2900:
2244:). In practice however, the 3D points may be represented in terms of coordinates relative to an arbitrary coordinate system (X1', X2', X3'). Assuming that the camera coordinate axes (X1, X2, X3) and the axes (X1', X2', X3') are of Euclidean type (orthogonal and isotropic), there is a unique Euclidean 3D transformation (rotation and translation) between the two coordinate systems. In other words, the camera is not necessarily at the origin looking along the
2736:
25:
3337:
1628:
1256:
1014:
2895:{\displaystyle \mathbf {y} \sim \mathbf {C} _{0}\,\mathbf {x} =\left({\begin{array}{c|c}\mathbf {I} &\mathbf {0} \end{array}}\right)\,\left({\begin{array}{c|c}\mathbf {R} &\mathbf {t} \\\hline \mathbf {0} &1\end{array}}\right)\mathbf {x} '=\left({\begin{array}{c|c}\mathbf {R} &\mathbf {t} \end{array}}\right)\,\mathbf {x} '}
2073:
3211:
2474:
is a 3-dimensional translation vector. When the first matrix is multiplied onto the homogeneous representation of a 3D point, the result is the homogeneous representation of the rotated point, and the second matrix performs instead a translation. Performing the two operations in sequence, i.e. first
1658:
The camera matrix derived here may appear trivial in the sense that it contains very few non-zero elements. This depends to a large extent on the particular coordinate systems which have been chosen for the 3D and 2D points. In practice, however, other forms of camera matrices are common, as will be
1442:
3132:
1047:
3679:
2662:
2978:, it assumes focal length = 1 and that image coordinates are measured in a coordinate system where the origin is located at the intersection between axis X3 and the image plane and has the same units as the 3D coordinate system. The resulting image coordinates are referred to as
807:
1430:
1937:
541:
2535:
2396:
2337:
2966:
3172:
This implies that the camera center (in its homogeneous representation) lies in the null space of the camera matrix, provided that it is represented in terms of 3D coordinates relative to the same coordinate system as the camera matrix refers to.
3332:{\displaystyle \mathbf {C} _{N}=\mathbf {R} \,\left({\begin{array}{c|c}\mathbf {I} &\mathbf {R} ^{-1}\,\mathbf {t} \end{array}}\right)=\mathbf {R} \,\left({\begin{array}{c|c}\mathbf {I} &-{\tilde {\mathbf {n} }}\end{array}}\right)}
1623:{\displaystyle \mathbf {C} ={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&{\frac {1}{f}}&0\end{pmatrix}}\sim {\begin{pmatrix}f&0&0&0\\0&f&0&0\\0&0&1&0\end{pmatrix}}}
3593:
1776:. This means that the camera center (and only this point) cannot be mapped to a point in the image plane by the camera (or equivalently, it maps to all points on the image as every ray on the image goes through this point).
1251:{\displaystyle {\begin{pmatrix}y_{1}\\y_{2}\\1\end{pmatrix}}\sim {\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&{\frac {1}{f}}&0\end{pmatrix}}\,{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\1\end{pmatrix}}}
3025:
3604:
2590:
1759:
1009:{\displaystyle {\begin{pmatrix}y_{1}\\y_{2}\\1\end{pmatrix}}={\begin{pmatrix}{\frac {f}{x_{3}}}x_{1}\\{\frac {f}{x_{3}}}x_{2}\\1\end{pmatrix}}\sim {\begin{pmatrix}x_{1}\\x_{2}\\{\frac {x_{3}}{f}}\end{pmatrix}}}
3528:
1325:
3137:
This is also, again, the coordinates of the camera center, now relative to the (X1',X2',X3') system. This can be seen by applying first the rotation and then the translation to the 3-dimensional vector
1846:
1292:
376:
254:
2068:{\displaystyle \mathbf {C} _{0}={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\end{pmatrix}}=\left({\begin{array}{c|c}\mathbf {I} &\mathbf {0} \end{array}}\right)}
1909:
429:
778:
2481:
2342:
2283:
407:, it too can be regarded as a projective element. This means that it has only 11 degrees of freedom since any multiplication by a non-zero scalar results in an equivalent camera matrix.
3371:
3167:
603:
3484:
2692:
2911:
3203:
3017:
2728:
2227:
718:
653:
3457:
3435:
2582:
2560:
2472:
2421:
2172:
2098:
1687:
1653:
1317:
1039:
401:
279:
215:
189:
3413:
2447:
2275:
2198:
2150:
2124:
157:
1810:
799:
330:
300:
2475:
the rotation and then the translation (with translation vector given in the already rotated coordinate system), gives a combined rotation and translation matrix
3127:{\displaystyle \mathbf {n} ={\begin{pmatrix}-\mathbf {R} ^{-1}\,\mathbf {t} \\1\end{pmatrix}}={\begin{pmatrix}{\tilde {\mathbf {n} }}\\1\end{pmatrix}}}
3381:
Given the mapping produced by a normalized camera matrix, the resulting normalized image coordinates can be transformed by means of an arbitrary 2D
3674:{\displaystyle \mathbf {C} =\mathbf {H} \,\mathbf {C} _{N}=\mathbf {H} \,\left({\begin{array}{c|c}\mathbf {R} &\mathbf {t} \end{array}}\right)}
3539:
2657:{\displaystyle \mathbf {x} =\left({\begin{array}{c|c}\mathbf {R} &\mathbf {t} \\\hline \mathbf {0} &1\end{array}}\right)\mathbf {x} '}
3736:
1699:
2584:
are precisely the rotation and translations which relate the two coordinate system (X1,X2,X3) and (X1',X2',X3') above, this implies that
1425:{\displaystyle \mathbf {C} ={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&{\frac {1}{f}}&0\end{pmatrix}}}
3717:
108:
1815:
1764:
This is also the homogeneous representation of the 3D point which has coordinates (0,0,0), that is, the "camera center" (aka the
1261:
338:
223:
217:
be a representation of the image of this point in the pinhole camera (a 3-dimensional vector). Then the following relation holds
3492:
674:
To derive the camera matrix, the expression above is rewritten in terms of homogeneous coordinates. Instead of the 2D vector
46:
42:
2240:
coordinate system, that is, a coordinate system which has its origin at the camera center (the location of the pinhole of a
1851:
89:
61:
3382:
2803:
2607:
2490:
2351:
2292:
536:{\displaystyle {\begin{pmatrix}y_{1}\\y_{2}\end{pmatrix}}={\frac {f}{x_{3}}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}}
68:
2905:
Consequently, the camera matrix which relates points in the coordinate system (X1',X2',X3') to image coordinates is
2530:{\displaystyle \left({\begin{array}{c|c}\mathbf {R} &\mathbf {t} \\\hline \mathbf {0} &1\end{array}}\right)}
2391:{\displaystyle \left({\begin{array}{c|c}\mathbf {I} &\mathbf {t} \\\hline \mathbf {0} &1\end{array}}\right)}
2332:{\displaystyle \left({\begin{array}{c|c}\mathbf {R} &\mathbf {0} \\\hline \mathbf {0} &1\end{array}}\right)}
3386:
723:
35:
3385:. This includes 2D translations and rotations as well as scaling (isotropic and anisotropic) but also general 2D
75:
3345:
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192:
2961:{\displaystyle \mathbf {C} _{N}=\left({\begin{array}{c|c}\mathbf {R} &\mathbf {t} \end{array}}\right)}
57:
3533:
Inserting the above expression for the normalized image coordinates in terms of the 3D coordinates gives
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3294:
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3179:
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2935:
2858:
2774:
2704:
2203:
2042:
2730:, the mapping from the coordinates in the (X1,X2,X3) system to homogeneous image coordinates becomes
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801:. First, we write the homogeneous image coordinates as expressions in the usual 3D coordinates.
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The two operations of rotation and translation of 3D coordinates can be represented as the two
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to the 2D image coordinates of the point's projection onto the image plane, according to the
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82:
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and instead of equality we consider equality up to scaling by a non-zero number, denoted
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1927:
The camera matrix derived above can be simplified even further if we assume that
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a concatenation of a 3D rotation matrix and a 3-dimensional translation vector.
24:
1019:
Finally, also the 3D coordinates are expressed in a homogeneous representation
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3169:
and the result is the homogeneous representation of 3D coordinates (0,0,0).
3588:{\displaystyle \mathbf {y} '=\mathbf {H} \,\mathbf {C} _{N}\,\mathbf {x} '}
3373:
is the 3D coordinates of the camera relative to the (X1',X2',X3') system.
1754:{\displaystyle \mathbf {n} ={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}}
1915:
image plane (even though, if the image plane is taken to be a
18:
2236:
So far all points in the 3D world have been represented in a
1841:{\displaystyle \mathbf {y} \sim \mathbf {C} \,\mathbf {x} }
1287:{\displaystyle \mathbf {y} \sim \mathbf {C} \,\mathbf {x} }
371:{\displaystyle \mathbf {y} =k\,\mathbf {C} \,\mathbf {x} .}
249:{\displaystyle \mathbf {y} \sim \mathbf {C} \,\mathbf {x} }
302:
sign implies that the left and right hand sides are equal
1923:
Normalized camera matrix and normalized image coordinates
3523:{\displaystyle \mathbf {y} '=\mathbf {H} \,\mathbf {y} }
3437:
which maps the homogeneous normalized image coordinates
3019:
described above, is spanned by the 4-dimensional vector
2990:
Again, the null space of the normalized camera matrix,
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1904:{\displaystyle \mathbf {y} =(y_{1}\,y_{2}\,0)^{\top }}
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from 3D points in the world to 2D points in an image.
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This produces the most general form of camera matrix
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matrix and a 3-dimensional vector. The camera matrix
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403:is involved in the mapping between elements of two
49:. Unsourced material may be challenged and removed.
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3459:to the homogeneous transformed image coordinates
2701:Assuming also that the camera matrix is given by
1911:. This corresponds to a point at infinity in the
720:we consider the projective element (a 3D vector)
659:is the camera's focal length for which we assume
609:relative to a camera centered coordinate system,
3389:. Such a transformation can be represented as a
1436:and the corresponding camera matrix now becomes
2974:This type of camera matrix is referred to as a
2694:is the homogeneous representation of the point
1919:, no corresponding intersection point exists).
415:The mapping from the coordinates of a 3D point
3708:Richard Hartley and Andrew Zisserman (2003).
1319:is the camera matrix, which here is given by
8:
773:{\displaystyle \mathbf {y} =(y_{1},y_{2},1)}
1041:and this is how the camera matrix appears:
2174:here is divided into a concatenation of a
306:for a multiplication by a non-zero scalar
3710:Multiple View Geometry in computer vision
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663:> 0. Furthermore, we also assume that
655:are the resulting image coordinates, and
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109:Learn how and when to remove this message
2698:in the coordinate system (X1',X2',X3').
3366:{\displaystyle {\tilde {\mathbf {n} }}}
3162:{\displaystyle {\tilde {\mathbf {n} }}}
1689:derived in the previous section has a
191:be a representation of a 3D point in
7:
47:adding citations to reliable sources
1768:; the position of the pinhole of a
1655:itself being a projective element.
598:{\displaystyle (x_{1},x_{2},x_{3})}
1896:
1633:The last step is a consequence of
195:(a 4-dimensional vector), and let
14:
1848:is well-defined and has the form
162:which describes the mapping of a
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3198:{\displaystyle \mathbf {C} _{N}}
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3012:{\displaystyle \mathbf {C} _{N}}
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2723:{\displaystyle \mathbf {C} _{0}}
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2222:{\displaystyle \mathbf {C} _{0}}
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23:
16:Computer vision geometry concept
1693:which is spanned by the vector
34:needs additional citations for
3712:. Cambridge University Press.
3357:
3315:
3153:
3103:
2229:is sometimes referred to as a
1892:
1863:
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735:
707:
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642:
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553:
1:
3479:{\displaystyle \mathbf {y} '}
3176:The normalized camera matrix
2687:{\displaystyle \mathbf {x} '}
713:{\displaystyle (y_{1},y_{2})}
648:{\displaystyle (y_{1},y_{2})}
281:is the camera matrix and the
3452:{\displaystyle \mathbf {y} }
3430:{\displaystyle \mathbf {H} }
2980:normalized image coordinates
2577:{\displaystyle \mathbf {t} }
2555:{\displaystyle \mathbf {R} }
2467:{\displaystyle \mathbf {t} }
2416:{\displaystyle \mathbf {R} }
2167:{\displaystyle \mathbf {C} }
2093:{\displaystyle \mathbf {I} }
1779:For any other 3D point with
1682:{\displaystyle \mathbf {C} }
1648:{\displaystyle \mathbf {C} }
1312:{\displaystyle \mathbf {C} }
1034:{\displaystyle \mathbf {x} }
396:{\displaystyle \mathbf {C} }
274:{\displaystyle \mathbf {C} }
210:{\displaystyle \mathbf {y} }
184:{\displaystyle \mathbf {x} }
3737:Geometry in computer vision
3387:perspective transformations
2126:identity matrix. Note that
3753:
605:are the 3D coordinates of
131:(camera) projection matrix
3408:{\displaystyle 3\times 3}
2442:{\displaystyle 3\times 3}
2270:{\displaystyle 4\times 4}
2193:{\displaystyle 3\times 3}
2145:{\displaystyle 3\times 4}
2119:{\displaystyle 3\times 3}
381:Since the camera matrix
152:{\displaystyle 3\times 4}
2976:normalized camera matrix
1805:{\displaystyle x_{3}=0}
794:{\displaystyle \,\sim }
325:{\displaystyle k\neq 0}
295:{\displaystyle \,\sim }
193:homogeneous coordinates
3675:
3589:
3524:
3480:
3453:
3431:
3409:
3367:
3333:
3205:can now be written as
3199:
3163:
3128:
3013:
2962:
2896:
2724:
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2658:
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2556:
2531:
2468:
2443:
2417:
2392:
2333:
2271:
2223:
2194:
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2146:
2120:
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2069:
1905:
1842:
1806:
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649:
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372:
326:
296:
275:
250:
211:
185:
153:
3676:
3590:
3525:
3481:
3454:
3432:
3410:
3377:General camera matrix
3368:
3334:
3200:
3164:
3129:
3014:
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2725:
2689:
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421:pinhole camera model
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339:
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224:
199:
173:
137:
43:improve this article
3696:Camera resectioning
2986:The camera position
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1667:The camera matrix
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149:
3360:
3318:
3156:
3106:
1524:
1407:
1258: or
1171:
996:
915:
884:
491:
405:projective spaces
119:
118:
111:
93:
3744:
3723:
3680:
3678:
3677:
3672:
3670:
3666:
3662:
3655:
3641:
3633:
3632:
3627:
3620:
3612:
3594:
3592:
3591:
3586:
3584:
3580:
3573:
3572:
3567:
3560:
3552:
3548:
3529:
3527:
3526:
3521:
3519:
3513:
3505:
3501:
3485:
3483:
3482:
3477:
3475:
3471:
3458:
3456:
3455:
3450:
3448:
3436:
3434:
3433:
3428:
3426:
3414:
3412:
3411:
3406:
3372:
3370:
3369:
3364:
3362:
3361:
3356:
3351:
3338:
3336:
3335:
3330:
3328:
3324:
3320:
3319:
3314:
3309:
3301:
3287:
3279:
3275:
3271:
3265:
3264:
3256:
3248:
3234:
3226:
3225:
3220:
3204:
3202:
3201:
3196:
3194:
3193:
3188:
3168:
3166:
3165:
3160:
3158:
3157:
3152:
3147:
3133:
3131:
3130:
3125:
3123:
3122:
3108:
3107:
3102:
3097:
3083:
3082:
3068:
3062:
3061:
3053:
3033:
3018:
3016:
3015:
3010:
3008:
3007:
3002:
2967:
2965:
2964:
2959:
2957:
2953:
2949:
2942:
2926:
2925:
2920:
2901:
2899:
2898:
2893:
2891:
2887:
2880:
2876:
2872:
2865:
2849:
2845:
2839:
2835:
2826:
2817:
2810:
2796:
2792:
2788:
2781:
2765:
2759:
2758:
2753:
2744:
2729:
2727:
2726:
2721:
2719:
2718:
2713:
2693:
2691:
2690:
2685:
2683:
2679:
2663:
2661:
2660:
2655:
2653:
2649:
2643:
2639:
2630:
2621:
2614:
2598:
2583:
2581:
2580:
2575:
2573:
2561:
2559:
2558:
2553:
2551:
2536:
2534:
2533:
2528:
2526:
2522:
2513:
2504:
2497:
2473:
2471:
2470:
2465:
2463:
2448:
2446:
2445:
2440:
2422:
2420:
2419:
2414:
2412:
2397:
2395:
2394:
2389:
2387:
2383:
2374:
2365:
2358:
2338:
2336:
2335:
2330:
2328:
2324:
2315:
2306:
2299:
2276:
2274:
2273:
2268:
2228:
2226:
2225:
2220:
2218:
2217:
2212:
2199:
2197:
2196:
2191:
2173:
2171:
2170:
2165:
2163:
2151:
2149:
2148:
2143:
2125:
2123:
2122:
2117:
2099:
2097:
2096:
2091:
2089:
2074:
2072:
2071:
2066:
2064:
2060:
2056:
2049:
2033:
2032:
1952:
1951:
1946:
1910:
1908:
1907:
1902:
1900:
1899:
1886:
1885:
1875:
1874:
1859:
1847:
1845:
1844:
1839:
1837:
1831:
1823:
1811:
1809:
1808:
1803:
1795:
1794:
1760:
1758:
1757:
1752:
1750:
1749:
1707:
1688:
1686:
1685:
1680:
1678:
1654:
1652:
1651:
1646:
1644:
1629:
1627:
1626:
1621:
1619:
1618:
1538:
1537:
1525:
1517:
1450:
1431:
1429:
1428:
1423:
1421:
1420:
1408:
1400:
1333:
1318:
1316:
1315:
1310:
1308:
1293:
1291:
1290:
1285:
1283:
1277:
1269:
1257:
1255:
1254:
1249:
1247:
1246:
1232:
1231:
1218:
1217:
1204:
1203:
1185:
1184:
1172:
1164:
1097:
1096:
1082:
1081:
1068:
1067:
1040:
1038:
1037:
1032:
1030:
1015:
1013:
1012:
1007:
1005:
1004:
997:
992:
991:
982:
976:
975:
962:
961:
941:
940:
926:
925:
916:
914:
913:
901:
895:
894:
885:
883:
882:
870:
857:
856:
842:
841:
828:
827:
800:
798:
797:
792:
779:
777:
776:
771:
760:
759:
747:
746:
731:
719:
717:
716:
711:
706:
705:
693:
692:
654:
652:
651:
646:
641:
640:
628:
627:
604:
602:
601:
596:
591:
590:
578:
577:
565:
564:
542:
540:
539:
534:
532:
531:
524:
523:
510:
509:
492:
490:
489:
477:
472:
471:
464:
463:
450:
449:
402:
400:
399:
394:
392:
377:
375:
374:
369:
364:
358:
346:
331:
329:
328:
323:
301:
299:
298:
293:
280:
278:
277:
272:
270:
255:
253:
252:
247:
245:
239:
231:
216:
214:
213:
208:
206:
190:
188:
187:
182:
180:
158:
156:
155:
150:
114:
107:
103:
100:
94:
92:
51:
27:
19:
3752:
3751:
3747:
3746:
3745:
3743:
3742:
3741:
3727:
3726:
3720:
3707:
3704:
3687:
3664:
3663:
3656:
3643:
3622:
3603:
3602:
3575:
3562:
3543:
3538:
3537:
3496:
3491:
3490:
3466:
3461:
3460:
3439:
3438:
3417:
3416:
3391:
3390:
3379:
3344:
3343:
3322:
3321:
3302:
3289:
3273:
3272:
3251:
3249:
3236:
3215:
3210:
3209:
3183:
3178:
3177:
3140:
3139:
3117:
3116:
3110:
3109:
3088:
3077:
3076:
3070:
3069:
3048:
3038:
3024:
3023:
2997:
2992:
2991:
2988:
2951:
2950:
2943:
2930:
2915:
2910:
2909:
2882:
2874:
2873:
2866:
2853:
2840:
2833:
2832:
2827:
2819:
2818:
2811:
2798:
2790:
2789:
2782:
2769:
2748:
2735:
2734:
2708:
2703:
2702:
2674:
2669:
2668:
2644:
2637:
2636:
2631:
2623:
2622:
2615:
2602:
2589:
2588:
2564:
2563:
2542:
2541:
2520:
2519:
2514:
2506:
2505:
2498:
2485:
2480:
2479:
2454:
2453:
2450:rotation matrix
2425:
2424:
2403:
2402:
2381:
2380:
2375:
2367:
2366:
2359:
2346:
2341:
2340:
2322:
2321:
2316:
2308:
2307:
2300:
2287:
2282:
2281:
2253:
2252:
2238:camera centered
2207:
2202:
2201:
2176:
2175:
2154:
2153:
2128:
2127:
2102:
2101:
2100:here denotes a
2080:
2079:
2058:
2057:
2050:
2037:
2027:
2026:
2021:
2016:
2011:
2005:
2004:
1999:
1994:
1989:
1983:
1982:
1977:
1972:
1967:
1957:
1941:
1936:
1935:
1925:
1917:Euclidean plane
1891:
1877:
1866:
1850:
1849:
1814:
1813:
1786:
1781:
1780:
1744:
1743:
1737:
1736:
1730:
1729:
1723:
1722:
1712:
1698:
1697:
1669:
1668:
1665:
1663:Camera position
1635:
1634:
1613:
1612:
1607:
1602:
1597:
1591:
1590:
1585:
1580:
1575:
1569:
1568:
1563:
1558:
1553:
1543:
1532:
1531:
1526:
1514:
1509:
1503:
1502:
1497:
1492:
1487:
1481:
1480:
1475:
1470:
1465:
1455:
1441:
1440:
1415:
1414:
1409:
1397:
1392:
1386:
1385:
1380:
1375:
1370:
1364:
1363:
1358:
1353:
1348:
1338:
1324:
1323:
1299:
1298:
1260:
1259:
1241:
1240:
1234:
1233:
1223:
1220:
1219:
1209:
1206:
1205:
1195:
1188:
1179:
1178:
1173:
1161:
1156:
1150:
1149:
1144:
1139:
1134:
1128:
1127:
1122:
1117:
1112:
1102:
1091:
1090:
1084:
1083:
1073:
1070:
1069:
1059:
1052:
1046:
1045:
1021:
1020:
999:
998:
983:
978:
977:
967:
964:
963:
953:
946:
935:
934:
928:
927:
917:
905:
897:
896:
886:
874:
862:
851:
850:
844:
843:
833:
830:
829:
819:
812:
806:
805:
782:
781:
751:
738:
722:
721:
697:
684:
676:
675:
668:
632:
619:
611:
610:
582:
569:
556:
548:
547:
526:
525:
515:
512:
511:
501:
494:
481:
466:
465:
455:
452:
451:
441:
434:
428:
427:
413:
383:
382:
337:
336:
308:
307:
283:
282:
261:
260:
222:
221:
197:
196:
171:
170:
135:
134:
123:computer vision
115:
104:
98:
95:
58:"Camera matrix"
52:
50:
40:
28:
17:
12:
11:
5:
3750:
3748:
3740:
3739:
3729:
3728:
3725:
3724:
3718:
3703:
3700:
3699:
3698:
3693:
3686:
3683:
3682:
3681:
3669:
3661:
3657:
3654:
3650:
3649:
3646:
3640:
3636:
3631:
3626:
3619:
3615:
3611:
3596:
3595:
3583:
3579:
3571:
3566:
3559:
3555:
3551:
3547:
3531:
3530:
3518:
3512:
3508:
3504:
3500:
3474:
3470:
3447:
3425:
3404:
3401:
3398:
3378:
3375:
3359:
3355:
3340:
3339:
3327:
3317:
3313:
3306:
3303:
3300:
3296:
3295:
3292:
3286:
3282:
3278:
3270:
3263:
3260:
3255:
3250:
3247:
3243:
3242:
3239:
3233:
3229:
3224:
3219:
3192:
3187:
3155:
3151:
3135:
3134:
3121:
3115:
3112:
3111:
3105:
3101:
3094:
3093:
3091:
3086:
3081:
3075:
3072:
3071:
3067:
3060:
3057:
3052:
3047:
3044:
3043:
3041:
3036:
3032:
3006:
3001:
2987:
2984:
2969:
2968:
2956:
2948:
2944:
2941:
2937:
2936:
2933:
2929:
2924:
2919:
2903:
2902:
2890:
2886:
2879:
2871:
2867:
2864:
2860:
2859:
2856:
2852:
2848:
2844:
2838:
2831:
2828:
2825:
2821:
2820:
2816:
2812:
2809:
2805:
2804:
2801:
2795:
2787:
2783:
2780:
2776:
2775:
2772:
2768:
2764:
2757:
2752:
2747:
2743:
2717:
2712:
2682:
2678:
2665:
2664:
2652:
2648:
2642:
2635:
2632:
2629:
2625:
2624:
2620:
2616:
2613:
2609:
2608:
2605:
2601:
2597:
2572:
2550:
2540:Assuming that
2538:
2537:
2525:
2518:
2515:
2512:
2508:
2507:
2503:
2499:
2496:
2492:
2491:
2488:
2462:
2438:
2435:
2432:
2411:
2399:
2398:
2386:
2379:
2376:
2373:
2369:
2368:
2364:
2360:
2357:
2353:
2352:
2349:
2327:
2320:
2317:
2314:
2310:
2309:
2305:
2301:
2298:
2294:
2293:
2290:
2266:
2263:
2260:
2242:pinhole camera
2231:canonical form
2216:
2211:
2189:
2186:
2183:
2162:
2141:
2138:
2135:
2115:
2112:
2109:
2088:
2076:
2075:
2063:
2055:
2051:
2048:
2044:
2043:
2040:
2036:
2031:
2025:
2022:
2020:
2017:
2015:
2012:
2010:
2007:
2006:
2003:
2000:
1998:
1995:
1993:
1990:
1988:
1985:
1984:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1962:
1960:
1955:
1950:
1945:
1924:
1921:
1898:
1894:
1890:
1884:
1880:
1873:
1869:
1865:
1862:
1858:
1836:
1830:
1826:
1822:
1801:
1798:
1793:
1789:
1770:pinhole camera
1766:entrance pupil
1762:
1761:
1748:
1742:
1739:
1738:
1735:
1732:
1731:
1728:
1725:
1724:
1721:
1718:
1717:
1715:
1710:
1706:
1677:
1664:
1661:
1643:
1631:
1630:
1617:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1592:
1589:
1586:
1584:
1581:
1579:
1576:
1574:
1571:
1570:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1548:
1546:
1541:
1536:
1530:
1527:
1523:
1520:
1515:
1513:
1510:
1508:
1505:
1504:
1501:
1498:
1496:
1493:
1491:
1488:
1486:
1483:
1482:
1479:
1476:
1474:
1471:
1469:
1466:
1464:
1461:
1460:
1458:
1453:
1449:
1434:
1433:
1419:
1413:
1410:
1406:
1403:
1398:
1396:
1393:
1391:
1388:
1387:
1384:
1381:
1379:
1376:
1374:
1371:
1369:
1366:
1365:
1362:
1359:
1357:
1354:
1352:
1349:
1347:
1344:
1343:
1341:
1336:
1332:
1307:
1295:
1294:
1282:
1276:
1272:
1268:
1245:
1239:
1236:
1235:
1230:
1226:
1222:
1221:
1216:
1212:
1208:
1207:
1202:
1198:
1194:
1193:
1191:
1183:
1177:
1174:
1170:
1167:
1162:
1160:
1157:
1155:
1152:
1151:
1148:
1145:
1143:
1140:
1138:
1135:
1133:
1130:
1129:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1107:
1105:
1100:
1095:
1089:
1086:
1085:
1080:
1076:
1072:
1071:
1066:
1062:
1058:
1057:
1055:
1029:
1017:
1016:
1003:
995:
990:
986:
980:
979:
974:
970:
966:
965:
960:
956:
952:
951:
949:
944:
939:
933:
930:
929:
924:
920:
912:
908:
904:
899:
898:
893:
889:
881:
877:
873:
868:
867:
865:
860:
855:
849:
846:
845:
840:
836:
832:
831:
826:
822:
818:
817:
815:
790:
769:
766:
763:
758:
754:
750:
745:
741:
737:
734:
730:
709:
704:
700:
696:
691:
687:
683:
666:
644:
639:
635:
631:
626:
622:
618:
594:
589:
585:
581:
576:
572:
568:
563:
559:
555:
544:
543:
530:
522:
518:
514:
513:
508:
504:
500:
499:
497:
488:
484:
480:
475:
470:
462:
458:
454:
453:
448:
444:
440:
439:
437:
423:, is given by
412:
409:
391:
379:
378:
367:
363:
357:
352:
349:
345:
321:
318:
315:
291:
269:
257:
256:
244:
238:
234:
230:
205:
179:
164:pinhole camera
148:
145:
142:
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
3749:
3738:
3735:
3734:
3732:
3721:
3719:0-521-54051-8
3715:
3711:
3706:
3705:
3701:
3697:
3694:
3692:
3691:3D projection
3689:
3688:
3684:
3667:
3644:
3634:
3629:
3613:
3601:
3600:
3599:
3581:
3569:
3553:
3549:
3536:
3535:
3534:
3506:
3502:
3489:
3488:
3487:
3472:
3402:
3399:
3396:
3388:
3384:
3376:
3374:
3325:
3304:
3290:
3280:
3276:
3261:
3258:
3237:
3227:
3222:
3208:
3207:
3206:
3190:
3174:
3170:
3119:
3113:
3089:
3084:
3079:
3073:
3058:
3055:
3045:
3039:
3034:
3022:
3021:
3020:
3004:
2985:
2983:
2981:
2977:
2972:
2954:
2931:
2927:
2922:
2908:
2907:
2906:
2888:
2877:
2854:
2850:
2846:
2836:
2829:
2799:
2793:
2770:
2766:
2755:
2745:
2733:
2732:
2731:
2715:
2699:
2697:
2680:
2650:
2640:
2633:
2603:
2599:
2587:
2586:
2585:
2523:
2516:
2486:
2478:
2477:
2476:
2451:
2436:
2433:
2430:
2384:
2377:
2347:
2325:
2318:
2288:
2280:
2279:
2278:
2264:
2261:
2258:
2249:
2247:
2243:
2239:
2234:
2232:
2214:
2187:
2184:
2181:
2139:
2136:
2133:
2113:
2110:
2107:
2061:
2038:
2034:
2029:
2023:
2018:
2013:
2008:
2001:
1996:
1991:
1986:
1979:
1974:
1969:
1964:
1958:
1953:
1948:
1934:
1933:
1932:
1930:
1922:
1920:
1918:
1914:
1888:
1882:
1878:
1871:
1867:
1860:
1824:
1812:, the result
1799:
1796:
1791:
1787:
1777:
1775:
1771:
1767:
1746:
1740:
1733:
1726:
1719:
1713:
1708:
1696:
1695:
1694:
1692:
1662:
1660:
1659:shown below.
1656:
1615:
1609:
1604:
1599:
1594:
1587:
1582:
1577:
1572:
1565:
1560:
1555:
1550:
1544:
1539:
1534:
1528:
1521:
1518:
1511:
1506:
1499:
1494:
1489:
1484:
1477:
1472:
1467:
1462:
1456:
1451:
1439:
1438:
1437:
1417:
1411:
1404:
1401:
1394:
1389:
1382:
1377:
1372:
1367:
1360:
1355:
1350:
1345:
1339:
1334:
1322:
1321:
1320:
1270:
1243:
1237:
1228:
1224:
1214:
1210:
1200:
1196:
1189:
1181:
1175:
1168:
1165:
1158:
1153:
1146:
1141:
1136:
1131:
1124:
1119:
1114:
1109:
1103:
1098:
1093:
1087:
1078:
1074:
1064:
1060:
1053:
1044:
1043:
1042:
1001:
993:
988:
984:
972:
968:
958:
954:
947:
942:
937:
931:
922:
918:
910:
906:
902:
891:
887:
879:
875:
871:
863:
858:
853:
847:
838:
834:
824:
820:
813:
804:
803:
802:
788:
764:
761:
756:
752:
748:
743:
739:
732:
702:
698:
694:
689:
685:
672:
670:
662:
658:
637:
633:
629:
624:
620:
608:
587:
583:
579:
574:
570:
566:
561:
557:
528:
520:
516:
506:
502:
495:
486:
482:
478:
473:
468:
460:
456:
446:
442:
435:
426:
425:
424:
422:
418:
410:
408:
406:
365:
350:
347:
335:
334:
333:
319:
316:
313:
305:
289:
232:
220:
219:
218:
194:
167:
165:
161:
146:
143:
140:
132:
128:
127:camera matrix
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3709:
3597:
3532:
3380:
3341:
3175:
3171:
3136:
2989:
2979:
2975:
2973:
2970:
2904:
2700:
2695:
2666:
2539:
2400:
2250:
2245:
2237:
2235:
2230:
2077:
1928:
1926:
1778:
1773:
1763:
1666:
1657:
1632:
1435:
1296:
1018:
673:
664:
660:
656:
606:
545:
416:
414:
380:
258:
168:
130:
126:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
3702:References
3383:homography
1913:projective
1691:null space
411:Derivation
69:newspapers
3400:×
3358:~
3316:~
3305:−
3259:−
3154:~
3104:~
3056:−
3046:−
2746:∼
2434:×
2277:matrices
2262:×
2185:×
2137:×
2111:×
1897:⊤
1825:∼
1540:∼
1271:∼
1099:∼
943:∼
789:∼
317:≠
290:∼
233:∼
144:×
99:July 2010
3731:Category
3685:See also
3582:′
3550:′
3503:′
3473:′
2889:′
2847:′
2681:′
2651:′
1772:) is at
3415:matrix
2152:matrix
83:scholar
3716:
3342:where
2667:where
2401:where
2248:axis.
2078:where
1297:where
669:> 0
546:where
304:except
259:where
160:matrix
85:
78:
71:
64:
56:
2423:is a
1929:f = 1
133:is a
90:JSTOR
76:books
3714:ISBN
2562:and
2452:and
2339:and
169:Let
62:news
129:or
121:In
45:by
3733::
3486::
2982:.
2233:.
1931::
671:.
332::
125:a
3722:.
3668:)
3660:t
3653:R
3645:(
3639:H
3635:=
3630:N
3625:C
3618:H
3614:=
3610:C
3578:x
3570:N
3565:C
3558:H
3554:=
3546:y
3517:y
3511:H
3507:=
3499:y
3469:y
3446:y
3424:H
3403:3
3397:3
3354:n
3326:)
3312:n
3299:I
3291:(
3285:R
3281:=
3277:)
3269:t
3262:1
3254:R
3246:I
3238:(
3232:R
3228:=
3223:N
3218:C
3191:N
3186:C
3150:n
3120:)
3114:1
3100:n
3090:(
3085:=
3080:)
3074:1
3066:t
3059:1
3051:R
3040:(
3035:=
3031:n
3005:N
3000:C
2955:)
2947:t
2940:R
2932:(
2928:=
2923:N
2918:C
2885:x
2878:)
2870:t
2863:R
2855:(
2851:=
2843:x
2837:)
2830:1
2824:0
2815:t
2808:R
2800:(
2794:)
2786:0
2779:I
2771:(
2767:=
2763:x
2756:0
2751:C
2742:y
2716:0
2711:C
2696:P
2677:x
2647:x
2641:)
2634:1
2628:0
2619:t
2612:R
2604:(
2600:=
2596:x
2571:t
2549:R
2524:)
2517:1
2511:0
2502:t
2495:R
2487:(
2461:t
2437:3
2431:3
2410:R
2385:)
2378:1
2372:0
2363:t
2356:I
2348:(
2326:)
2319:1
2313:0
2304:0
2297:R
2289:(
2265:4
2259:4
2246:z
2215:0
2210:C
2188:3
2182:3
2161:C
2140:4
2134:3
2114:3
2108:3
2087:I
2062:)
2054:0
2047:I
2039:(
2035:=
2030:)
2024:0
2019:1
2014:0
2009:0
2002:0
1997:0
1992:1
1987:0
1980:0
1975:0
1970:0
1965:1
1959:(
1954:=
1949:0
1944:C
1893:)
1889:0
1883:2
1879:y
1872:1
1868:y
1864:(
1861:=
1857:y
1835:x
1829:C
1821:y
1800:0
1797:=
1792:3
1788:x
1774:O
1747:)
1741:1
1734:0
1727:0
1720:0
1714:(
1709:=
1705:n
1676:C
1642:C
1616:)
1610:0
1605:1
1600:0
1595:0
1588:0
1583:0
1578:f
1573:0
1566:0
1561:0
1556:0
1551:f
1545:(
1535:)
1529:0
1522:f
1519:1
1512:0
1507:0
1500:0
1495:0
1490:1
1485:0
1478:0
1473:0
1468:0
1463:1
1457:(
1452:=
1448:C
1432:,
1418:)
1412:0
1405:f
1402:1
1395:0
1390:0
1383:0
1378:0
1373:1
1368:0
1361:0
1356:0
1351:0
1346:1
1340:(
1335:=
1331:C
1306:C
1281:x
1275:C
1267:y
1244:)
1238:1
1229:3
1225:x
1215:2
1211:x
1201:1
1197:x
1190:(
1182:)
1176:0
1169:f
1166:1
1159:0
1154:0
1147:0
1142:0
1137:1
1132:0
1125:0
1120:0
1115:0
1110:1
1104:(
1094:)
1088:1
1079:2
1075:y
1065:1
1061:y
1054:(
1028:x
1002:)
994:f
989:3
985:x
973:2
969:x
959:1
955:x
948:(
938:)
932:1
923:2
919:x
911:3
907:x
903:f
892:1
888:x
880:3
876:x
872:f
864:(
859:=
854:)
848:1
839:2
835:y
825:1
821:y
814:(
768:)
765:1
762:,
757:2
753:y
749:,
744:1
740:y
736:(
733:=
729:y
708:)
703:2
699:y
695:,
690:1
686:y
682:(
667:3
665:x
661:f
657:f
643:)
638:2
634:y
630:,
625:1
621:y
617:(
607:P
593:)
588:3
584:x
580:,
575:2
571:x
567:,
562:1
558:x
554:(
529:)
521:2
517:x
507:1
503:x
496:(
487:3
483:x
479:f
474:=
469:)
461:2
457:y
447:1
443:y
436:(
417:P
390:C
366:.
362:x
356:C
351:k
348:=
344:y
320:0
314:k
268:C
243:x
237:C
229:y
204:y
178:x
147:4
141:3
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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