Numerical Geometry of Images: Theory, Algorithms, and Applications

221 Pages·2004·5.95 MB·English

by  Ron Kimmel (auth.)

Numerical Geometry of Images
Springer Science+Business Media, LLC
Ron Kimmel
Numerical Geometry
of Images
Theory, Algorithms,
and Applications
, Springer
RonKimmel
Department of Computer Science
Technion-Israel Institute ofTechnology
Haifa 32000
Israel
ron(jcs.technion.ac.il
www.cs.technion.ac.ilI-ron
Couer: lllustration by Michael and Alex Bronstein.
Library of Congress Cataloging-in-Publication Data
Kimmel, Ron.
Numerical geometry of images : theory, algorithms, and applications I Ron Kimmel.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4684-9535-5 ISBN 978-0-387-21637-9 (eBook)
DOI 10.1007/978-0-387-21637-9
1. Geometry, Differential-Data processing. 1. Title.
QA641.K56 2003
516.3'6--dc21 20003052963
Printed on acid-free paper.
" 2004 Springer Science+-Business Media New York
Originally published by Springer-Verlag New York,lnc. in 2004
All rights reserved. This work may not be translated or copied in whole or in part with·
ou t the written permission of the publisher (Springer Science+Business Media, LLC) except
for brief excerpts in connection with review$ ar seholaTly analysis. Use in eonnection
with any form of information storage and retrieval. electronic adaptation. computer
software, or by similar or dissimilar methodology now known or hereafter developed is
forbidden.
The use in this publication of trade names, trademarks, service marks, and similar
terms, even if they are not identified as such, is not to be taken as an expression of
opinion as to whether or not they are subject to proprietary rights.
987654321 SPIN 10889579
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To my family
Preface
Perception of visual information occupies about 70% of our cortex activity.
Apparently, about the same percentage of visual content occupies commu
nication networks over the world. Thus, it is not surprising that as scien
tists we find it fascinating to process visual information and explore related
fields in which we evaluate and model the formation of images and try to
interpret their content. For example, we try to understand the process of
visual perception, and invent new methods of analysis of visual data, such
as our perception of color. What is a color image? And how can it be effi
ciently described, processed, and analyzed? In our study we need some sort
of vocabulary and machinery to first formulate the problems that explain
natural phenomena, and then find efficient solutions. This book provides a
new view point on some classical problems, and uses classical tools to solve
new problems. The variational geometric methods and resulting numerical
schemes that we review in this book have already proven to be useful in var
ious industrial applications where image analysis of complicated scenarios
is involved.
Intended Use
This book is intended as a graduate textbook for engineers and computer
science students. Unlike other monographs in this area, which assume deep
prior knowledge, we provide the required mathematical basis using a se
quence of exercises that lead the reader to an understanding of the tools
we use. This is true for calculus of variations and differential geometry, as
well as numerical analysis methods.
viii PREFACE
Prerequisites
We assume knowledge in basic calculus, linear algebra, and geometry. Be
yond that, the basics are reviewed and taught through a sequence of exer
cises, with solutions to selected ones. These enable a smooth introduction
of advanced methods and related applications.
Features
The book introduces modern and classical mathematical tools to the field of
image analysis and processing. It has been taught as a one-semester course
for graduate and undergraduate students at the Technion, Israel and as an
invited tutorial in various forms by industrial companies and universities.
The focus of the book is more on the tools rather than the problems, and
in many cases problems from other disciplines are given as examples.
Roughly speaking, the book is divided into two main parts. The first
five chapters are an introduction to the mathematical and numerical tools
we use for the algorithms and applications that are presented in the second
part of the book. The mathematical language we use is heavily influenced
by differential geometry of curves and surfaces, as well as calculus of vari
ations. Therefore, we start with brief introductions to these two powerful
mathematical tools in Chapters 1 and 2. Next, Chapter 3 adds motion to
the curves and surfaces and allows them to dynamically propagate. It also
studies the properties of the evolving curves, such as curvature flow, also
known as the "geometric heat equation," and affine flow. We can use dy
namic curves as steepest descent processes of geometric models that we
meet again later in the book. For example, the curvature flow is the result
of arclength minimization of a planar curve, yet it also plays a major role
in "total variation" minimization in image processing, as we meet again
in Chapter 10. The link between propagating curves and dynamic images
is the Osher-Sethian level set formulation introduced in Chapter 4. This
modern formulation of curve evolution and general front propagation is a
natural bridge between image processing and dynamics of embedded prop
agating curves. Chapter 5 brings some numerical considerations of working
with the level set formulation.
Mathematical morphology is the main theme of Chapter 6. This field
deals with set operations on geometric structures. In simple words, it is the
algebra of shapes. We touch upon this field only through its connection to
curve evolution, and relate to this aspect of the whole theory. If we restrict
some of the shapes to be convex, then the basic morphology operations
are closely related to the problem of computing distances. Efficient and
accurate methods of computing distances in various geometric scenarios
bring us into the fast marching algorithms presented in Chapter 7. Next,
PREFACE ix
we are ready to explore the classic problem of shape from shading, which
is presented in Chapter 8. The shape from shading is a toy problem in
the field of computer vision by which we link various methods that we
introduce and study in previous chapters. We show how this problem can
be formulated as a solution of an eikonal equation and various classical and
modern numerical solutions are reviewed.
After this exercise we tackle one of the major problems in image anal
ysis, known as the "segmentation problem." Chapter 9 introduces a set of
variational solutions to the segmentation problem in which geometry plays
a dominant part. It shows, for example, that thresholding is a result of min
imal variation criterion that can be coupled with other integral measures
such as the "geodesic active contour" model. We move from curves to sur
faces by introducing the concept of image as a surface in Chapter 10. The
Beltrami framework considers images as surfaces embedded in the hybrid
spatial-intensity space. This framework becomes interesting when consider
ing color images as surfaces in a five-dimensional space: two dimensions for
the location in the image and three dimensions that capture the geomet
ric structure of our color perception. We show how to selectively smooth
images while minimizing the area of the image surface. Finally, chapter 11
tackles one of the most challenging problems in image analysis, the face
recognition problem. It reviews a new method I recently discovered and
explored with my students Asi Elad and Michael and Alexander Bronstein.
The bending invariant signature is defined and used to match isometric sur
faces. It translates the problem of matching flexible surfaces into a simpler
one of matching rigid objects. Equipped with such a powerful tool we are
able to distinguish between identical twins under varying facial expressions
based on a three-dimensional image of their facial surface. Each chapter
includes graphic sketches, images, figures, and exercises at the end, with
solutions to selected exercises in Chapter 12.
Intellectual Support
The book's Web Site includes slides for both students and instructors, as
well as color illustrations. The solutions to selected exercises appear at the
end of the book. New exercises will be added annually, at the course Web
Site at: http://www.cs.technion.ac.il/ ... ron/ng.html.
Acknowledgment
I learned and helped to develop some of the numerical methods and geo
metrical algorithms covered in these lecture notes while collaborating with
Alfred M. Bruckstein from the Technion, Israel, James A. Sethian from
x PREFACE
UC Berkeley, Ravi Malladi from Berkeley Labs., David Adalsteinsson from
the University of North Carolina at Chapel Hill, Laurent Cohen from Paris
University Dauphine, Guillermo Sapiro from the University of Minnesota,
Vicent Caselles from the University of Barcelona, Doron Shaked from HP
Labs, Israel, and Nahum Kiryati and Nir Sochen from Tel-Aviv University.
I am grateful to these brilliant researchers with whom I had the pleasure to
collaborate. Nir Sochen, James Sethian, Alfred M. Bruckstein and Joachim
Weickert gave valuable comments on early drafts.
I thank my graduate students Alexander Brook, Gil Zigelman, Alon
Spira, Adi Liav and the students and the participants from the HP Labo
ratory and Applied Materials research group, of the "Numerical Geometry
of Images" course held at the Technion during the winter semesters of
1999 through 2003, for their valuable comments and careful reading of the
manuscript. I thank Yoav Schechner and Didi Sazbon from the Technion,
for correcting typos in early versions of this manuscript. During the last
year parts of the course were given as tutorials, short seminars, and invited
lectures in Sydney, Australia, Copenhagen, Denmark, and all over Japan,
France, and California. I thank all the participants who made valuable
comments that helped me rewrite and refresh some of the chapters. I am
grateful to Yana Katz, Hadas Heier, and Deborah Miller for their time and
effort in LaTeXing and shaping my notes.
Feedback
Comments, suggestions, and corrections are most welcome by E-mail:
[e-mail]
Ron Kimmel
Technion-lIT, Haifa, Israel
July 2003
Contents
Preface vii
1 Introduction 1
1.1 Mathematical Tools and Machinery. 2
1.2 Applications 4
1.3 Exercises.......... 13
2 Basic Differential Geometry 17
2.1 Introduction to Differential Geometry in the Plane 17
2.2 Invariant Signatures. . . . . . . . . . . . . . 20
2.3 Calculus of Variations in Parametric Form. 26
2.4 Geometry of Surfaces . . . . . . . . . . . . . 27
2.5 A Brief Introduction to Intrinsic Geometry 31
2.6 Exercises ................... . 33
3 Curve and Surface Evolution 35
3.1 Evolution .......... . 35
3.2 Properties of Curve Evolution 40
3.3 Surface Evolution 45
3.4 Exercises ........... . 48
4 The Osher-Sethian Level Set Method 50
4.1 The Eulerian Formulation . . . . 51
4.2 From Curve to Image Evolution . 55
4.3 Exercises.............. 59
5 The Level Set Method: Numerical Considerations 61
5.1 Finite Difference Approximation .......... . 61