Bayro-Corrochano E., Scheuermann G. Geometric Algebra Computing: in Engineering and Computer Science
Подождите немного. Документ загружается.
Index
A
Active-vision, 300
Additive split, 16
Algebra
geometric
representation, 397
Analytic signal, 169, 185, 249, 250, 254, 257
Antipodal inversion, 86
Aperture, 286, 289
Approximation, 62
Associative memories, 211
geometric, 216, 221
Automatic theorem proving, 498
B
Bergman-Hodge decomposition, 350
Bézier
cubic, 495, 503
surface, 503
Bigebra software, 458
Biomolecule
crystal, 397
Bivector orthogonalization, 142, 160
Bivector representation of a hue, 148
Blade, 192
Blades, 403, 457
CGA, 45
Bravais
symbol, 394, 396
system, 396
BSC, 402
Buchberger
algorithm, 495
theorem, 497
Bundle Adjustment, 278, 295
C
Camera, 278, 286, 289
Camera Calibration, 287, 289
Cartan representation, 414
Cartan–Dieudonné, 386
Catadioptric, panoramic, 277, 278, 286, 289
Cauchy integral, 164, 167, 181
metrodynamical, 182
Cauchy–Riemann operator, anisotropic, 182
Caustic, 503
Cayley transform, 86
CGA, 7, 36
implementation, 51
Circular convolution, 406
Classification, 235
geometric algebra, 215
Clean-up memory, 405
Clifford, 53, 386
Fourier kernel, 96
Fourier transform, 95
inverse, 96
windowed, 93
Gabor filter, 103
Gabor filters, 94, 97
Gabor transform, 97
geometric product, 386
group, 387
module, 95
monomial, 387
window daughter function, 96
window function, 96
Clifford algebra, 6, 93, 108, 110, 111, 213, 498
Clifford analysis, 107–111, 118, 119, 166
Clifford product, 192
CLIFFORD software, 458
Clifford–Fourier transform, 107, 108, 110, 111,
113, 139, 143, 145
E. Bayro-Corrochano, G. Scheuermann (eds.), Geometric Algebra Computing,
DOI 10.1007/978-1-84996-108-0, © Springer-Verlag London Limited 2010
519
520 Index
Clifford–Hermite polynomial, 109
Clops, 435
CLUCalc, 66
Clustering, 237
Code generation, 467
Coherent bivector, 86
Color
scheme, 393
stereo option, 393
Color image processing, 249, 268, 269
Commutativity, 96
Computer vision, 105, 277
Cone of view, 277, 278, 283, 286
Confluent hypergeometric function, 116
Conformal
group, 389
model, 388
point, 194
sphere, 194
Conformal algebra, 498
Conformal Clifford algebra, 72
Conformal geometric algebra, 7, 36, 54, 72,
279, 480
Conformal Grassmann–Cayley algebra, 72
Conformal model, 7, 71
Conformal split, 15
Conformal transformation, 74
Contourlet transform, 266
Contraction, 44
Convolution
Clifford, 124
Coordinate value, 235
Crystal, 385, 398
biomolecule, 397
cell, 388
cell choice, 393
class, 387, 390
general element, 392
International Tables of Crystallography, 393
lattice
vector, 397
locus, 392
oblique, 395
rectangular, 395
structure, 392
symmetry, 385
system, 390
view, 393
Crystal cell
cubic, 388
hexagonal, 388, 396
hexagonally centered, 396
monoclinic, 388
inclined, 396
orthogonal, 396
oblique, 395
orthorhombic, 388, 396
rectangular, 395
rhombohedral, 388
tetragonal, 388, 396
triclinic, 388, 396
trigonal, 388, 396
Crystal system
hexagonally centered, 396
monoclinic
inclined, 396
oblique, 396
orthogonal, 396
rectangular, 396
Crystallographic, 385
group, 385
Crystallography
International Tables Online, 393
CUDA, 66, 477
CWFT
delay, 99
left linearity, 97
modulation, 99
orthogonality relation, 100
parity, 98
reconstruction formula, 100
reproducing kernel, 101
reversion, 98
shift
frequency, 99
spatial, 99
switching, 98
Cycle, 435
cover, 435
enumerating, 437
Hamiltonian, 435, 438
proper, 435
Cylindrical
Fourier transform, 107, 108, 111–114,
117–119
D
Delta product, 465
Density
decay, 317
function, 304
Dickson’s Lemma, 497
Dilator, 195
Dirac operator, 109, 111, 166
anisotropic, 182
Dirac–Hestenes
equation, 397
Directional filter bank, 249, 266
Index 521
Distance
Euclidean, 388
Distance to ellipse, 509
Distributed representation, 401
Distributions, 176
Division algorithm
general, 496
Dual of a group, 136, 155
Dualization, 43
E
Ego-center, 300
Elbow
of a robot arm, 498
Elimination
ideal
second, 502
order, 502, 503
Elimination Theory, 498
Ellipse, 284
EM algorithm, 235
Energy density, 97
Entangled bivector, 86
Envelope, 498, 501
singularities of, 503
Equation
Dirac–Hestenes, 397
Oseen, 369
Equations
forecasting, 378
Navier–Stokes, 367
shallow water, 378
Equidistant
curve, 498
surface, 498
Euclidean
distance, 388
group, 389
Euclidean distance, 416
Euclidean geometry, 8, 38
Exponential map, 74
Exponential map of a Lie group, 157
Extension
Theorem, 510
Theory, 498
Exterior exponential, 75
F
Face recognition, 249–251, 269, 272
Factorization, 457
FastJoin, 462
Feature extraction, 233
Fermat
cubic, 495
curve, 503
Filter banks, 105
Filtering
orientation, 304
proximity, 304
Fluid dynamics, 353
Fluid flow analysis, 121
Forward kinematic problem, 498
Fourier analysis, 107
Fourier series
quaternion, 128
Fourier transform, 107, 110–112, 118, 137, 156
biquaternion, 130
Clifford, 95, 124
discrete Clifford, 125
quaternion, 126
FPGA, 66, 477
Frequencies filtering, 148
Frieze group, 394
Funk–Hecke theorem, 109, 114, 116
G
Gaalop, 66, 477
Gabor filters
Clifford, 94
quaternionic, 94
Gaigen software, 457, 466
Gaussian
window function, 103
Gaussian function, 97
Gaussian mixture model, 232
Gegenbauer polynomial, 115, 116
Generalized Cauchy–Riemann operator, 330
Generator
geometric, 395
geometric algebra, 391
product, 391
screw, 395
space group, 389
Geometric algebra, 93, 122, 192, 213, 232, 402
and neural networks, 191
classification, 215
conformal, 36, 194, 213
Geometric data, 233
Geometric feature, 232
Geometric product, 192
Geometry
Euclidean, 8, 38
GGNF
Generalized Gradient Vector Flow, 197
522 Index
GGVF
energy functional, 197
neural networks and, 198
Grade
of blade, 481
Grade involution, 387
Gram–Schmidt orthogonalization, 471, 473
Graph, 435
circumference, 435, 439
component, 435
finite, 435
girth, 440
matching, 436
regular, 435
representation, 301
Grassmann, 53, 386
algebra, 499
basis monomial, 499
outer product, 386
Grassmann–Cayley algebra, 72
Gröbner basis, 495, 515
reduced, 498
theory, 495
Group
cell point groups, 386
Clifford, 387
conformal, 389
crystallographic, 385
Euclidean, 389
frieze, 395
generator, 389
hexagonal point, 387
layer, 396
monoclinic, 398
monoclinic oblique, 398
monoclinic rectangular, 398
triclinic, 398
triclinic oblique, 398
layer group, 396
GA generator, 396
geometric notation, 396
Lipschitz, 387
monoclinic, 389
monoclinic point group, 387
orthogonal, 389
point group, 385, 387, 390, 397
presentation, 387
relation, 387
rod, 396
rod group
GA generator, 396
geometric notation, 396
monoclinic, 397
monoclinic inclined, 397
monoclinic orthogonal, 397
triclinic, 397
rotation subgroup, 387
selection, 390
space group, 385, 390, 397
subperiodic, 385, 386, 394
frieze, 394
layer, 394
rod, 394
symbol, 389
visualization, 385
Group morphism from R
2
to Spin(4), 141
Group representation, 155
H
Hamming distance, 416
Hand-written digit, 240
Handshaking Lemma, 436
Hardy space, 165, 167, 181
Hermann–Maugin, 387
Hermitean conjugation, 108
Hermitean inner product, 108
Hestenes, 53
Hilbert basis theorem, 497
Hilbert transform, 163, 166, 167, 181, 249, 250,
253, 254, 257
anisotropic, 181
Holomorphy
D
a
-, 349
L-, 347
Homothety transformation, 321
HRR, 402
Humanoid robot, 300
Hypercomplex derivative, 330
Hypercomplex Fourier transform, 152
Hyperholomorphic constant, 341
Hyperplane, 386
I
Ideal
finitely generated, 496
of relations, 514
prime, 514
Image analysis, 105
Image compression, 105
Image recognition, 249, 251, 268
Image registration, 249, 250, 261, 262, 268
Implicitization problem, 498
Inner product, 234, 403
Integral transform, 107, 118, 119
Interaction
visual, 391
International Tables of Crystallography, 393
Interpolation function, 514
Index 523
Intersection, 279, 283, 286, 288, 291
Invariant
generating, 498
Invariant feature, 249, 250, 262, 264, 265
Inverse Clifford–Fourier transform, 143, 145
Inverse Fourier transform, 157
Inverse kinematic problem, 498
Inverse kinematics, 485
Inversion, 386
Involution
grade, 387
main, 387
IPNS
Inner Product Null Space, 193
J
Join, 457
K
Kummer’s function, 116, 117
L
Laplace operator, 109, 111
Layer group, 394
Legendre polynomial, 110, 115
Lexicographic order, 503
Lie algebra morphism, 158
Lie algebra morphism from R
2
to R
2
4,0
, 141
LIFT, 458
Lipschitz
group, 387
Local geometric properties, 340
Localization, 97
Location
hypotheses, 309
Location information, 93
M
Main involution, 387
Matching, 441
Matrix representation, 413
Meet, 457, 474
Model
conformal, 388
Model-base
self-localization, 300
Monoclinic, 387, 389
Monogenic extension, 171
Monogenic function, 109, 166
Monogenic-conformal mappings, 328, 332
Monomial
Clifford, 387
ideal, 497
order, 496
graded inverse lex, 496
graded reverse lex, 496
lexicographic (lex), 496
Motion estimation, 278, 292, 295
Mouse
3D connexion, 392
three-dimensional, 392
Multi-dimensional, 232
Multivector, 192
N
Nil–Clifford algebra, 433
Nilpotent adjacency matrix, 437
Numerical analysis
approximation, 373
stability, 374
Numerical stability, 465
O
Offset, 507
Operator
Cauchy–Fueter, 352
Dirac, 351
generalized intersection, 300
OPNS
Outer Product Null Space, 193
Optical flow, 249, 250, 257, 258, 260–262
Optics, 105
Optimization, 218, 220
Orthogonal
group, 389
Orthographic
projection, 393
view, 393
Outer product, 234, 457
P
Path, 435
Pattern association, 211
Pattern classification, 211, 215, 221
Pattern matching, 125
Pattern recognition, 105
Perception–action, 299
Perception–recognition, 300
Percepts
fusion, 303
matching, 303
Phase congruency, 249, 250, 262–265
Pixel, 278, 283
Pixel gray-level, 287
Plemelj–Sokhotzki formulae
anisotropic, 165, 183
524 Index
Point group, 385
hexagonal, 387
Polygon
regular, 387
Polytope, 320
face cells, 320
ridges, 320
vertices, 320
Pose
6D, 300
estimation, 310
Problem
magnetic Benard, 375
nonlinear Stokes, 355
stationary Navier–Stokes, 356
stationary Stokes, 354
Product
inner, 44
outer, 42
Projection
Bergman, 351
orthogonal, 44
orthographic, 393
Projections
general Plemelj, 348
Q
Quasi-conformal mappings, 336
Quaternion algebra, 249–251, 253, 254
Quaternion phase, 257, 264
Quaternion wavelet, 249–252, 254, 255, 257,
258, 263, 265, 269, 272
Quaternionic analysis, 348
discrete structures, 352
quaternionic operator calculus, 351
Quaternionic Fourier transform, 153
Quaternionic Vahlen matrices, 78
Quaternions, 348
Gabor filters, 94
Questionnaire, 242
R
Reduced quaternions, 329
Reflection, 385, 386, 389
composition, 386
diagonal glide, 394
glide, 394
hyperplane, 386
rotary, 386
Reverse
multivector, 193
Rigid body motion, 55, 195
Ring of invariants
of a finite group, 514
Robotics, 31
Rod group, 394
Rodrigues matrix, 510, 511
Roles and fillers, 404
Rotation, 385, 386, 389
screw, 394
subgroup, 387
Rotation, translation, 282, 283
Rotor, 195
S
S-polynomial, 497
Scalar sum, 434
Score functions, 292
Screw
generator, 395
Screw motion, 74
Screw theory, 26
Semi-discretisation
Rothe method, 368
Semi-supervised learning, 237
Shell, 503
Signal representation, 93
Simulated annealing, 295
Software
ambient light, 393
animation, 391
cell choice, 393
cell type menu, 393
CLUCalc, 385
color scheme, 393
light source, 393
lighting menu, 393
multivector, 390
OpenGL, 385
rotation center, 392
SGV browser panel, 391
SGV general element selection, 392
SGV GUI, 390
SGV toolbar, 390, 391
three-dimensional graphics, 392
three-dimensional interaction, 392
virtual reality, 393
visual interaction, 391
XML input, 390
Space
ego reference frame, 303
physical, 301
radial, 317
radial normalized density, 323
visual, 301
world-model, 301
Index 525
Space group, 385, 390
frieze, 395
generator, 389
layer, 396
magnetic, 397
monoclinic, 389
orbit
noncharacteristic, 397
rod, 396
selection, 390
subperiodic, 386, 394, 397
magnetic, 397
symbol, 389
visualization, 385
Space Group Visualizer, 389
Spatial extension
limited, 105
Spatial vectors, 233
Spatial-frequency analysis, 93
Spectrogram, 97
Sphere
Gaussian, 314
restriction, 308
Spherical monogenic, 178
Spherical neighborhood, 215, 217
optimal, 217
Spherically separable, 221
patterns, 216
Spinor group, 139, 141, 160
Split
additive, 16
conformal, 15
Stereo
color, 393
Stereo matching, 249, 250, 258, 264, 265
Structure
topological and geometric, 300
Subperiodic
space group, 394
Subperiodic group
frieze, 395
layer, 396
layer group, 396
rod, 396
Subspace
configuration, 319
intersection, 300
restriction, 300
Subspace intersection, 457
Subspace union, 457
Subspaces, 457
Symbol
Bravais, 394, 396
geometric for group, 395
space group, 389
Symmetry, 385
active, 392
composition, 386
diagonal glide reflection, 394
glide reflection, 394
inversion, 386
operation, 392
operator, 385
translator, 389
reflection, 385, 386, 389
rotary reflection, 386
rotation, 385, 386, 389
screw rotation, 394
transformation, 386
translation, 385, 389
System
Bravais, 396
Syzygy
ideal, 512, 513
relation, 498
T
Tensor product, 416
Theorem
Cartan–Dieudonné, 7, 39, 386
Three-nil algebra, 433
Transform
Teodorescu, 352
Translation, 385, 389
Translator, 195, 389
Tree, 435
Triangular element, 515
Twist, 277, 283
Twisted Vahlen matrices, 78
V
Vahlen matrices, 78
Var ie ty
affine, 496
Versor, 41, 387
as exponential, 49
combination, 389
Ver tex
enumeration, 320
View
orthographic, 393
Virtual reality, 393
Visualization
ambient light, 393
cell choice, 393
cell type menu, 393
color scheme, 393
coordinate frame, 393
526 Index
interactive, 389
navigate, 393
protein, 397
space group, 385
view rotation, 392
W
Walk, 435
Window function
B-spline, 103
Windowed Fourier transform
Clifford, 93
X
XOR, 403