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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
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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: The Geometry of Multiple Images: The Laws That Govern the Formation of Multiple Images of a Scene and Some of Their Applications

: Faugeras O., Luong Q., Papadopoulo T.

:

This book formalizes and analyzes the relations between multiple views of a scene from the perspective of various types of geometries. A key feature is that it considers Euclidean and affine geometries as special cases of projective geometry.


: en

: /

:

ed2k: ed2k stats

: 1st edition

: 2001

: 644

: 16.11.2009

: | | ID
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$L_{\infty}$      see Affine line at infinity
$\mathbb{P}^{1}$      see Projective line
$\mathbb{P}^{2}$      see Projective plane
$\mathbb{P}^{3}$      see 3-D projective space
$\mathbb{P}^{n}$      see General projective space
$\mathbb{P}^{n}*$      see Dual projective space
$\mathcal{AG}_{n}$      see Affine group
$\mathcal{OG}(E)$      see Group of similarities
$\mathcal{O}(E)$      see Orthogonal group
$\mathcal{PLG}_{n}$      see Projective group
$\Pi_{\infty}$      see Affine plane at infinity Hyperplane
Adjoint matrix      xx 55 113 159 350
Affine basis      38 70 9597 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 516518 559 565567 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 516518
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 9698 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 1922 30 31 3739 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 4042 385386 390 395 398 405 538
Affine rigid displacement      4 41 42 55 75 124 207 208 238 260 503 529 530 532535 549 557559 562 563 570 574 583 584
Affine similarity      5 11 39 4244 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 380392 525529 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      295297 559560
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 $\mathbb{P}^{2}$      119 121
Angle of two lines of $\mathbb{P}^{3}$      14 121 213 397 398
Angle of two planes of $\mathbb{P}^{3}$      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 589591
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 521525 537 572 577 590
Camera obscura      178
Camera, calibration      56 57 236 239 243 582
Camera, calibration, classical      13 15 16 175 234239 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 394397 528 534 575 583
Canonical basis of $G_{1}(E_{3})$      159 160
Canonical basis of $G_{2}(E_{3})$      131 164
Canonical basis of $G_{2}(E_{4})$      131 134 153 154 165 169 171
Canonical basis of $G_{3}(E_{3})$      131
Canonical basis of $G_{3}(E_{4})$      131 136 163 169
Canonical basis of $G_{4}(E_{4})$      131
Canonical basis of $G_{k}(E)$      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 368370 389 390 513515 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 361362 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 161165 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 442444 465 584
Conic, absolute      xvii 39 44 55 57 69 115 117119 121 207 212 243 396 397 542 554 559 569 570 583585 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 7981 83 84 87 88 9193 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 151153 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 302307 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 103107 109112 120 121 124 213 266 369373 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 106114 116 118 128 146168 185 189 190 252 261 352 354 364 386 396 406 423 444 446 459
Eight-point algorithm      318 320 354
ellipse      55 232 350353 588
Ellipse of uncertainty      346349 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 332340 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 2830 35 44 48 106 260 261 266 268 276 288291 298 300 301 303 312 323 326 335 337 338 347349 354 355 367 369 376 378 379 392 406 412 415 418 422424 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 4345 49 52 53 57 60 260 262 264 265 269 274276 278 279 281 287 288 290293 297 306 308 311 320 322 325 335 337 346348 355 356 365 367 371 374 377 379 385 395 403 404 407 412 415 416 418 429431 440 444 448 449 452 454 462 490492 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 395396 399 401 405 406 468 572 574 575 578581 590
Euclidean space      see Affine space Euclidean
Euclidean stratum      361 393403 525529 560 568
Euclidean transformation      see Affine rigid displacement
Extensor      129 130132 134 136138 140144 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      516518
Factorization method, perspective      518521
Focal length      12 14 58 178 209 219 228 231 234 243 244 398 548 561 572 573 576 590 591
Focal plane      197 225227 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 262264 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 412415 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 316358 525
Fundamental matrix, parametrization      266 269 321325 483 489
Fundamental matrix, uncertainty      341346 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 3335 37 43 44 46 48 51 55 67 83 86 89 92 99101 108 111 120 124 159 192194 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 530532 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 372376
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 4143 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 571574 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 5960 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 553558 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 227229 241 243 312
Outlier      329 330 332 334 335 338 357 482 497
Panoramic view      see Mosaic
Para-perspective      220227 229232 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 292300 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 106109 111
Pencil of lines      106108 110 146 266 442 444 447 474 476 477
Pencil of planes      106 107 109111 116 147 275
Permutation      103 129 130
Permutation, signature      129
Perspective camera      19 198 226 255 516 518521
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 176180 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 137140 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 588590
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