Handbook of Geometric Computing: Applications in Pattern Recognition, Computer Vision, Neuralcomputing, and Robotics

774 Pages·2005·11.27 MB·English

by  Eduardo Bayro Corrochano

Handbook of Geometric Computing
Eduardo Bayro Corrochano
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Applications in Pattern Recognition,
Computer Vision, Neuralcomputing,
and Robotics
With 277 Figures, 67 in color, and 38 Tables
123
Prof. Dr. Eduardo Bayro Corrochano
Cinvestav
Unidad Guadalajara
Ciencias de la Computación
P. O. Box 31-438
Plaza la Luna, Guadalajara
Jalisco 44550
México
[e-mail]
Library of Congress Control Number: 2004118329
ACM Computing Classification (1998): I.4, I.3, I.5, I.2, F. 2.2
ISBN-10 3-540-20595-0 Springer Berlin Heidelberg New York
ISBN-13 978-3-540-20595-1 Springer Berlin Heidelberg New York
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Preface
One importantgoalofhumancivilizationisto build intelligentmachines,not
necessarily machines that can mimic our behavior perfectly, but rather ma-
chines that can undertake heavy, tiresome, dangerous, and even inaccessible
(forman)labortasks.Computersareagoodexampleofsuchmachines.With
their ever-increasing speeds and higher storage capacities, it is reasonable to
expect that in the future computers will be able to perform even more useful
tasks for man and society than they do today, in areas such as health care,
automated visual inspection or assembly, and in making possible intelligent
man–machine interaction. Important progress has been made in the develop-
mentofcomputerizedsensorsandmechanicaldevices.Forinstance,according
toMoore’slaw,thenumberoftransistorsonachiproughlydoubleseverytwo
years – as a result, microprocessors are becoming faster and more powerful
and memory chips can store more data without growing in size.
Developmentswith respectto concepts,unifiedtheory,andalgorithmsfor
buildingintelligentmachineshavenotoccurredwiththesamekindoflightning
speed.However,theyshouldnotbemeasuredwiththesameyardstick,because
the qualitative aspects of knowledge development are far more complex and
intricate. In 1999, in his work on building anthropomorphic motor systems,
Rodney Brooks noted: “A paradigm shift has recently occurred – computer
performance is no longer a limiting factor. We are limited by our knowledge
of what to build.” On the other hand, at the turn of the twenty-firstcentury,
it would seem we collectively know enough about the human brain and we
have developed sufficiently advanced computing technology that it should be
possible for us to find ways to construct real-time, high-resolution, verifiable
models for significant aspects of human intelligence.
Just as greatstrides in the dissemination of human knowledge were made
possiblebytheinventionofthe printingpress,inthe samewaymodernscien-
tificdevelopmentsareenhancedtoagreatextentbycomputertechnology.The
Internet now plays an important role in furthering the exchange of informa-
tion necessaryfor establishing cooperationbetweendifferent researchgroups.
Unfortunately,thetheoryforbuildingintelligentmachinesorperception-and-
VI Preface
action systems is still in its infancy. We cannot blame a lack of commitment
on the part of researchers or the absence of revolutionary concepts for this
state of affairs. Remarkably useful ideas were proposed as early as the mid-
nineteenth century, when Babbage was building his first calculating engines.
Sincethen,usefulconceptshaveemergedinmathematics,physics,electronics,
andmechanicalengineering–allbasicfieldsforthedevelopmentofintelligent
machines. In its time, classical mechanics offered many of the necessary con-
ceptual tools. In our own time, Lie group theory and Riemann differential
geometry play a largerole in modern mathematics and physics.For instance,
as a representationtool, symmetry, a visual primitive probably unattentively
encoded, may provide an important avenue for helping us understand per-
ceptual processes.Unfortunately, the application of these concepts in current
work on image processing, neural computing, and robotics is still somewhat
limited. Statistical physics and optimization theory have also proven to be
useful in the fields of numerical analysis, nonlinear dynamics, and, recently,
in neural computing. Other approaches for computing under conditions of
uncertainty, like fuzzy logic and tensor voting, have been proposed in recent
years. As we can see, since Turing’s pioneering 1950 work on determining
whether machines areintelligent,the developmentof computersfor enhanced
intelligence has undergone great progress.
Thisnewhandbooktakesadecisivestepinbringingtogetherinonevolume
varioustopicshighlighting the geometricaspectsnecessaryfor imageanalysis
and processing, perception, reasoning, decision making, navigation, action,
andautonomouslearning.Unfortunately,evenwithgrowingfinancialsupport
for researchand the enhanced possibilities for communication brought about
by the Internet, the various disciplines within the research community are
still divorced from one another, still working in a disarticulated manner. Yet
the effort to build perception–action systems requires flexible concepts and
efficientalgorithms,hopefully developedinanintegratedandunifiedmanner.
Itis ourhope that this handbookwillencourageresearchersto worktogether
onproposalsandmethodologiessoastocreatethenecessarysynergyformore
rapid progress in the building of intelligent machines.
Structure and Key Contributions
Thehandbookconsistsofninepartsorganizedbydiscipline,sothatthereader
can form anunderstanding of how work among the variousdisciplines is con-
tributing to progress in the area of geometric computing. Understanding in
each individual field is a fundamental requirement for the development of
perception-action systems. In this regard, a tentative list of relevant topics
might include:
• brain theory and neuroscience
• learning
• neurocomputing, fuzzy computing, and quantum computing
Preface VII
• image analysis and processing
• geometric computing under uncertainty
• computer vision
• sensors
• kinematics, dynamics, and elastic couplings
• fuzzy and geometric reasoning
• control engineering
• robot manipulators, assembly, MEMS, mobile robots, and humanoids
• path planning, navigation,reaching, and haptics
• graphic engineering, visualization, and virtual reality
• medical imagery and computer-aided surgery
We have collected contributions from the leading experts in these diverse
areas of study and have organized the chapters in each part to address low-
level processingfirst before moving on to the more complex issues of decision
making. In this way, the reader will be able to clearly identify the current
state ofresearchfor eachtopic andits relevancefor the directionandcontent
offutureresearch.Bygatheringthisworktogetherundertheumbrellaofbuil-
dingperception–actionsystems,weareabletoseethateffortstowardthatgoal
are flourishing in each of these disciplines and that they are becoming more
interrelatedandareprofitingfromdevelopmentsintheotherfields.Hopefully,
in the near future, we will see all of these fields interacting even more closely
in the construction of efficient and cost-effective autonomous systems.
Part I Neuroscience
In Chapter 1 Haluk Öğmen reviews the fundamental properties of the pri-
mate visual system, highlighting its maps and pathways as spatio-temporal
informationencodingandprocessingstrategies.Heshowsthatretinotopicand
spatial-frequencymapsrepresentthegeometryofthefusionbetweenstructure
andfunctioninthenervoussystem,andthatmagnocellularandparvocellular
pathways can resolve the trade-off between spatial and temporal deblurring.
In Chapter 2 Hamid R. Eghbalnia, Amir Assadi, and Jim Townsend a-
nalyze the important visual primitive of symmetry, probably unattentively
encoded, which can have a central role in addressing perceptual processes.
The authors argue that biological systems may be hardwired to handle fil-
tering with extreme efficiency.They believe that it may be possible to appro-
ximate this filtering, effectively preserving all the important temporal visual
features, by using current computer technology. For learning, they favor the
use ofbidirectionalassociativememories,usinglocalinformationin the spirit
of a local-to-globalapproachto learning.
VIII Preface
Part II Neural Networks
In Chapter 3 Hyeyoung Park, Tomoko Ozeki, and Shun-ichi Amari choose
a geometric approach to provide intuitive insights on the essential properties
of neural networks and their performance. Taking into account Riemann’s
structureofthe manifoldofmultilayerperceptrons,they designgradientlear-
ning techniques for avoidingalgebraicsingularitiesthat havea greatnegative
influence on trajectories of learning. They discuss the singular structure of
neuromanifolds and pose an interesting problem of statistical inference and
learning in hierarchical models that include singularities.
InChapter4GerhardRitterandLaurentiuIancupresentanewparadigm
for neuralcomputing using the lattice algebraframework.They developmor-
phological auto-associative memories and morphological feed-forward net-
works based on dendritic computing. As opposed to traditional neural net-
works,theirmodelsdonotneedhiddenlayersforsolvingnon-convexproblems,
but rather they converge in one step and exhibit remarkable performance in
both storage and recall.
In Chapter 5 Tijl De Bie, Nello Cristianini, and Roman Rosipal de-
scribe a large class of pattern-analysis methods based on the use of genera-
lized eigenproblems and their modifications. These kinds of algorithms can
be used for clustering, classification,regression,and correlationanalysis.The
chapter presents all these algorithms in a unified framework and shows how
they can all be coupled with kernels and with regularization techniques in
order to produce a powerful class of methods that compare well with those
of the support-vector type. This study provides a modern synthesis between
several pattern-analysis techniques.
Part III Image Processing
In Chapter 6 Jan J. Koenderink sketches a framework for image processing
that is coherent and almost entirely geometric in nature. He maintains that
thetimeisripeforestablishingimageprocessingasasciencethatdepartsfrom
fundamental principles, one that is developed logically and is free of hacks,
unnecessaryapproximations,andmereshowpiecesonmathematicaldexterity.
In Chapter 7 Alon Spira, Nir Sochen, and Ron Kimmel describe ima-
ge enhancement using PDF-based geometric diffusion flows. They start with
variationalprinciplesforexplainingtheoriginoftheflows,andthisgeometric
approach results in some nice invariance properties. In the Beltrami frame-
work,theimageisconsideredtobeanembeddedmanifoldinthespace-feature
manifold, so that the requiredgeometric filters for the flows in gray-leveland
colorimagesortexturewilltakeintoaccounttheinducedmetric.Thischapter
presents numerical schemes and kernels for the flows that enable an efficient
and robust implementation.
In Chapter 8 Yaobin Mao and Guanrong Chen show that chaos theory
is an excellent alternative for producing a fast, simple, and reliable image-
encryption scheme that has a high degree of security. The chapter describes
Preface IX
a practical and efficient chaos-based stream-cipher scheme for still images.
Fromanengineer’sperspective,thechaosimage-encryptiontechnologyisvery
promisingforthereal-timeimagetransferandhandlingrequiredforintelligent
discerning systems.
Part IV Computer Vision
In Chapter 9 Kalle Åström is concerned with the geometry and algebra
of multiple one-dimensional projections in a 2D environment. This study is
relevantfor 1Dcameras,for understanding the projectionoflines inordinary
vision, and, on the application side, for understanding the ordinary vision of
vehicles undergoing planar motion. The structure-of-motion problem for 1D
cameras is studied at length, and all cases with non-missing data are solved.
Cases with missing data are more difficult; nevertheless, a classification is
introduced and some minimal cases are solved.
InChapter 10 Anders Heydendescribes in-depth, n-viewgeometrywith
allthe computationalaspects requiredfor achieving stratifiedreconstruction.
He starts with camera modeling and a review of projective geometry. He de-
scribesthemulti-viewtensorsandconstraintsandtheassociatedlinearrecon-
struction algorithms. He continues with factorization and bundle adjustment
methods and concludes with auto-calibration methods.
InChapter 11AmnonShashuaandLiorWolfintroducea generalization
of the classical collineation of Pn. The m-view tensors for Pn referred to as
homographytensorsarestudiedindetailforthecasen=3,4inwhichtheindi-
vidual points are allowed to move while the projective change of coordinates
takes place. The authors show that without homographytensors a recovering
of the alignment requires statistical methods of sampling, whereas with the
tensor approach both stationary and moving points can be considered alike
and part of a global transformation can be recovered analytically from some
matchingpointsacrossmviews.Ingeneral,thehomographytensorsareuseful
for recovering linear models under linear uncertainty.
InChapter 12AbhijitOgale,CorneliaFermüllerandYiannis Aloimonos
examinetheproblemofinstantaneousfindingofobjectsmovingindependently
inavideoobtainedbyamovingcamerawitharestrictedfieldofview.Inthis
problem,theimagemotioniscausedbythecombinedeffectofcameramotion,
scene depth, and the independent motions of objects. The authors present a
classification of moving objects and discuss detection methods; the first class
isdetectedusingmotionclustering,theseconddependsonordinaldepthfrom
occlusionsandthethirdusescardinalknowledgeofthedepth.Robustmethods
for deducing ordinal depth from occlusions are also discussed.
X Preface
Part V Perception and Action
In Chapter 13 Eduardo Bayro-Corrochano presents a framework of con-
formal geometric algebra for perception and action. As opposed to standard
projective geometry, in conformal geometric algebra, using the language of
spheres,planes, lines, and points, one can deal simultaneously with incidence
algebraoperations(meetandjoin)andconformaltransformationsrepresented
effectively using bivectors. This mathematical system allows us to keep our
intuitions and insights into the geometry of the problem at hand and it helps
ustoreduceconsiderablythecomputationalburdenoftherelatedalgorithms.
Conformalgeometric algebra,with its powerful geometric representationand
rich algebraic capacity to provide a unifying geometric language, appears
promising for dealing with kinematics, dynamics, and projective geometry
problems without the need to abandon a mathematical system. In general,
this can be a great advantage in applications that use stereo vision, range
data, lasers, omnidirectionality, and odometry-based robotic systems.
Part VI Uncertainty in Geometric Computations
In Chapter 14 Kenichi Kanatani investigates the meaning of “statistical
methods” for geometric inference on image points. He traces back the ori-
ginoffeatureuncertaintytoimage-processingoperationsforcomputervision,
andhediscussestheimplicationsofasymptoticanalysiswithreferenceto“ge-
ometric fitting” and “geometric model selection.” The author analyzes recent
progress in geometric fitting techniques for linear constraints and semipara-
metric models in relation to geometric inference.
In Chapter 15 Wolfgang Förstner presents an approach for geometric
reasoning in computer vision performed under uncertainty. He shows that
the great potential of projective geometry and statistics can be integrated
easily for propagating uncertainty through reasoning chains. This helps to
make decisions on uncertain spatial relations and on the optimal estimation
of geometricentities andtransformations.The chapter discusses the essential
link between statistics and projective geometry, and it summarizes the basic
relations in 2D and 3D for single-view geometry.
In Chapter 16 Gérard Medioni, Philippos Mordohai, and Mircea Nico-
lescu presenta tensor voting frameworkfor computer visionthat can address
awiderangeofmiddle-levelvisionproblemsinaunifiedway.Thisframework
isbasedonadatarepresentationformalismthatusessecond-ordersymmetric
tensors and an information propagationmechanism that uses a tensor voting
scheme. The authors show that their approach is suitable for stereo and mo-
tion analysis because it can detect perceptual structures based solely on the
smoothness constraint without using any model. This property allows them
to treat the arbitrary surfaces that are inherent in non-trivial scenes.