计算机视觉中的多视图几何原理与应用

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"《计算机视觉中的多视点几何》第二版是由Richard Hartley和Andrew Zisserman合著的一本经典著作，主要关注计算机视觉领域中理解真实世界场景结构的关键问题。该书源自于项目ive几何和摄影测量技术，针对如何从多个视角图像中解析物体的三维结构进行了深入探讨。书中核心内容包括相机投影矩阵的几何原理及其代数表示，这些矩阵用于描述光线如何从相机到三维空间的映射过程。接着，作者引入了基本矩阵和三焦距张量的概念，它们是处理多视图问题的重要工具。基本矩阵用于确定两个或更多相机之间的一对对应点的相对位置，而三焦距张量则进一步扩展了这一概念，用于处理三个或更多相机的情况。理论部分详细阐述了这些矩阵的计算方法，并通过实际案例进行解释，使读者能够理解并应用这些技术来重建场景。此外，书中还提供了丰富的背景材料，确保熟悉线性代数和基本数值方法的读者能够跟上讨论，直接从书中实现算法。新版本在原有基础上做了扩展，增添了对全书关键思想的全新介绍，增加了新的示例和附录，反映了近年来计算机视觉领域的最新进展。特别地，版权信息强调了未经许可复制内容需遵守相关法律法规，并提醒读者外部网站链接可能存在变动，但出版社不保证其内容的持续准确。《计算机视觉中的多视点几何》是研究和实践者必备的参考书，它不仅涵盖了多视点几何的基本理论，还提供了实用的技术手段，对于解决现实世界的计算机视觉问题具有重要的指导意义。"

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,

4 1 Introduction – a Tour of Multiple View Geometry

there is no concept of parallelism of lines, since parallel lines (or planes in the three-

dimensional case) are ones that meet at inﬁnity. However, in projective space, there is

no concept of which points are at inﬁnity – all points are created equal. We say that

parallelism is not a concept of projective geometry. It is simply meaningless to talk

about it.

In order for such a concept to make sense, we need to pick out some particular line,

and decide that this is the line at inﬁnity. This results in a situation where although

all points are created equal, some are more equal than others. Thus, start with a blank

sheet of paper, and imagine that it extends to inﬁnity and forms a projective space

. What we see is just a small part of the space, that looks a lot like a piece of the

ordinary Euclidean plane. Now, let us draw a straight line on the paper, and declare

that this is the line at inﬁnity. Next, we draw two other lines that intersect at this

distinguished line. Since they meet at the “line at inﬁnity” we deﬁne them as being

parallel. The situation is similar to what one sees by looking at an inﬁnite plane. Think

of a photograph taken in a very ﬂat region of the earth. The points at inﬁnity in the

plane show up in the image as the horizon line. Lines, such as railway tracks show

up in the image as lines meeting at the horizon. Points in the image lying above the

horizon (the image of the sky) apparently do not correspond to points on the world

plane. However, if we think of extending the corresponding ray backwards behind the

camera, it will meet the plane at a point behind the camera. Thus there is a one-to-one

relationship between points in the image and points in the world plane. The points at

inﬁnity in the world plane correspond to a real horizon line in the image, and parallel

lines in the world correspond to lines meeting at the horizon. From our point of view,

the world plane and its image are just alternative ways of viewing the geometry of a

projective plane, plus a distinguished line. The geometry of the projective plane and a

distinguished line is known as afﬁne geometry and any projective transformation that

maps the distinguished line in one space to the distinguished line of the other space is

known as an afﬁne transformation.

By identifying a special line as the “line at inﬁnity” we are able to deﬁne parallelism

of straight lines in the plane. However, certain other concepts make sense as well, as

soon as we can deﬁne parallelism. For instance, we may deﬁne equalities of intervals

between two points on parallel lines. For instance, if A, B, C and D are points, and

the lines AB and CD are parallel, then we deﬁne the two intervals AB and CD to

have equal length if the lines AC and BD are also parallel. Similarly, two intervals on

the same line are equal if there exists another interval on a parallel line that is equal to

both.

Euclidean geometry. By distinguishing a special line in a projective plane, we gain

the concept of parallelism and with it afﬁne geometry.Afﬁne geometry is seen as

specialization of projective geometry, in which we single out a particular line (or plane

– according to the dimension) and call it the line at inﬁnity.

Next, we turn to Euclidean geometry and show that by singling out some special

feature of the line or plane at inﬁnity afﬁne geometry becomes Euclidean geometry. In

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