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Faugeras O., Luong Q., Papadopoulo T. — The Geometry of Multiple Images: The Laws That Govern the Formation of Multiple Images of a Scene and Some of Their Applications |
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Предметный указатель |
see “Affine line at infinity”
see “Projective line”
see “Projective plane”
see “3-D projective space”
see “General projective space”
see “Dual projective space”
see “Affine group”
see “Group of similarities”
see “Orthogonal group”
see “Projective group”
see “Affine plane at infinity” “Hyperplane
Adjoint matrix xx 55 113 159 350
Affine basis 38 70 95—97 165 197 386 393
Affine basis, change of 72 317 381 565
Affine basis, orthonormal 42 75 280 390 393 530
Affine camera 198 216 221 225 229 241 242 244 257 292 381 468 516—518 559 565—567 590
Affine coordinates 5 12 13 72 95 97 100 197 198 204 232 293 320 322 327 352 376 384 387 390 393 396 398 460 516—518
Affine group 68 69 71 72 75 99 124 361 388 503
Affine invariant xv 11 19 43 65 68 99 106 120 123 125 360 387 391 404
Affine line 94 96—98 105 119
Affine line at infinity 11 55 94 97 118 119 197 200 227 240 269 279 351 559
Affine line, parallel 5 9 20 97 98 100 202 388
Affine morphism 71 75 99 226 296
Affine parallax 387
Affine plane 94 96 97 118 197
Affine plane at infinity xvii 10 19—22 30 31 37—39 44 52 55 95 97 98 100 118 121 124 197 200 202 207 213 225 256 278 280 281 380 385 389 396 454 527 530 532 538 542 559 565 570 571 583 585 589
Affine plane, parallel 100
Affine point 55 97 165
Affine point at infinity 5 6 9 19 68 87 94 97 99 101 105 106 121 124 196 199 200 202 225 290 376 387 391
Affine projection matrix 225 227 229 459 461 518
Affine reconstruction 38 40—42 385—386 390 395 398 405 538
Affine rigid displacement 4 41 42 55 75 124 207 208 238 260 503 529 530 532—535 549 557—559 562 563 570 574 583 584
Affine similarity 5 11 39 42—44 52 55 69 75 117 118 120 121 124 243 296 361 395 397 399 403 404 503 548 576
Affine space 19 38 94 97 106 119 164 174 196 199 207 240 242 380
Affine space, Euclidean 6 15 75 207 310
Affine space, general 68 69 72 73 75 93 95 99 105 116 123
Affine stratum 360 380—392 525—529 568
Affine subspace 73 94 97 99
Affine subspace, parallel 68 73 94 99 196
Affine transformation 36 37 39 43 44 55 61 68 71 99 106 120 123 124 198 208 210 240 244 256 259 297 317 318 329 385 387 395 403 404
Affine viewing situation 295—297 559—560
Algebra 132 140 143 156 157
angle 4 41 65 124 208 211 399 577
Angle of two hyperplanes 69
Angle of two lines of 119 121
Angle of two lines of 14 121 213 397 398
Angle of two planes of 121 122
Angle of two vectors 337 456
Angle of view 203 232 234
Aspect ratio 14 16 41 55 208 210 229 235 560 564 565 570 576 589—591
Augmented reality 579 584
Bas-relief ambiguity 298 302
Baseline 23 479 538
Basis see “Vector basis” “Affine “Projective
Bilinear 25 44 47 265 423 449 450
Bilinear constraint see “Epipolar constraint”
Bilinear deformation 238
Bilinear interpolation 235
Bilinear map 74 132 143 161 423 442 443 446 448 451
Bracket 129 141 167 272 292 294
Bundle adjustment 16 239 328 403 521—525 537 572 577 590
Camera obscura 178
Camera, calibration 56 57 236 239 243 582
Camera, calibration, classical 13 15 16 175 234—239 242 335 336 340 403 576
Camera, coordinate system 12 14 42 179 182 216 224 240 292 558
Camera, coordinate system, normalized 55 179 202 203 281 298 394—397 528 534 575 583
Canonical basis of 159 160
Canonical basis of 131 164
Canonical basis of 131 134 153 154 165 169 171
Canonical basis of 131
Canonical basis of 131 136 163 169
Canonical basis of 131
Canonical basis of 131 150
Canonical form, affine 382 389 394 398 525
Canonical form, Euclidean 393 529 567
Canonical form, projective 35 362 363 510
Canonical frame, affine 383 386 387 389
Canonical frame, Euclidean 394 396
Canonical frame, projective 364 368—370 389 390 513—515 529 530 533 569
Canonical projection 78 79 83 93
Canonical representation 536
Canonical representation of n views 52 53 503
Canonical representation of two views 43 361—362 404
Canonical representation, affine 39 54 526
Canonical representation, Euclidean 42 54 58 528
Canonical representation, projective 33 54 58 512 513 515 521 526
Cap-product 150 153 154 167 186 187 201
Cap-product of a vector and a covector 148 158 162 181 183
Catadioptric system 290
Center of mass 11 68 226
Centroid 226 318 517
Cheiral inequalities 569
Cholesky decomposition 213 545 568
Circular points 55 69 117 118 120 121 215 240 559 591
Cobasis xxii 150 154 158 159 161—165 168 169 171 181 186
Cofactor matrix xx 252
Compatible Fundamental matrices 417 507
Complexified affine space 76 117 118
Complexified projective space 117 118 119
Complexified vector space 76 116
conic 55 67 113 116 117 303 328 353 358 442—444 465 584
Conic, absolute xvii 39 44 55 57 69 115 117—119 121 207 212 243 396 397 542 554 559 569 570 583—585 588
Conic, absolute, image of xv 55 211 212 214 215 239 400 542 553 570 590
Conic, dual 55 114 215 349 352 353 396 542 570
Contravariant index 46 470
Convex hull 68 407
Coordinate vector 8 78 79—81 83 84 87 88 91—93 108 112 114 117 132 134 137 165 168 186 197 198 261 274 275 322 346 349 351 369 389 518
Covariance matrix xxiv 243 324 327 341 342 345 346 349 352 356 564
Covariant index 46 470
Covector xxii 148 149 151—153 156 158 162 181 187
Cramer relation 153 436 449 506
Critical configuration 58 468 529
Critical sequence 564 583 584 586 587 591
Critical surface 27 300 302—307 312
Cross-product 8 35 50 159 165 267 431 443 447 458 462 464 474 487
Cross-product, matrix form xxiii 8 57 160
Cross-product, representation of a 2-D line 8 159 262 474 475
Cross-product, representation of a 2-D point 92 97 110 159 161
Cross-product, representation of a 3-D plane xxiii 164
Cross-product, tensor form 162 470
Cross-ratio xv xxii 11 67 101 103—107 109—112 120 121 124 213 266 369—373 376 391 397 434
Cyclopean viewing 294 295
Cyclotorsion 298 300
Decentering distortion 234 235
Direction of a hyperplane 68 99 119
Direction of a line 6 9 19 20 37 68 94 98 99 119 196 199 200 202 205 216 220 225 226 241 383 389 390 586
Direction of a plane 10 95 97 197 204 239 243 491
Disparity 312 377 381 382 384 405 575
Dot product see “Inner product”
Duality 7 8 10 55 66 89 106—114 116 118 128 146—168 185 189 190 252 261 352 354 364 386 396 406 423 444 446 459
Eight-point algorithm 318 320 354
ellipse 55 232 350—353 588
Ellipse of uncertainty 346—349 351 352
Epipolar band 349 352 355 356
Epipolar constraint 25 44 47 51 235 259 260 261 269 285 287 293 301 307 308 311 318 328 329 357 367 372 375 389 415 440 464 504 505 536 559 572 577
Epipolar geometry xv xvi 25 260 288 290 291 298 332—340 346 354 355 358 372 378 401 406 418 429 529 533
Epipolar geometry, affine 294 297 312
Epipolar homography 259 266 267 269 285 310 320 322 323 346 367 418 433 553 554
Epipolar line 23 24 26 28—30 35 44 48 106 260 261 266 268 276 288—291 298 300 301 303 312 323 326 335 337 338 347—349 354 355 367 369 376 378 379 392 406 412 415 418 422—424 430 433 434 440 444 448 452 508 512 547 553 560 561
Epipolar pencil see “Pencil of epipolar lines”
Epipolar plane 23 106 260 263 266 277 294 358 412 422 423 425 437 512 553 588
Epipolar projection 276 277
Epipole 26 28 30 33 37 39 43—45 49 52 53 57 60 260 262 264 265 269 274—276 278 279 281 287 288 290—293 297 306 308 311 320 322 325 335 337 346—348 355 356 365 367 371 374 377 379 385 395 403 404 407 412 415 416 418 429—431 440 444 448 449 452 454 462 490—492 497 507 509 510 512 513 520 527 533 536 553 554 560
Essential matrix xxiii 273 282 284 285 290 297 307 310 312 317 355 358 453 551 557 573
Euclidean calibration 540
Euclidean coordinates 6 122 180 212
Euclidean geometry xiv xv xvii 4 9 11 65 69 89 116 124 309
Euclidean invariant xv 11 19 43 55 64 77 118 119 121 125 361 397 399 404
Euclidean parallax 396
Euclidean reconstruction 31 32 41 42 54 56 243 395—396 399 401 405 406 468 572 574 575 578—581 590
| Euclidean space see “Affine space Euclidean”
Euclidean stratum 361 393—403 525—529 560 568
Euclidean transformation see “Affine rigid displacement”
Extensor 129 130—132 134 136—138 140—144 146 148 150 152 153 155 156 168 195 265
Extensor, split of 140 152
Exterior algebra of the join 132 156
Exterior algebra of the meet 143 156
External parameters see “Extrinsic parameters”
Extrinsic parameters 14 41 196 207 210 227 238 240 243 281 285 395 530 573 576 577 580
F-matrix see “Fundamental matrix”
Factorization method 242
Factorization method, affine 516—518
Factorization method, perspective 518—521
Focal length 12 14 58 178 209 219 228 231 234 243 244 398 548 561 572 573 576 590 591
Focal plane 197 225—227 294 296 364 371 387
Frobenius norm 320 338 518
Fronto-parallel 219 226 296 298
Fundamental matrix xv xvi xxiii 24 28 31 35 39 44 51 54 57 261 262—264 269 270 272 279 281 282 287 291 301 306 308 311 312 318 322 324 335 346 349 355 358 362 365 368 370 372 376 378 381 393 406 412—415 431 439 463 465 491 496 507 513 520 526 533 538 548 550 551 555 557 559 560 563 573 591
Fundamental matrix, affine 293 294 296 297 299 559
Fundamental matrix, estimation 27 316—358 525
Fundamental matrix, parametrization 266 269 321—325 483 489
Fundamental matrix, uncertainty 341—346 352
Fundamental points 304 306 468
Fundamental theorem of projective geometry, First 86 101
Fundamental theorem of projective geometry, Second 104 266
Gaussian noise 477
Grassman — Cayley algebra xiv xvi
Grassmann — Cayley algebra 92 143 147 170 171 182 264 466
Group see “Affine group” “Projective “Orthogonal
Group of rotations 580 585
Group of similarities 75 77 120 121
Group, general xiv 67 123
Hodge operator 155 160 163 169 181 262 389
Homogeneous coordinates see “Projective coordinates”
Homography 16 17 20 21 24 31 33—35 37 43 44 46 48 51 55 67 83 86 89 92 99—101 108 111 120 124 159 192—194 200 227 235 239 240 252 266 302 310 312 353 356 358 361 364 369 380 390 395 403 404 406 434 459 487 489 503 520 530—532 538 567
Homography, infinity 38 39 56 279 385 403 454 526 542 544 545 547 590
Homography, planar 22 28 29 33 37 49 52 53 252 253 270 273 274 286 365 374 407 437 513 514
Homography, rectifying 373 376
Horizon line 5 204
Hyper-ellipsoid of uncertainty 345
Hyperbola 55 349 351 353 356
Hyperboloid of one sheet 302
Hyperplane at infinity see “Projective hyperplane at infinity”
Image plane see “Retinal plane”
Image rectification 288 372—376
Implicit functions theorem 342
incidence 34 92 124 225
Infinity morphism see “Projective morphism infinity”
Inner product 74 149 158 162 416
integral xxii 155 156 265
Internal parameters 393 (see also “Intrinsic parameters”)
Intrinsic homography 49 463
Intrinsic line 277
Intrinsic parameters xxiii 14 19 21 25 31 41—43 52 56 57 60 196 208 210 212 215 219 227 229 231 238 240 242 243 255 280 281 286 287 291 297 310 312 335 354 376 380 393 395 398 404 454 461 503 528 530 533 537 538 542 544 545 547 548 550 558 560 564 565 568 571—574 576 582 590
Intrinsic planar morphism 425 426
Intrinsic plane 30 38 277 278 363 436 507 513 515
Intrinsic representation of a plane 278 280 310 311 362 527 530
Invariant xiv 11 46 59—60 67 124 292 360 543 585 “Affine “Euclidean
Inverse perspective projection 188
Inverse perspective projection, matrix 185 189 200 201 212 252 262 263 363
Isotropic cone 76 77
Isotropic lines 76 77
Jacobian 324 325 342 486 522 523
Join 129 155 170 265 389
Join of points 8 136 167 185
Join, antisymmetry 133
Join, associativity 133
Join, geometric interpretation 134
k-sequence 129
Kernel see “Nullspace”
Kruppa equations 58 346 553—558 560 564 566 573 580 589 590
Kruppa equations, degeneracy 559 587
Kruppa matrix xxiii 55 57 58 215 396 528 542 544 553 563 568 570
Laguerre formula for 2-D lines 120 213
Laguerre formula for 3-D lines 122
Laguerre formula for hyperplanes 69
Laguerre formula for planes 121
Least-median-of-squares 330 333 334 336 337 340 488
Levi-Civita tensor 161 470
Line at infinity see “Affine line at infinity”
LMedS see “Least-median-of-squares”
Longuet-Higgins equation 282 283 311 317 354
M-estimator 330 332 336 337 340
Median 332 334 564
Meet 153 155 171 264 265 295 422 435 515
Meet of a plane and a line 148
Meet of covectors 152
Meet of planes 163 167 182 185 196 272 295 419 423 445 448 504
Meet of two 2-D lines 8 146
Meet of two 3-D lines 147
Meet of two extensors 142
Meet, antisymmetry 143
Meet, associativity 143
Meet, geometric interpretation 143
Morphism see “Affine morphism” “Projective
Mosaic 23 255 258 259 311 548 550 551
Normal equations 473 522 524
Normalization 50
Normalization of line coordinates 471
Normalization of point coordinates 317
NullSpace 18 83 84 195 198 227 263 322 429 433 442 455 457 462 490 492 594
Numerical continuation 563
Obstacle detection 16 35 376 382
Optical axis 11 178 202 387 538 562
Optical center xxiii 11 18 22 23 35 44 78 84 178 180 182 185 195 199 201 227 252 253 260 277 286 303 363 387 411 424 440
Optical ray 11 19 20 23 47 78 84 104 178 183 186 191 195 199 213 262 264 293 445
Orbit 86 101 361 503
Orthogonal constraint 16
Orthogonal dual subspace 106 107
Orthogonal group 74 75
Orthogonal lines 166 205 243 298 390 398 463
Orthogonal matrix 14 18 43 57 69 77 115 193 212 231 284 373 404 455 517 519 547 552 555 572
Orthogonal pixel grid 14 55 58 208 215
Orthogonal planes 122 401 479
Orthogonal retinal axes 208
Orthogonal vectors 74 166 290 455 486 490 491 509 517 519 543
Orthographic projection 19 192 219 222 224 227—229 241 243 312
Outlier 329 330 332 334 335 338 357 482 497
Panoramic view see “Mosaic”
Para-perspective 220—227 229—232 241 536
parabola 55 351 353
Parallax effect 250 251
Parallel camera see “Affine camera”
Parallel projection 19 198 199 217 218 220 225 227 241 244 262 292—300 381 459 565
Parallelism xv 4 5 11 16 38 40 43 64 67 68 99 124 175 216 220 222 225 256 293 404
Part at infinity 93 94 99 117
Pencil of epipolar lines 106 266 267 269 289 294 311 367 368 433
Pencil of epipolar planes 106
Pencil of hyperplanes 67 68 69 106—109 111
Pencil of lines 106—108 110 146 266 442 444 447 474 476 477
Pencil of planes 106 107 109—111 116 147 275
Permutation 103 129 130
Permutation, signature 129
Perspective camera 19 198 226 255 516 518—521
Perspective distortion 11 177 193 194 205 232 241 243 375
Perspective projection 5 13 18 19 84 89 92 104 177 179 198 199 222 223 233 240 241 262 279 297 370 381 390 391 397 406 459 549
Perspective projection, matrix xxiii 13 20 21 30 31 41 58 85 180 182 184 191 197 219 238 242 262 263 280 285 296 302 420 421 441 462 486 503 507 509 510 513 514 519 521 522 525 529 548
Perspective transformation see “Perspective projection”
Pinhole camera 11 12 19 78 83 176—180 185 195 198 202 219 232 233 240 242 244 260 261 361 403 549
Pixel 14 19 180 183 195 200 208 210 212 215 233 234 255 259 262 264 276 317 339 346 349 368 379 386 396 407 415 440 482 495 505 521 572
Pixel coordinates 13 27 180 187 197 208 220 235 237 260 280 282 286 289 364 369 371 377 396 403 460 504 519 555 575 588
Pixel uncertainty 327 329 337 341 525 564
Plane at infinity see “Affine plane at infinity”
Pluecker coordinates 132 137 156 165 194 419
Pluecker coordinates of a 3-D line 135 164 165 195
Pluecker product 139 154 167 168 194 195 264 420
Pluecker relations 132 137—140 165
Pluecker relations, algebraically independent 138
Point at infinity see “Affine point at infinity”
Principal point 12 178 208 210 212 228 231 234 235 238 243 244 398 561 562 564 576 588—590
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