Geometric Computing: for Wavelet Transforms, Robot Vision, Learning, Control and Action

625 Pages·2010·20.52 MB·English

by  Eduardo Bayro-Corrochano

Geometric Computing
Eduardo Bayro-Corrochano
Geometric Computing
For Wavelet Transforms, Robot Vision,
Learning, Control and Action
ABC
EduardoBayro-Corrochano
CINVESTAV
UnidadGuadalajara
Dept.ElectricalEng.&ComputerScience
Av.Científica1145
45010ColoniaElBajío
Zapópan,JAL
México
[e-mail]
http://www.gdl.cinvestav.mx/edb
ISBN978-1-84882-928-2 e-ISBN978-1-84882-929-9
DOI10.1007/978-1-84882-929-9
SpringerLondonDordrechtHeidelbergNewYork
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MyThree DedicationStrophes
I.To thesocialfightersNelsonMandela who
helpedtoeliminatetheAfricanapartheidand
Evo Moraleswho worked to eliminatethe
AndeanIndianapartheid.Ama Sua,Ama
Qhella,AmaLlulla,Ama Llunk’a.
II.To allscientistswho don’tworkforthe
developmentofweapons andtechnology
destinedtooccupyand dominatecountries;
forallwho workfor education,health,water,
alternativeenergy,preservationofthe
environmentand thewelfareof thepoor
people.
III.To mybeloved wifeJoannaJablonska
andtwo setsof myadoredchildren:Esteban,
Fabio,VinzenzandSilvana;andNikolai,
ClaudioandGladys.
Foreword
Geometricalgebra(GA)isapowerfulnewmathematicalsystemforcomputational
geometry.AlthoughitsoriginscanbetracedbacktoHermannGrassmann(1844),
its developmentasa languageforspace–timegeometrywith applicationsto all of
physicsdidnotbeginuntil1966.Suddenly,intheyear2000itwasrecognizedthat
aspecializedversioncalledconformalgeometricalgebra(CGA)wasideallysuited
forcomputationalEuclideangeometry.CGAhasthegreatadvantagethatgeometric
primitives(point,lineplane,circle,sphere)canbedirectlyrepresented,compared,
and manipulated without coordinates, so there is an immediate correspondence
between algebraic objects and geometric figures. Moreover, CGA enhances and
smoothly integrates the classical methods of projective, affine, and metric geom-
etrywiththemorespecializedmethodsofquaternions,screwtheory,andrigidbody
mechanics. Applications to computer science and engineering have accumulated
rapidlyinthelastfewyears.Thisbookassemblesdiverseaspectsofgeometrical-
gebrainoneplacetoserveasageneralreferenceforapplicationstorobotics.Then,
itdemonstratesthepowerandefficiencyofthesystemwithspecificapplicationsto
ahostofproblemsrangingfromcomputervisiontomechanicalcontrol.Perceptive
readers will recognize many places where the treatment can be extended or im-
proved.Thus,thisbookisaworkinprogress,anditshigherpurposewillbeserved
ifitstimulatesfurtherresearchanddevelopment.
Physics&AstronomyDepartment DavidHestenes
ArizonaStateUniversity
September2009
vii
Preface
This book presents the theory and applications of an advanced mathematical
language called geometric algebra that greatly helps to express the ideas and
concepts,andtodevelopalgorithmsinthebroaddomainofrobotphysics.
Inthehistoryofscience,withoutessentialmathematicalconcepts,theorieswould
have notbeen developedat all. We can observethat in variousperiodsof the his-
tory of mathematics and physics, certain stagnation occurred; from time to time,
thankstonewmathematicaldevelopments,astonishingprogresstookplace.Inad-
dition, we see that the knowledge became unavoidably fragmented as researchers
attempted to combine different mathematical systems. Each mathematical system
bringsaboutsomepartsofgeometry;however,together,thesesystemsconstitutea
highly redundantsystem due to an unnecessarymultiplicity of representationsfor
geometricconcepts.Theauthorexpectsthatduetohispersistenteffortstobringto
thecommunitygeometricalgebraforapplicationsasameta-languageforgeometric
reasoning,inthenearfuturetremendousprogressinroboticsshouldtakeplace.
Whatisgeometricalgebra?Whyareitsapplicationssopromising?Whyshould
researchers, practitioners, and students make the effort to understand geometric
algebraanduse it?We wanttoanswerallthese questionsandconvincethereader
that becoming acquainted with geometric algebra for applications is a worthy
undertaking.
The history of geometricalgebra is unusualand quite surprising.In the 1870s,
William Kingdon Clifford introduced his geometric algebra, building on the ear-
lier works of Sir William Rowan Hamilton and Hermann Gunther Grassmann. In
Clifford’swork, we perceivethathe intendedto describethe geometricproperties
of vectors, planes, and higher-dimensionalobjects. Most physicists encounter the
algebra in the guise of Pauli and Dirac matrix algebras of quantum theory. Many
roboticists or computergraphic engineersuse quaternionsfor 3D rotation estima-
tionandinterpolation,asapointwiseapproachistoodifficultforthemtoformulate
homogeneous transformations of high-order geometric entities. They resort often
to tensorcalculusformultivariablecalculus.Since roboticsand engineeringmake
use of the developmentsof mathematical physics, many beliefs are automatically
inherited; for instance, some physicists come away from a study of Dirac theory
with the view that Clifford’s algebra is inherently quantum-mechanical.The goal
ofthisbookistoeliminatethese kindsofbeliefsbygivinga clearintroductionof
geometric algebra and showing this new and promising mathematical framework
ix
x Preface
tomultivectorsandgeometricmultiplicationinhigherdimensions.Inthisnewge-
ometric language, most of the standard matter taught to roboticists and computer
scienceengineerscanbeadvantageouslyreformulatedwithoutredundanciesandin
a highly condensed fashion. Geometric algebra allows us to generalize and trans-
ferconceptsandtechniquestoawiderangeofdomainswithlittleextraconceptual
work.Leibnizdreamedofageometriccalculussystemthatdealsdirectlywithgeo-
metricobjectsratherthanwithsequencesofnumbers.Itisclearthatbyincreasing
the dimensionofthe geometricspaceandthe generalizationofthe transformation
group, the invariance of the operations with respect to a reference frame will be
moreandmoredifficult.Leibniz’sinvariancedreamisfulfilledforthenDclassical
geometriesusingthecoordinate-freeframeworkofgeometricalgebra.
Theaimofthisbookispreciseandwellplanned.Itis notmerelyanexpose´ of
mathematicaltheory;rather,theauthorintroducesthetheoryandnewtechniquesof
geometric algebra by showing their applications in diverse domainsranging from
neuralcomputingandroboticstomedicalimageprocessing.
Acknowledgments
EduardoJose´ Bayro Corrochanowould like to thank the Center for Research and
AdvancedStudies(CINVESTAV,Guadalajara,Mexico)andthe ConsejoNacional
deCienciayTecnolog´ıa(SEP-CONACYT,Mexico)fortheirsupportofthisbook.
IamalsoverygratefultomyformerPh.D.studentsJulioZamora-Esquivel,Nancy
Arana-Daniel, Jorge Rivera Rovelo, Leo Reyes Hendrick, Luis Eduardo Falco´n,
Carlos Lo´pez-Franco, and Rube´n Machucho Cadena for fruitful discussions and
technical cooperation. Their creative suggestions, criticism, and patient research
workweredecisiveforthecompletionofthisbook.Inthegeometricalgebracom-
munity, first of all I am indebted to David Hestenes for all his amazing work in
developing the modern subject of geometric algebra and his constant encourage-
ment to me for tackling problems in robot physics. Also, I am very thankful to
Garret Sobczyk, Eckard Hitzer, Dietmar Hildebrand, and Joan Lasenby for their
supportand constructive suggestions. Finally, I am very thankfulto the people of
Mexico,whopayformysalary,andwhichmadeitpossibleformeto accomplish
thiscontributiontoscientificknowledge.
CINVESTAV,Guadalajara,Mexico EduardoBayroCorrochano
27August2009
Howto UseThis Book
Thissectionbeginsbybrieflydescribingtheorganizationandcontentofthechapters
andtheirinterdependency.Thenitexplainshowreaderscanuse thebookforself-
studyorfordeliveringaregulargraduatecourse.
Preface xi
Chapter Organization
– Part I: Fundamentals of Geometric Algebra. Chapter 1 gives an outline of
geometricalgebra.Afterpreliminarydefinitions,wediscusshowtohandlelinear
algebraandsimplexes,andmultivectorcalculusisbrieflyillustratedwithMaxwell
and Dirac equations. In Chap. 2, we explain the computational advantages of
geometric algebra for modeling and solving problems in robotics, computer
vision,artificialintelligence,neuralcomputing,andmedicalimageprocessing.
– PartII:Euclidean,Pseudo-Euclidean,LieandIncidenceAlgebrasandConfor-
mal Geometries. Chapter 3 beginsby explainingthe geometric algebra models
in2D,3D,and4D.Chapter4presentsthekinematicsofpoints,lines,andplanes
using 3D geometric algebra and motor algebra. Chapter 5 examines Lie group
theory,Lie algebra, and the algebra of incidenceusing the universalgeometric
algebrageneratedbyreciprocalnullcones.Chapter6isdevotedtotheconformal
geometric algebra, explaining the representation of geometric objects and ver-
sors. Differentgeometric configurationsare studied, includingsimplexes, flats,
plunges, and carriers. We briefly discuss the promising use of ruled surfaces.
Chapter7discussesthemainissuesoftheimplementationofcomputerprograms
forgeometricalgebra.
– Part III: Geometric Computing for Image Processing, Computer Vision, and
Neurocomputing.Chapter8presentsacompletestudyofthestandardandnew
CliffordwaveletandFouriertransforms.Chapter9usesgeometricalgebratech-
niquestoformulatethen-viewgeometryofcomputervisionandtheformationof
3Dprojectiveinvariantsforbothpointsandlinesinmultipleimages.Weextend
theseconceptsforomnidirectionalvisionusingstereographicmappingontothe
unit sphere. In Chap. 10, we present the geometric multilayer perceptrons and
Cliffordsupportvectormachinesforclassification,regression,andrecurrence.
– Part IV: Geometric Computing of Robot Kinematics and Dynamics. Chapter
11 presents a study of the kinematics of robot mechanisms using a language
basedon points, lines, planes, and spheres.In Chap. 12,the dynamicsof robot
manipulatorsis treated, simplifying the representation of the tensors of Euler–
Lagrange equations. The power of geometric algebra over matrix algebra and
tensorcalculusisconfirmedwiththeseworks.
– PartV:ApplicationsI:ImageProcessing,ComputerVision,andNeurocomput-
ing.Chapter13showsapplicationsofLieoperatorsforkeypointdetection,the
quaternionFouriertransformforspeechrecognition,andthequaternionwavelet
transform for optical flow estimation. In Chap. 14, we use projective invari-
antsfor3Dshapeandmotionreconstruction,androbotnavigationusingn-view
camerasandomnidirectionalvision.Chapter15usestensorvotingandgeomet-
ricalgebrato estimatenonrigidmotion.Chapter16presentsexperimentsusing
real data for robot object recognition,interpolation,and the implementation of
aCliffordSVDrecurrentsystem.Chapter17showstheuseofageometricself-
organizingneuralnettosegment2Dcontoursand3Dshapes.
– PartVI:ApplicationsII:RoboticsandMedicalRobotics.Chapter18isdevoted
to line motion estimation using SVD and extended Kalman filter techniques.
Chapter19presentsatrackerendoscopecalibrationandthecalibrationofsensors
xii Preface
Fig. 0.1 Chapter interdependence: fundamentals ! theory of applications ! applications !
appendix
withrespecttoa robotframe.Here,we use purelya languageoflinesandmo-
tors. Chapter 20 illustrates visual-guided grasping tasks using representations
and geometric constraints developed and found using conformal geometric al-
gebra. Chapter 21 describes a 3D map reconstruction and relocalization using
conformal geometric entities exploiting the Hough space. Chapter 22 presents
the application of marching spheres for 3D medical shape representation and
registration.
– Part VII: Appendix. Chapter 23 includes an outline of Clifford algebra. The
reader can find concepts and definitions related to classic Clifford algebra and
relatedalgebras:Gibbsvectoralgebra,exterioralgebras,andGrassmann–Cayley
algebras.
Interdependence oftheBook Chapters
TheinterdependenceofthebookchaptersisshowninFig.0.1.Essentially,thereare
fourgroupsofchapters:
– Fundamentalsofgeometricalgebra:Chaps.1,3,and4.Chapter6isoptional.
– Theoryoftheapplicationsareasusingthegeometricalgebraframework.
– Chapter8Clifford–Fourierandwavelettransforms
– Chapter9Computervision