Table Of ContentPDE AND LEVEL SETS
Algorithmic Approaches to
Static and Motion Imagery
TOPICS IN BIOMEDICAL ENGINEERING
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PDE and Level Sets: Algorithmic Approaches to Static and Motion Imagery
Edited by Jasjit S. Suri and Swamy Laxminarayan
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PDE AND LEVEL SETS
Algorithmic Approaches to
Static and Motion Imagery
Edited by
Jasjit S. Suri, Ph.D.
Philips Medical Systems, Inc.
Cleveland, Ohio, USA
and
Swamy Laxminarayan, Ph.D.
New Jersey Institute of Technology
Newark, New Jersey, USA
KLUWER ACADEMIC PUBLISHERS
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eBook ISBN: 0-306-47930-3
Print ISBN: 0-306-47353-4
©2004 Kluwer Academic Publishers
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Jasjit Suri would like to dedicate this book
to his parents and especially to his late mother
for her immortal softness and encouragements.
Swamy Laxminarayan would like to dedicate this book
to his late sister, Ramaa, whose death at the tender age of 16
inspired his long career in biomedical engineering.
Contributors
Jasjit S. Suri, Ph.D. Philips Medical Systems, Inc., Cleveland, Ohio, USA
Jianbo Gao, Ph.D. KLA-Tencor, Milpitas, California, USA
Jun Zhang, Ph.D. University of Wisconsin, Milwaukee, Wisconsin, USA
Weison Liu, Ph.D. University of Wisconsin, Milwaukee, Wisconsin, USA
Alessandro Sarti, Ph.D. University of Bologna, Bologna, Italy
Xioaping Shen, Ph.D. University of California, Davis, California, USA
Laura Reden, B.S. Philips Medical Systems, Inc., Cleveland, Ohio, USA
David Chopp, Ph.D. Northwestern University, Chicago, Illinois, USA
Swamy Laxminarayan, Ph.D. New Jersey Institute of Technology, Newark, New Jer-
sey, USA
vii
The Editors
Dr. Jasjit S. Suri received his B.S. in computer engineering with distinction from MACT,
Bhopal, M.S. in computer sciences from the University of Illinois, and Ph.D. in electrical
engineering from the University of Washington, Seattle. He has been working in the field of
computer engineering/imaging sciences for more than 18 years, and has published more
than 85 papers on image processing. He is a lifetime member of various research engineer-
ing societies, including Tau Beta Pi and Eta Kappa Nu, Sigma Xi, the New York Academy
of Sciences, EMBS, SPIE, ACM and is also a senior member of IEEE. He is also on the
editorial board/reviewer of several international journals, including Real-Time Imaging,
Pattern Analysis and Applications, Engineering in Medicine and Biology Society, Radiol-
ogy, JCAT, IEEE-ITB and IASTED. He has chaired image processing sessions at several
international conferences and has given more than 30 international presentations. Dr. Suri
has written a book on medical imaging covering cardiology, neurology, pathology, and
mammography imaging. He also holds and has filed several US patents. Dr. Suri has been
listed in Who’s Who five times (World, Executive and Mid-West), is a recipient of Presi-
dent’s Gold Medal in 1980, and has been awarded more than 50 scholarly and extra-
curricular awards during his career. Dr. Suri’s major interest are: computervision,graphics
and image processing (CVGIP), object-oriented programming, and image guided surgery.
Dr. Suri has been with Picker/Marconi/Philips Medical Systems Inc., Cleveland since
December 1998.
Dr. Swamy Laxminarayan is currently the Chief Information Officer at the National Louis
University (NLU) in Chicago. Prior to coming to NLU, he was an adjunct Professor of
Biomedical Engineering at the New Jersey Institute of Technology, Newark, New Jersey
and a Clinical Associate Professor of Medical Informatics and Director and Chair of
VocalTec University. Until recently, he was the Director of Health Care Information
Services as well as Director of Bay Networks, authorized educational center at NextJen
Internet, Princeton, New Jersey. He also serves as a visiting Professor of Biomedical
Information Technology at the University of Brno, Slovak Republic, and an Honorary
Professor at Tsinghua University, China. He is an internationally recognized scientist,
engineer, and educator with over 200 technical publications in areas as wide ranging as
biomedical information technology, computation biology, signal and image processing,
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x PDE and Level Sets
biotechnology, and physiological system modeling. He has been involved in Internet and
information technology application for well over a decade with significant contributions in
the applications of the disciplines in medicine and health care. Dr. Laxminaryan has won
numerous international awards and has lectured widely as an invited speaker in over 35
countries. He has been closely associated with the IEEE Engineering and Medicine and
Biology Society in various administrative and executive committee roles, including his
previous appointments as a Vice President of the society and currently as Editor-in-Chief of
IEEE Transactions on Information Technology and Biomedicine. Among the many awards
and honors he has received, he is one of the 1995 recipients of the Purkynje Award, one of
Europe’s highest forms of recognition, given for pioneering contributions in cardiac and
neurophysiological modeling work and his international bioengineering leadership. In
1994, he was inducted into the College of Fellows of the American Institute of Medical and
Biological Engineering (AIMBE) for “outstanding contributions to advanced computing
and high performance communication applications in biomedical research and education.”
He recently became the recipient of the IEEE 3rd Millennium Medal.
Preface
Chapter 1 is for readers who have less background in partial differential equations (PDEs).
It contains materials which will be useful in understanding some of the jargonrelated to the
rest of the chapters in this book. A discussion about the classification of the PDEs is
presented. Here, we outline the majoranalyticalmethods. Later in the chapter, we introduce
the most important numerical techniques, namely the finitedifference method and finite
element method. In the last section we briefly introduce the level set method. We hope the
reader will be able to extrapolate the elements presented here to initiate an understanding of
the subject on his or her own.
Chapter 2 presents a brief survey of the modern implementation of the level set
method beginning with its roots in hyperbolic conservation laws and Hamilton-Jacobi
equations. Extensions to the level set method, which enable the method to solve a broad
range of interface motion problems, are also detailed including reinitialization, velocity
extensions, and coupling withfinite element methods. Several examples showing different
implementation issues and ways to exploit the level set representation framework are
described.
Level sets have made a tremendous impact on medical imagery due to its ability to
performtopology preservation and fastshape recovery. In chapter 3, we introduce a class of
geometric deformable models, also known as level sets. In an effort to facilitate a clear and
full understanding of these powerful state-of-the-art applied mathematical tools, this chap-
ter attempts to explore these geometric methods, their implementations, and integration of
regularizers to improve the robustness of these topologically independent propagating
curves and surfaces. This chapter first presents the origination of level sets, followed by the
taxonomy of level sets. We then derive the fundamental equation of curve/surfaceevolution
and zero-level curves/surfaces. The chapter then focuses on the first core class of level sets,
known as “level sets without regularizers.” This class presents five prototypes: gradient,
edge, area-minimization, curvature-dependent and application driven. The next section is
devoted to second core class of level sets, known as “level sets with regularizers.” In this
class, we present four kinds: clustering-based, Bayesian bi-directional classifier-based,
shape-based, and coupled constrained-based. An entire section is dedicated to optimization
and quantification techniques for shape recovery when used in the level-set framework.
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Finally, the chapter concludes with the general merits and demerits on level sets, and the
future of level sets in medical image segmentation.
Chapter 4 focuses on the partial differential equations (PDEs), as these have domi-
nated image processing research recently. The three main reasons for their success are: (1)
their ability to transform a segmentation modeling problem into a partial differential
equation framework and their ability to embed and integrate different regularizers into
these models; (2) their ability to solve PDEs in the level set framework using finite
difference methods; and (3) their easy extension to a higher dimensional space. This
chapter is an attempt to understand the power of PDEs to incorporate into geometric
deformable models for segmentation of objects in 2-D and 3-D in static and motion
imagery. The chapter first presents PDEs and their solutionsapplied to image diffusion. The
main concentration of this chapter is to demonstrate the usage of regularizers, PDEs and
level sets to achieve image segmentation in static and motion imagery. Lastly, we cover
miscellaneous applications such as mathematical morphology, computation of missing
boundaries for shape recovery, and low pass filtering, all under the PDE framework. The
chapter concludes with the merits and the demerits of PDE and level set-based framework
techniques for segmentation modeling. The chapter presents a variety of examples covering
both synthetic and real world images.
In chapter 5, we describe a new algorithm for color image segmentation and a novel
approach for image sequence segmentation using PDE framework. The color image
segmentation algorithm can be used for image sequence intraframe segmentation, and it
gives accurate region boundaries. Because this method produces accurateboundaries, the
accuracy of motion boundaries of the image sequence segmentation algorithms may be
improved when it is integrated in the sequence segmentation framework. To implement this
algorithm, we have also developed a new multi-resolution technique,called the “Narrow
Band”, which is significantly faster than both single resolution and traditional multi-
resolution methods. As a color image segmentation technique, it is unsupervised, and its
segmentation is accurate at the object boundaries. Since it uses the Markov Random Field
(MRF) and mean field theory, the segmentation results are smooth and robust. This is then
demonstrated by showing good results obtained in dermatoscopic images and image
sequence frames. We then present a new approach to the image sequence segmentation that
contains three parts: (i) global motion compensation, (ii) robust frame differencing and
(iii) curve evolution. In the global motion compensation, we adopt a fast method, which
needs only a sparse set of pixels evenly distributed in the image frames. Block-matching
and regression are used to classify the sparse set of pixels into inliers and outliers according
to the affine model. With the regression, the inliers of the sparse set, which are related to the
global motion, is determined iteratively. For the robust frame differencing, we used a local
structuretensor field, which robustly represents the object motion characteristics. With the
level set curve evolution, the algorithm can detect all the moving objects and circle out the
objects’ outside contours. The approach discussed in this chapter is computationally effi-
cient, does not require a dense motion field and is insensitive to global/background motion
and to noise. Its efficacy is demonstrated on both TV and surveillance video.
In chapter 6, we describe a novel approach to image sequence segmentation and its
real-time implementation. This approach uses the 3-D structure tensor to produce a more
robust frame difference and uses curve evolution to extract whole (moving) objects. The
algorithm is implemented on a standard PC running the MS Windows operating system
