Geometric Algebra Applications Vol. I: Computer Vision, Graphics and Neurocomputing

753 Pages·2019·10.97 MB·English

by  Bayro-Corrochano, Eduardo

Eduardo
Bayro-Corrochano
Geometric
Algebra
Applications

Vol. I

Computer Vision,
Graphics
and
Neurocomputing
123
Geometric Algebra Applications Vol. I
Eduardo Bayro-Corrochano
Geometric Algebra
Applications Vol. I
Computer Vision, Graphics
and Neurocomputing
123
Eduardo Bayro-Corrochano
Electrical Engineering and
Computer ScienceDepartment
CINVESTAV, CampusGuadalajara
Jalisco, Mexico
ISBN978-3-319-74828-3 ISBN978-3-319-74830-6 (eBook)
https://doi.org/10.1007/978-3-319-74830-6
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My Three Dedication Strophes
I. To the social fighters Nelson Mandela who
achieved the elimination of the African
apartheid and Evo Morales Aima who
worked to eliminate the Andean Indian
apartheid according the quechua saying Ama
Sua, Ama Qhella, Ama Llulla, Ama Llunk’a.
II. To all scientists who do not work for the
development of weapons and technology
destined to occupy and dominate countries;
for all who work for education, health,
preservation of the environment and social
justice.
III. To my beloved wife Joanna Jablonska
and my first wife Mechthild Kaiser for their
constant and patient support during all these
years of my scientific work; and to my adored
children: Esteban, Fabio, Vinzenz, Silvana,
Nikolai, Claudio and Gladys.
Preface
This book presents the theory and applications of an advanced mathematical lan-
guagecalledgeometricalgebrathatgreatlyhelpstoexpresstheideasandconcepts
and to develop algorithms in the broad domain of robot physics.
In the history of science, without essential mathematical concepts, theories
wouldhavenotbeendevelopedatall.Wecanobservethatinvariousperiodsofthe
historyofmathematicsandphysics,certainstagnationoccurred;fromtimetotime,
thanks to new mathematical developments, astonishing progress took place. In
addition,weseethattheknowledgebecameunavoidablyfragmentedasresearchers
attempted to combine different mathematical systems. Each mathematical system
bringsaboutsomepartsofgeometry;however,together,thesesystemsconstitutea
highly redundant system due to an unnecessary multiplicity of representations for
geometricconcepts.Theauthorexpectsthatduetohispersistenteffortstobringto
thecommunitygeometricalgebraforapplicationsasametalanguageforgeometric
reasoning, in the near future tremendous progress in computer vision, machine
learning, and robotics should take place.
What is geometric algebra? Why is its application so promising? Why should
researchers, practitioners, and students make the effort to understand geometric
algebra and use it? We want to answer all these questions and convince the reader
that becoming acquainted with geometric algebra for applications is a worthy
undertaking.
The history of geometric algebra is unusual and quite surprising. In the 1870s,
WilliamKingdonCliffordintroducedhisgeometricalgebra,buildingontheearlier
works of Sir William Rowan Hamilton and Hermann Gunther Grassmann. In
Clifford’s work, we perceive that he intended to describe the geometric properties
of vectors, planes, and higher-dimensional objects. Most physicists encounter the
algebra in the guise of Pauli and Dirac matrix algebras of quantum theory. Many
roboticists or computer graphic engineers use quaternions for 3D rotation estima-
tionandinterpolation,asapointwiseapproachistoodifficultforthemtoformulate
homogeneoustransformationsofhigh-ordergeometricentities.Theyresortoftento
tensorcalculusformultivariablecalculus.Sinceroboticsandengineeringmakeuse
of the developments of mathematical physics, many beliefs are automatically
vii
viii Preface
inherited; for instance, some physicists come away from a study of Dirac theory
withtheviewthatClifford’salgebraisinherentlyquantummechanical.Thegoalof
this book is to eliminate these kinds of beliefs by giving a clear introduction of
geometricalgebraandshowingthisnewandpromisingmathematicalframeworkto
multivectors and geometric multiplication in higher dimensions. In this new geo-
metric language, most of the standard matter taught to roboticists and computer
scienceengineerscanbeadvantageouslyreformulatedwithoutredundanciesandin
a highly condensed fashion. The geometric algebra allows us to generalize and
transfer concepts and techniques to a wide range of domains with little extra
conceptual work. Leibniz dreamed of a geometric calculus system that deals
directly with geometric objects rather than with sequences of numbers. It is clear
that by increasing the dimension of the geometric space and the generalization
of the transformation group, the invariance of the operations with respect to a
reference frame will be more and more difficult. Leibniz’s invariance dream is
fulfilled for the nD classical geometries using the coordinate-free framework of
geometric algebra.
The aim of this book is precise and well planned. It is not merely an expose of
mathematicaltheory;rather,theauthorintroducesthetheoryandnewtechniquesof
geometricalgebra,albeit byshowingtheirapplicationsindiversedomains ranging
from neurocomputing and robotics to medical image processing.
Guadalajara, Mexico Prof. Eduardo Bayro-Corrochano
August 2017
Acknowledgements
Eduardo José Bayro-Corrochano would like to thank the Center for Research and
AdvancedStudies(CINVESTAV,Guadalajara,Mexico)andtheConsejoNacional
deCienciayTecnologίa(SEP-CONACYT,Mexico)fortheirsupportofthisproject.
IamalsoverygratefultomyformerPh.D.studentsJulioZamora-Esquivel,Nancy
Arana-Daniel, Jorge Rivera Rovelo, Leo Reyes Hendrick, Luis Eduardo Falcón,
Carlos López-Franco, Rubén Machucho Cadena, Eduardo Ulises Moya-Sáanchez,
Eduardo Vázquez, Gehová López-González, Gerardo Altamirano-Gómez, and
Oscar Carbajal-Espinoza for fruitful discussions and technical cooperation. Their
creative suggestions, criticism, and patient research work were decisive for the
completion of this book. In the geometric algebra community, 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 encouragement to me for tackling
problems in robot physics. Also, I am very thankful to Garret Sobczyk, Rafal
Ablamowicz, Anthony Lasenby, Eckard Hitzer, Dietmar Hildebrand, and Joan
Lasenbyfortheirsupportandconstructivesuggestions.Finally,Iamverythankful
to the Mexican people, who pay my salary, which made it possible for me to
accomplish this contributionto scientific knowledge.
ix
Contents
1 Geometric Algebra for the Twenty-First Century Cybernetics . . . . 1
1.1 Cybernetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Roots of Cybernetics. . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Contemporary Cybernetics. . . . . . . . . . . . . . . . . . . 2
1.1.3 Cybernetics and Related Fields . . . . . . . . . . . . . . . 2
1.1.4 Cybernetics and Geometric Algebra . . . . . . . . . . . . 3
1.2 The Roots of Geometry and Algebra . . . . . . . . . . . . . . . . . . . 3
1.3 Geometric Algebra a Unified Mathematical Language. . . . . . . 5
1.4 What does Geometric Algebra Offer for Geometric
Computing? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.1 Coordinate-Free Mathematical System . . . . . . . . . . 7
1.4.2 Models for Euclidean and PseudoEuclidean
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.3 Subspaces as Computing Elements. . . . . . . . . . . . . 9
1.4.4 Representation of Orthogonal Transformations . . . . 9
1.4.5 Objects and Operators. . . . . . . . . . . . . . . . . . . . . . 10
1.4.6 Extension of Linear Transformations . . . . . . . . . . . 11
1.4.7 Signals and Wavelets in the Geometric Algebra
Framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.8 Kinematics and Dynamics . . . . . . . . . . . . . . . . . . . 12
1.5 Solving Problems in Perception and Action Systems. . . . . . . . 12
Part I Fundamentals of Geometric Algebra
2 Introduction to Geometric Algebra. . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 History of Geometric Algebra . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 What is Geometric Algebra? . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Basic Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2 Non-orthonormal Frames and Reciprocal
Frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
xi
xii Contents
2.2.3 Reciprocal Frames with Curvilinear Coordinates. . . 26
2.2.4 Some Useful Formulas . . . . . . . . . . . . . . . . . . . . . 27
2.3 Multivector Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Further Properties of the Geometric Product . . . . . . 30
2.3.2 Projections and Rejections. . . . . . . . . . . . . . . . . . . 32
2.3.3 Projective Split . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.4 Generalized Inner Product . . . . . . . . . . . . . . . . . . . 33
2.3.5 Geometric Product of Multivectors. . . . . . . . . . . . . 35
2.3.6 Contractions and the Derivation. . . . . . . . . . . . . . . 36
2.3.7 Hodge Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.8 Dual Blades and Duality in the Geometric
Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Multivector Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 Involution, Reversion and Conjugation
Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.2 Join and Meet Operations . . . . . . . . . . . . . . . . . . . 42
2.4.3 Multivector-valued Functions and the Inner
Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.4 The Multivector Integral . . . . . . . . . . . . . . . . . . . . 44
2.4.5 Convolution and Correlation of Scalar Fields . . . . . 44
2.4.6 Clifford Convolution and Correlation . . . . . . . . . . . 45
2.5 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5.1 Linear Algebra Derivations . . . . . . . . . . . . . . . . . . 47
2.6 Simplexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Differentiation, Linear, and Multilinear Functions
in Geometric Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.1 Differentiation by Vectors . . . . . . . . . . . . . . . . . . . 56
3.1.2 Differential Identities. . . . . . . . . . . . . . . . . . . . . . . 60
3.1.3 Multivector Derivatives, Differentials,
and Adjoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2 Linear Multivector Functions. . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 The Determinant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Adjoints and Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5 Eigenvectors and Eigenblades . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6 Symmetric and Skew-Symmetric Transformations. . . . . . . . . . 73
3.7 Traction Operators: Contraction and Protraction . . . . . . . . . . . 74
3.8 Tensors in Geometric Algebra . . . . . . . . . . . . . . . . . . . . . . . . 76
3.8.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.8.2 Orthonormal Frames and Cartesian Tensors . . . . . . 79
3.8.3 Non-orthonormal Frames and General Tensors . . . . 81
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86