pair of real

numbers, (x, y). We may add an extra coordinate to this pair, giving a triple (x, y, 1),

that we declare to represent the same point. This seems harmless enough, since we

can go back and forward from one representation of the point to the other, simply by

adding or removing the last coordinate. We now take the important conceptual step

of asking why the last coordinate needs to be 1 – after all, the others two coordinates

are not so constrained. What about a coordinate triple (x, y, 2). It is here that we

make a deﬁnition and say that (x, y, 1) and (2x, 2y, 2) represent the same point, and

furthermore, (kx,ky, k) represents the same point as well, for any non-zero value k.

Formally, points are represented by equivalence classes of coordinate triples, where

two triples are equivalent when they differ by a common multiple. These are called the

homogeneous coordinates of the point. Given a coordinate triple (kx,ky, k), we can

get the original coordinates back by dividing by k to get (x, y).

The reader will observe that although (x, y, 1) represents the same point as the co-

ordinate pair (x, y), there is no point that corresponds to the triple (x, y, 0).Ifwetry

to divide by the last coordinate, we get the point (x/0,y/0) which is inﬁnite. This is

how the points at inﬁnity arise then. They are the points represented by homogeneous

coordinates in which the last coordinate is zero.

Once we have seen how to do this for 2-dimensional Euclidean space, extending it

to a projective space by representing points as homogeneous vectors, it is clear that we

can do the same thing in any dimension. The Euclidean space IR

can be extended to

a projective space IP

by representing points as homogeneous vectors. It turns out that

the points at inﬁnity in the two-dimensional projective space form a line, usually called

the line at inﬁnity. In three-dimensions they form the plane at inﬁnity.

Homogeneity. In classical Euclidean geometry all points are the same. There is no

distinguished point. The whole of the space is homogeneous. When coordinates are

added, one point is seemingly picked out as the origin. However, it is important to

realize that this is just an accident of the particular coordinate frame chosen. We could

just as well ﬁnd a different way of coordinatizing the plane in which a different point

is considered to be the origin. In fact, we can consider a change of coordinates for the

Euclidean space in which the axes are shifted and rotated to a different position. We

may think of this in another way as the space itself translating and rotating to a different

position. The resulting operation is known as a Euclidean transform.

A more general type of transformation is that of applying a linear transformation

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