《计算机视觉中的多视图几何》经典第二版详解

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"《计算机视觉中的多视图几何(第二版)》是由Richard Hartley和Andrew Zisserman两位权威作者合著的经典之作。这本书是机器视觉领域的基石，特别关注于相机标定和多视图几何在计算机视觉中的应用。它是Cambridge University Press出版的，专为那些希望深入了解计算机视觉原理和技术的专业人士设计。该书详细探讨了如何通过多个视角的图像来理解三维空间，包括相机的内在参数、外在参数估计、投影模型、特征匹配、结构从运动等核心概念。第二版更新了技术发展和研究成果，为读者提供了丰富的理论框架和实践案例。书中不仅包含了理论阐述，还配以清晰的公式推导和实用的算法，使得学习者能够逐步掌握多视图处理的技术。相较于其他版本，如CSDN上可能存在的扫描版，这本书的文字版质量更佳，体积小巧，便于阅读和学习。同时，由于是正式出版物，版权保护严密，未经许可的复制行为可能会涉及到法律问题。如果你是学生、研究者或工程师，这本书是你提升计算机视觉技能、理解三维重建和相机校准不可或缺的参考书。为了获取最新信息和进一步资源，你可以访问出版社官网www.cambridge.org/9780521540513，那里可能有电子版、在线课程或其他补充材料。此外，注意书中的ISBN号码，它们对于查找图书、订阅电子资源或获得授权至关重要。《计算机视觉中的多视图几何(第二版)》是一本深度和广度兼备的教材，无论你是入门者还是资深从业者，它都将为你提供一个扎实的理论基础和实践经验，助力你在计算机视觉领域取得成功。"

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

1.1 Introduction – the ubiquitous projective geometry 3

to IR

, followed by a Euclidean transformation moving the origin of the space. We

may think of this as the space moving, rotating and ﬁnally stretching linearly possibly

by different ratios in different directions. The resulting transformation is known as an

afﬁne transformation.

The result of either a Euclidean or an afﬁne transformation is that points at inﬁn-

ity remain at inﬁnity. Such points are in some way preserved, at least as a set, by

such transformations. They are in some way distinguished, or special in the context of

Euclidean or afﬁne geometry.

From the point of view of projective geometry, points at inﬁnity are not any dif-

ferent from other points. Just as Euclidean space is uniform, so is projective space.

The property that points at inﬁnity have ﬁnal coordinate zero in a homogeneous co-

ordinate representation is nothing other than an accident of the choice of coordinate

frame. By analogy with Euclidean or afﬁne transformations, we may deﬁne a projec-

tive transformation of projective space. A linear transformation of Euclidean space IR

is represented by matrix multiplication applied to the coordinates of the point. In just

the same way a projective transformation of projective space IP

is a mapping of the

homogeneous coordinates representing a point (an (n +1)-vector), in which the coor-

dinate vector is multiplied by a non-singular matrix. Under such a mapping, points at

inﬁnity (with ﬁnal coordinate zero) are mapped to arbitrary other points. The points at

inﬁnity are not preserved. Thus, a projective transformation of projective space IP

represented by a linear transformation of homogeneous coordinates



= H

(n+1)×(n+1)

In computer vision problems, projective space is used as a convenient way of repre-

senting the real 3D world, by extending it to the 3-dimensional (3D) projective space.

Similarly images, usually formed by projecting the world onto a 2-dimensional repre-

sentation, are for convenience extended to be thought of as lying in the 2-dimensional

projective space. In reality, the real world, and images of it do not contain points at

inﬁnity, and we need to keep our ﬁnger on which are the ﬁctitious points, namely the

line at inﬁnity in the image and the plane at inﬁnity in the world. For this reason, al-

though we usually work with the projective spaces, we are aware that the line and plane

at inﬁnity are in some way special. This goes against the spirit of pure projective ge-

ometry, but makes it useful for our practical problems. Generally we try to have it both

ways by treating all points in projective space as equals when it suits us, and singling

out the line at inﬁnity in space or the plane at inﬁnity in the image when that becomes

necessary.

1.1.1 Afﬁne and Euclidean Geometry

We have seen that projective space can be obtained from Euclidean space by adding

a line (or plane) at inﬁnity. We now consider the reverse process of going backwards.

This discussion is mainly concerned with two and three-dimensional projective space.

Afﬁne geometry. We will take the point of view that the projective space is initially

homogeneous, with no particular coordinate frame being preferred. In such a space,

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