Table Of ContentLecture Notes in Computer Science 2145
EditedbyG.Goos,J.Hartmanis,andJ.vanLeeuwen
3
Berlin
Heidelberg
NewYork
Barcelona
HongKong
London
Milan
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Michael Leyton
A Generative
Theory of Shape
1 3
SeriesEditors
GerhardGoos,KarlsruheUniversity,Germany
JurisHartmanis,CornellUniversity,NY,USA
JanvanLeeuwen,UtrechtUniversity,TheNetherlands
Author
MichaelLeyton
RutgersUniversity
CenterforDiscreteMathematics&TheoreticalComputerScience(DIMACS)
NewBrunswick,NJ08854,USA
E-mail:mleyton@dimacs.rutgers.edu
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Leyton,Michael:
Agenerativetheoryofshape.MichaelLeyton.-
Berlin;Heidelberg;NewYork;Barcelona;HongKong;
London;Milan;Paris;Tokyo:Springer,2001
(Lecturenotesincomputerscience;2145)
ISBN3-540-42717-1
CRSubjectClassification(1998):I.4,I.3.5,I.2,J.6,J.2
ISSN0302-9743
ISBN3-540-42717-1Springer-VerlagBerlinHeidelbergNewYork
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Preface
The purpose of this book is to develop a generativetheory of shape that has
twopropertiesweregardasfundamentaltointelligence–(1)maximization of
transfer:wheneverpossible,newstructureshouldbedescribedasthetransfer
of existing structure; and (2) maximization of recoverability: the generative
operations in the theory must allow maximal inferentiability from data sets.
We shall show that, if generativity satisfies these two basic criteria of in-
telligence, then it has a powerful mathematical structure and considerable
applicability to the computational disciplines.
The requirement of intelligence is particularly important in the genera-
tion of complex shape. There are plenty of theories of shape that make the
generation of complex shape unintelligible. However, our theory takes the
opposite direction: we are concerned with the conversion of complexity into
understandability. In this, we will develop a mathematical theory of under-
standability.
The issue of understandability comes down to the two basic principles of
intelligence-maximizationoftransferandmaximizationofrecoverability.We
shall show how to formulate these conditions group-theoretically. (1) Maxi-
mization of transfer will be formulated in terms of wreath products. Wreath
products are groups in which there is an upper subgroup (which we will call
a control group) that transfers a lower subgroup (which we will call a fiber
group)ontocopiesofitself.(2)maximizationofrecoverabilityisinsuredwhen
the control group is symmetry-breaking with respect to the fiber group.
A major part of this book is the invention of classes of wreath-product
groups that describe, with considerable insight, the generation of complex
shape;e.g.,incomputervisionandcomputer-aideddesign.Thesenewgroups
willbe calledunfolding groups.As the name suggests,suchagroupworksby
unfolding the complex shape from a structural core. The core will be called
an alignment kernel. We shall see that any complex object can be described
VI
as having an alignment kernel, and that the object can be generated from
this kernelbytransferring structure fromthe kerneloutto become the parts
of the object.
A significant aspect of all the groups to be invented in this book, is that
they express the object-oriented nature of modern geometric programming.
In this way, the book develops an object-oriented theory of geometry. For
example, we will develop an algebraic theory of object-oriented inheritance.
Ourgenerativetheoryofshape issignificantlydifferentfromcurrentgen-
erative theories (such as that of Stiny and Gips) which are based on pro-
duction rules. In our theory, shape generation proceeds by group extensions.
Thealgebraictheorythereforehasaverydifferentcharacter.Brieflyspeaking:
Featurescorrespondtosymmetrygroups,andaddition offeaturescorresponds
to group extensions.
The major application areas in the book are visual perception, robotics,
andcomputer-aided design. In visualperception,our centralprinciple is that
anintelligentperceptualsystem(e.g.,thehumanperceptualsystem)isstruc-
tured as an n-fold wreath product G1(cid:1)wG2(cid:1)w...(cid:1)wGn. In previous publica-
tions, we have put forward several hundred pages of empirical psychological
evidence, to demonstrate the correctness of this view for the human sys-
tem. We shall see that the fact that the visual system is structured as a
wreath product, has powerful consequences on the way in which perception
organizes the world into cohesive structures. Chapter 5 shows how the per-
ceptual groupings can be systematically predicted from the wreath product
G1(cid:1)wG2(cid:1)w...(cid:1)wGn.
Chapter six develops a group theory of robot manipulators. We require
thegrouptheorytosatisfythreefundamentalconstraints:(1)Perceptualand
motorsystemsshouldberepresentationallyequivalent.(2)Thegrouplinking
basetoeffectorcannotbeSE(3)(whichisrigid)butagroupthatwewillcall
semi-rigid; i.e., allowing a breakdown in rigidity at a specific set of points.
(3) The group must encode the object-oriented structure.
The theory of robotic kinematics continues in two ways: (1) within the
theory of mechanical CAD in Chapter 14; and (2) in the theory of rela-
tive motion(in visualperception,computer animation,andphysics)givenin
Chapter 9.
Chapter ten begins the analysis of static CAD by developing a theory of
surface primitives, showing that, in accord with the theory of recoverability,
thestandardprimitivesofCAD(andvisualperception)canbesystematically
elaborated in terms of what we call iso-regular groups. Such groups are n-
foldwreathproductsG1(cid:1)wG2(cid:1)w...(cid:1)wGn,inwhicheachlevelGiisanisometry
groupandiscyclicoraone-parameterLie group.Togofromsuchstructures
to non-primitive objects, one then uses either the theory of splines given
later in the book, or the theory of unfolding groupsgiven in Chapters 11,12
and 13.
VII
Thebasicpropertiesofanunfoldinggrouparethatitisawreathproduct
in which the control group acts (1) selectively on only part of its fiber, and
(2)bymisalignment.Inmanysignificantcases,thefiberisthedirectproduct
G1×...×Gn ofthesymmetrygroupsGi oftheprimitives,i.e.,theiso-regular
groups,andanyfibercopycorrespondstoaconfigurationofobjects.Thefiber
copyinwhichtheobjectsymmetrygroupsG1,...,Gn aremaximallyaligned
with each other is called the alignment kernel. The action of the control
group, in transferring fibers onto each other is to successively misalign the
symmetry groups. This gives an unfolding effect.
Chapter 14 then presents a lengthy and systematic analysis of mechan-
ical CAD using the above theory. We work through the main stages of
MCAD/CAM: part-design, assembly, and machining. For example, in part-
design, we give an extensive algebraic analysis of sketching, alignment,
dimensioning, resolution, editing, sweeping, feature-addition, and intent-
management.
Chapter 15 then carries out an equivalent analysis of the stages of archi-
tectural CAD. Then, Chapter 16 gives an advanced algebraic theory of solid
structure, Chapter 17 gives a theory of spline-deformation as automorphic
actions on groups; and Chapter 18 provides an equivalent analysis for sweep
structures.
Chapter 20 examines the conservation laws of physics, in terms of our
generative theory, although the next volume will be devoted almost entirely
tothegeometricfoundationsofphysics.Chapter21givesatheoryofsequence
generation in music.
Finally, Chapters 2, 8, and 22, examine in detail the fundamental differ-
ences between our theory of geometry and Klein’s Erlanger program.Essen-
tially,inourtheory,therecoverabilityofgenerativeoperationsfromthedata
set means that the shape acts as a memory store for the operations. More
strongly,wewillarguethatgeometry is equivalent tomemorystorage.Thisis
fundamentally opposite to the Erlangerapproachin whichgeometricobjects
aredefinedasinvariantunderactions.Ifanobjectisinvariantunderactions,
the actions are not recoverable from the object. We demonstrate that our
approach to geometry is the appropriate one for modern computational dis-
ciplines suchascomputervisionandCAD,whereasthe Erlangerapproachis
inadequate and leads to incorrect results.
June 2001 Michael Leyton
Table of Contents
1. Transfer ................................................................1
1.1 Introduction ........................................................1
1.2 Complex Shape Generation .........................................2
1.3 Object-Oriented Theory of Geometry ...............................3
1.4 Transfer ............................................................4
1.5 Human Perception ..................................................6
1.6 Serial-Link Manipulators ...........................................18
1.7 Object-Oriented Inheritance .......................................20
1.8 Complex Shape Generation ........................................21
1.9 Design .............................................................21
1.10 Cognition and Transfer ...........................................22
1.11 Transfer in Differential Equations .................................23
1.12 Scientific Structure ...............................................24
1.13 Maximization of Transfer .........................................28
1.14 Primitive-Breaking ...............................................28
1.15 The Algebraic Description of Machines ............................30
1.16 Agent Self-Substitution ...........................................30
1.17 Rigorous Definition of Shape ......................................32
1.18 Rigorous Definition of Aesthetics .................................32
1.19 Shape Generation by Group Extensions ...........................34
2. Recoverability ........................................................35
2.1 Geometry and Memory ...........................................35
2.2 Practical Need for Recoverability ..................................35
2.3 Theoretical Need for Recoverability ...............................37
2.4 Data Sets .........................................................37
2.5 The Fundamental Recovery Rules .................................39
2.6 Design as Symmetry-Breaking .....................................44
2.7 Computational Vision and Symmetry-Breaking ....................45
2.8 Occupancy ........................................................47
2.9 External vs. Internal Inference ....................................48
2.10 Exhaustiveness and Internal Inference ............................49
X Table of Contents
2.11 Externalization Principle .........................................52
2.12 Choice of Metric in the Externalization Principle .................58
2.13 Externalization Principle and Environmental Dimensionality .....59
2.14 History Symmetrization Principle ................................61
2.15 Symmetry-to-Trace Conversion ...................................63
2.16 Roots ............................................................66
2.17 Inferred Order of the Generative Operations .....................67
2.18 Symmetry-Breaking vs. Asymmetry-Building .....................69
2.19 Against the Erlanger Program ...................................71
2.20 Memory .........................................................72
2.21 Regularity .......................................................73
2.22 Aesthetics .......................................................74
2.23 The Definition of Shape ..........................................75
3. Mathematical Theory of Transfer, I ................................77
3.1 Shape Generation by Group Extensions ...........................77
3.2 The Importance of n-Cubes in Computational Vision and CAD ...78
3.3 Stage 1: Defining Fibers and Control ..............................80
3.4 Stage 2: Defining the Fiber-Group Product ........................82
3.5 The Fiber-Group Product as a Symmetry Group ..................84
3.6 Defining the Action of the Fiber-Group Product on the Data Set ..84
3.7 Stage 3: Action of G(C) on the Fiber-Group Product ..............88
3.8 Transfer as an Automorphism Group ..............................89
3.9 Stage 4: Splitting Extension of the Fiber-Group Product
by the Control Group .............................................90
3.10 Wreath Products ................................................91
3.11 The Universal Embdedding Theorems ............................93
3.12 Nesting vs. Control-Nesting ......................................93
3.13 Stage 5: Defining the Action of G(F)(cid:1)wG(C) on F ×C ...........95
3.14 Control-GroupIndexes ...........................................97
3.15 Up-Keeping Effect of the Transfer Automorphisms ...............98
3.16 The Direct vs. the Indirect Representation ......................103
3.17 Transfer as Conjugation ........................................104
3.18 Conjugation and Recoverability .................................108
3.19 Infinite Control Sets ............................................109
3.20 The Full Structure ..............................................110
3.21 The Five Group Actions ........................................112
4. Mathematical Theory of Transfer, II .............................115
4.1 Introduction .....................................................115
4.2 The Iterated Wreath Product ....................................115
4.3 Opening Up .....................................................116
4.4 The Group Theory of Hierarchical Detection .....................118
4.5 Control-Nested τ-Automorphisms ................................123
4.6 The Wreath Modifier ............................................128
