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An Invitation to 3-D Vision
From Images to Models
Yi Ma
Jana Koˇseck´a
Stefano Soatto
Shankar Sastry
November 19, 2001
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This book is dedicated to people who love this book.
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Preface
This book is intended to give students at the advanced undergraduate or
introductory graduate level and researchers in computer vision, robotics,
and computer graphics a self-contained introduction to the geometry of
3-D vision: that is, the reconstruction of 3-D models of objects from a
collection of 2-D images. The only prerequisite for this book is a course in
linear algebra at the undergraduate level.
As timely research summary, two bursts of manuscripts were published
in the past on a geometric approach to computer vision: the ones that
were published in mid 1990’s on the geometry of two views (e.g., Faugeras
1993, Weng, Ahuja and Huang 1993, Maybank 1993), and the ones that
were recently published on the geometry of multiple views (e.g., Hartley
and Zisserman 2000, Faugeras, Luong and Papadopoulo 2001).
1
While a
majority of those manuscripts were to summarize up to date research re-
sults by the time they were published, we sense that now the time is ripe
for putting a coherent part of that material in a unified and yet simplified
framework which can be used for pedegogical purposes. Although the ap-
proach we are to take here deviates from those old ones and the techniques
we use are mainly linear algebra, this book nonetheless gives a comprehen-
sive coverage of what is known todate on the geometry of 3-D vision. It
1
To our knowledge, there are also two other books on computer vision currently in
preparation: Ponce and Forsyth (expected in 2002), Pollefeys and van Gool (expected
in 2002).
viii Preface
also builds on a homogeneous terminology a solid analytical foundation for
future research in this young field.
This book is organized as follows. Following a brief introduction, Part I
provides background materials for the rest of the book. Two fundamental
transformations in multiple view geometry, namely the rigid-body motion
and perspective projection, are introduced in Chapters 2 and 3 respectively.
Image formation and feature extraction are discussed in Chapter 4. The two
chapters in Part II cover the classic theory of two view geometry based on
the so-called epipolar constraint. Theory and algorithms are developed for
both discrete and continuous motions, and for both calibrated and uncal-
ibrated camera models. Although the epipolar constraint has been very
successful in the two view case, Part III shows that a more proper tool
for studying the geometry of multiple views is the so-called rank condi-
tion on the multiple view matrix, which trivially implies all the constraints
among multiple images that are known todate, in particular the epipolar
constraint. The theory culminates in Chapter 10 with a unified theorem
on a rank condition for arbitrarily mixed point, line and plane features. It
captures all possible constraints among multiple images of these geometric
primitives, and serves as a key to both geometric analysis and algorithmic
development. Based on the theory and conceptual algorithms developed in
early part of the book, Part IV develops practical reconstruction algorithms
step by step, as well as discusses possible extensions of the theory covered
in this book.
Different parts and chapters of this book have been taught as a one-
semester course at the University of California at Berkeley, the University
of Illinois at Urbana-Champaign, and the George Mason University, and as
a two-quater course at the University of California at Los Angles. There is
apparantly adequate material for lectures of one and a half semester or two
quaters. Advanced topics suggested in Part IV or chosen by the instructor
can be added to the second half of the second semester if a two-semester
course is offered. Given below are some suggestions for course development
based on this book:
1. A one-semester course: Appendix A and Chapters 1 - 7 and part of
Chapters 8 and 13.
2. A two-quater course: Chapters 1 - 6 for the first quater and Chapters
7 - 10 and 13 for the second quater.
3. A two-semester course: Appendix A and Chapters 1 - 7 for the first
semester; Chapters 8 - 10 and the instructor’s choice of some advanced
topics from chapters in Part IV for the second semester.
Exercises are provided at the end of each chapter. They consist of three
types: 1. drill exercises that help student understand the theory covered
in each chapter; 2. programming exercises that help student grasp the
algorithms developed in each chapter; 3. exercises that guide student to
