2327:
2784:
1477:
2720:
2016:
2393:
514:
2322:{\displaystyle {\begin{aligned}x_{1}&={\textbf {H}}I={\begin{bmatrix}h_{1}&h_{2}&h_{3}\end{bmatrix}}{\begin{bmatrix}1\\j\\0\end{bmatrix}}=h_{1}+jh_{2}\\x_{2}&={\textbf {H}}J={\begin{bmatrix}h_{1}&h_{2}&h_{3}\end{bmatrix}}{\begin{bmatrix}1\\-j\\0\end{bmatrix}}=h_{1}-jh_{2}\end{aligned}}}
1507:
Zhang's method is a camera calibration method that uses traditional calibration techniques (known calibration points) and self-calibration techniques (correspondence between the calibration points when they are in different positions). To perform a full calibration by the Zhang method, at least three
1121:
are also important although they cannot be included in the linear camera model described by the intrinsic parameter matrix. Many modern camera calibration algorithms estimate these intrinsic parameters as well in the form of non-linear optimisation techniques. This is done in the form of optimising
2814:
Selby's camera calibration method addresses the auto-calibration of X-ray camera systems. X-ray camera systems, consisting of the X-ray generating tube and a solid state detector can be modelled as pinhole camera systems, comprising 9 intrinsic and extrinsic camera parameters. Intensity based
2715:{\displaystyle {\begin{aligned}x_{1}^{T}\omega x_{1}&=\left(h_{1}+jh_{2}\right)^{T}\omega \left(h_{1}+jh_{2}\right)\\&=\left(h_{1}^{T}+jh_{2}^{T}\right)\omega \left(h_{1}+jh_{2}\right)\\&=h_{1}^{T}\omega h_{1}+j\left(h_{2}^{T}\omega h_{2}\right)\\&=0\end{aligned}}}
1508:
different images of the calibration target/gauge are required, either by moving the gauge or the camera itself. If some of the intrinsic parameters are given as data (orthogonality of the image or optical center coordinates), the number of images required can be reduced to two.
304:
833:
2756:. This involves calculating the camera's position and orientation relative to a known object in the scene. The process typically requires identifying specific points in the calibration pattern and solving for the camera's rotation and translation vectors.
2730:
Tsai's algorithm, a significant method in camera calibration, involves several detailed steps for accurately determining a camera's orientation and position in 3D space. The procedure, while technical, can be generally broken down into three main stages:
1238:
2768:. In this stage, the algorithm refines the lens distortion coefficients, addressing radial and tangential distortions. Further optimization of internal and external camera parameters is performed to enhance the calibration accuracy.
2815:
registration based on an arbitrary X-ray image and a reference model (as a tomographic dataset) can then be used to determine the relative camera parameters without the need of a special calibration body or any ground-truth data.
2743:
stage, where a series of images are captured by the camera. These images, often featuring a known calibration pattern like a checkerboard, are used to estimate intrinsic camera parameters such as focal length and optical center.
105:
The camera projection matrix is derived from the intrinsic and extrinsic parameters of the camera, and is often represented by the series of transformations; e.g., a matrix of camera intrinsic parameters, a 3 × 3
1808:
1434:
is used, light from the environment is focused on an image plane and captured. This process reduces the dimensions of the data taken in by the camera from three to two (light from a 3D scene is stored on a 2D image). Each
509:{\displaystyle {\begin{bmatrix}wu\\wv\\w\end{bmatrix}}=K\,{\begin{bmatrix}R&T\end{bmatrix}}{\begin{bmatrix}x_{w}\\y_{w}\\z_{w}\\1\end{bmatrix}}=M{\begin{bmatrix}x_{w}\\y_{w}\\z_{w}\\1\end{bmatrix}}}
568:
724:
2398:
2021:
960:
914:
2008:
1957:
1136:
2946:
1394:
1882:
1855:
691:
2771:
This structured approach has positioned Tsai's
Algorithm as a pivotal technique in both academic research and practical applications within robotics and industrial metrology.
693:
are the coordinates of the source of the light ray which hits the camera sensor in world coordinates, relative to the origin of the world. By dividing the matrix product by
1668:
1271:
which denote the coordinate system transformations from 3D world coordinates to 3D camera coordinates. Equivalently, the extrinsic parameters define the position of the
1695:
1906:
1058:
2385:
2358:
1112:
1085:
1014:
987:
1828:
1715:
1535:. Subsequently, self-calibration techniques are applied to obtain the image of the absolute conic matrix. The main contribution of Zhang's method is how to, given
638:
237:
161:
1265:
594:
1735:
1633:
1613:
1593:
1573:
1553:
1529:
1414:
1333:
1313:
1293:
1034:
856:
711:
614:
288:
1743:
110:, and a translation vector. The camera projection matrix can be used to associate points in a camera's image space with locations in 3D world space.
2906:
3001:
Roger Y. Tsai, "A Versatile Camera
Calibration Technique for High-Accuracy 3D Machine Vision Metrology Using Off-the-Shelf TV Cameras and Lenses,"
2933:, In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 432–437, Fort Collins, CO, USA, June 1999
2926:
1447:
There are many different approaches to calculate the intrinsic and extrinsic parameters for a specific camera setup. The most common ones are:
38:
3058:
3018:
2953:", Proceedings of the Symposium on Close-Range Photogrammetry (pp. 1-18), Falls Church, VA: American Society of Photogrammetry, (1971)
97:
where the camera projection matrices of two cameras are used to calculate the 3D world coordinates of a point viewed by both cameras.
2943:
2887:
1295:
is the position of the origin of the world coordinate system expressed in coordinates of the camera-centered coordinate system.
522:
2859:
867:
828:{\displaystyle K={\begin{bmatrix}\alpha _{x}&\gamma &u_{0}\\0&\alpha _{y}&v_{0}\\0&0&1\end{bmatrix}}}
73:
247:(i.e. they have an additional last component, which is initially, by convention, a 1), which is the most common notation in
3068:
1644:
1532:
1451:
3042:
2944:
Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry
1233:{\displaystyle {}{\begin{bmatrix}R_{3\times 3}&T_{3\times 1}\\0_{1\times 3}&1\end{bmatrix}}_{4\times 4}}
2913:, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol.22, No.11, pages 1330–1334, 2000
919:
873:
1962:
1911:
244:
89:
3063:
53:
2903:
3025:, Australasian Physical & Engineering Science in Medicine, Vol.34, No.3, pages 391–400, 2011
2839:
1338:
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643:
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2834:
264:
23:
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26:
approximating the camera that produced a given photograph or video; it determines which incoming
87:
The classic camera calibration requires special objects in the scene, which is not required in
2984:
2883:
2829:
1673:
1123:
863:
3037:
1891:
1043:
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31:
30:
is associated with each pixel on the resulting image. Basically, the process determines the
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992:
965:
3022:
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2910:
1813:
1700:
1424:
1417:
619:
170:
121:
107:
2965:"Self-Calibration of a Moving Camera from Point Correspondences and Fundamental Matrices"
1244:
858:
contains 5 intrinsic parameters of the specific camera model. These parameters encompass
573:
84:
refer to the determination of only the extrinsic and intrinsic parameters, respectively.
1720:
1618:
1598:
1578:
1558:
1538:
1514:
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1318:
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841:
696:
599:
273:
2924:"On plane-based camera calibration: a general algorithm, singularities, applications'"
2783:
1476:
1439:
on the image plane therefore corresponds to a shaft of light from the original scene.
3052:
2844:
1272:
268:
94:
43:
3016:"Patient positioning with X-ray detector self-calibration for image guided therapy"
1037:
859:
1803:{\displaystyle I,J={\begin{bmatrix}1&\pm j&0\end{bmatrix}}^{\mathrm {T} }}
1114:
represent the principal point, which would be ideally in the center of the image.
1016:
are the inverses of the width and height of a pixel on the projection plane and
1122:
the camera and distortion parameters in the form of what is generally known as
3015:
2980:
1060:
represents the skew coefficient between the x and the y axis, and is often 0.
252:
61:
specify the camera image format (focal length, pixel size, and image origin).
2988:
27:
2964:
1315:
is often mistakenly considered the position of the camera. The position,
248:
37:
Usually, the camera parameters are represented in a 3 × 4
1555:
poses of the calibration target, extract a constrained intrinsic matrix
1531:
between the calibration target and the image plane is determined using
596:
by convention are the x and y coordinates of the pixel in the camera,
76:
or be restricted for the estimation of the intrinsic parameters only.
1511:
In a first step, an approximation of the estimated projection matrix
1431:
1436:
713:, the theoretical value for the pixel coordinates can be found.
2778:
1471:
16:
Process of estimating the parameters of a pinhole camera model
563:{\displaystyle M=K\,{\begin{bmatrix}R&T\end{bmatrix}}}
93:. Camera resectioning is often used in the application of
2752:
Following initial calibration, the algorithm undertakes
2795:
1488:
2255:
2205:
2109:
2059:
1765:
1423:
Camera calibration is often used as an early stage in
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739:
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383:
359:
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2366:
2339:
2019:
1965:
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coordinates. In both cases, they are represented in
3045:- Augmented reality lecture at TU Muenchen, Germany
2904:"A flexible new technique for camera calibration'"
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2002:
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1335:, of the camera expressed in world coordinates is
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155:
1884:of course means they are also projected onto the
962:represent focal length in terms of pixels, where
22:is the process of estimating the parameters of a
2882:. Cambridge University Press. pp. 155–157.
616:is the intrinsic matrix as described below, and
3038:Zhang's Camera Calibration Method with Software
1275:and the camera's heading in world coordinates.
640:form the extrinsic matrix as described below.
2878:Richard Hartley and Andrew Zisserman (2003).
8:
290:is used to denote a projective mapping from
239:is used to represent a 3D point position in
2963:Luong, Q.-T.; Faugeras, O.D. (1997-03-01).
2774:
2360:while substituting our new expression for
2880:Multiple View Geometry in Computer Vision
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2969:International Journal of Computer Vision
955:{\displaystyle \alpha _{y}=f\cdot m_{y}}
909:{\displaystyle \alpha _{x}=f\cdot m_{x}}
3003:IEEE Journal of Robotics and Automation
2870:
2003:{\displaystyle x_{2}^{T}\omega x_{2}=0}
1952:{\displaystyle x_{1}^{T}\omega x_{1}=0}
1117:Nonlinear intrinsic parameters such as
72:, although that term may also refer to
57:(position and orientation) while the
7:
163:to represent a 2D point position in
2189:
2043:
1655:
2775:Selby's method (for X-ray cameras)
2010:. The circular points project as
1869:
1865:
1842:
1838:
1794:
1463:Selby's method (for X-ray cameras)
1389:{\displaystyle C=-R^{-1}T=-R^{T}T}
14:
1877:{\displaystyle \Omega _{\infty }}
1850:{\displaystyle \Omega _{\infty }}
686:{\displaystyle x_{w},y_{w},z_{w}}
2942:Abdel-Aziz, Y.I., Karara, H.M. "
2782:
1475:
2860:Rational polynomial coefficient
3005:, Vol. RA-3, No.4, August 1987
220:
174:
144:
125:
74:photometric camera calibration
1:
1663:{\displaystyle {\textbf {H}}}
64:This process is often called
2739:The process begins with the
1888:of the absolute conic (IAC)
1810:lie on both our probe plane
1452:Direct linear transformation
66:geometric camera calibration
3059:Geometry in computer vision
3085:
3014:Boris Peter Selby et al.,
1830:and on the absolute conic
2922:P. Sturm and S. Maybank,
2725:
2766:refinement of parameters
2760:Refinement of Parameters
1690:{\displaystyle x_{\pi }}
1635:calibration parameters.
118:In this context, we use
2981:10.1023/A:1007982716991
2764:The final phase is the
2333:We can actually ignore
1901:{\displaystyle \omega }
1053:{\displaystyle \gamma }
245:homogeneous coordinates
114:Homogeneous coordinates
90:camera auto-calibration
34:of the pinhole camera.
2716:
2381:
2354:
2323:
2004:
1953:
1902:
1878:
1851:
1824:
1804:
1731:
1711:
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1609:
1589:
1569:
1549:
1525:
1467:
1410:
1390:
1329:
1309:
1289:
1261:
1234:
1108:
1081:
1054:
1040:in terms of distance.
1030:
1010:
983:
956:
910:
868:camera principal point
852:
829:
707:
687:
634:
610:
590:
564:
510:
284:
233:
157:
2840:Eight-point algorithm
2717:
2382:
2380:{\displaystyle x_{1}}
2355:
2353:{\displaystyle x_{2}}
2324:
2005:
1954:
1903:
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1852:
1825:
1805:
1732:
1712:
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1610:
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1526:
1411:
1391:
1330:
1310:
1290:
1262:
1235:
1109:
1107:{\displaystyle v_{0}}
1082:
1080:{\displaystyle u_{0}}
1055:
1031:
1011:
1009:{\displaystyle m_{y}}
984:
982:{\displaystyle m_{x}}
957:
911:
853:
830:
708:
688:
635:
611:
591:
565:
511:
285:
234:
158:
3069:Stereophotogrammetry
2850:Pinhole camera model
2835:Augmented virtuality
2394:
2364:
2337:
2017:
1963:
1912:
1892:
1861:
1834:
1823:{\displaystyle \pi }
1814:
1744:
1740:The circular points
1721:
1710:{\displaystyle \pi }
1701:
1674:
1650:
1619:
1599:
1579:
1559:
1539:
1515:
1400:
1339:
1319:
1299:
1279:
1269:extrinsic parameters
1245:
1137:
1130:Extrinsic parameters
1091:
1064:
1044:
1020:
993:
966:
920:
874:
842:
725:
717:Intrinsic parameters
697:
644:
633:{\displaystyle R\,T}
620:
600:
574:
523:
305:
274:
265:pinhole camera model
232:{\displaystyle ^{T}}
171:
156:{\displaystyle ^{T}}
122:
82:interior orientation
78:Exterior orientation
59:intrinsic parameters
49:extrinsic parameters
24:pinhole camera model
2855:Perspective-n-Point
2741:initial calibration
2735:Initial Calibration
2676:
2637:
2568:
2547:
2415:
1980:
1929:
1697:on a "probe plane"
1260:{\displaystyle R,T}
870:. The parameters
864:image sensor format
589:{\displaystyle u,v}
20:Camera resectioning
3043:Camera Calibration
3021:2023-11-10 at the
2949:2019-08-02 at the
2929:2016-03-04 at the
2909:2015-12-03 at the
2825:3D pose estimation
2794:. You can help by
2712:
2710:
2662:
2623:
2554:
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2401:
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2350:
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2317:
2280:
2244:
2131:
2098:
2000:
1966:
1949:
1915:
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1800:
1786:
1727:
1707:
1687:
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1487:. You can help by
1406:
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70:camera calibration
51:define the camera
2830:Augmented reality
2812:
2811:
2191:
2045:
1730:{\displaystyle x}
1670:that maps points
1657:
1643:Assume we have a
1628:{\displaystyle T}
1608:{\displaystyle R}
1588:{\displaystyle n}
1568:{\displaystyle K}
1548:{\displaystyle n}
1524:{\displaystyle H}
1505:
1504:
1409:{\displaystyle R}
1328:{\displaystyle C}
1308:{\displaystyle T}
1288:{\displaystyle T}
1124:bundle adjustment
1029:{\displaystyle f}
851:{\displaystyle K}
706:{\displaystyle w}
609:{\displaystyle K}
283:{\displaystyle M}
263:Referring to the
215:
202:
189:
139:
133:
39:projection matrix
3076:
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2999:
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2726:Tsai's algorithm
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1496:December 2008
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1483:This section
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95:stereo vision
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44:camera matrix
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2803:October 2011
2800:
2796:adding to it
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2387:as follows:
2332:
1885:
1857:. Lying on
1739:
1642:
1510:
1506:
1493:
1489:adding to it
1484:
1454:(DLT) method
1446:
1429:
1422:
1268:
1241:
1133:
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1038:focal length
860:focal length
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518:
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255:transforms.
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101:Formulation
41:called the
3053:Categories
2902:Z. Zhang,
2866:References
1717:to points
1645:homography
1639:Derivation
1533:DLT method
1443:Algorithms
259:Projection
253:rigid body
68:or simply
2989:1573-1405
2678:ω
2639:ω
2575:ω
2480:ω
2417:ω
2300:−
2266:−
1982:ω
1931:ω
1896:ω
1870:∞
1866:Ω
1843:∞
1839:Ω
1818:π
1774:±
1705:π
1683:π
1371:−
1357:−
1349:−
1223:×
1198:×
1178:×
1160:×
1048:γ
940:⋅
925:α
894:⋅
879:α
780:α
755:γ
744:α
28:light ray
3019:Archived
2947:Archived
2927:Archived
2907:Archived
2819:See also
1267:are the
249:robotics
1908:, thus
1430:When a
1396:(since
1036:is the
2987:
2886:
1432:camera
866:, and
519:where
214:
201:
188:
138:
132:
47:. The
1886:image
1437:pixel
1416:is a
296:pixel
292:world
241:world
165:pixel
2985:ISSN
2884:ISBN
1959:and
1615:and
1087:and
989:and
916:and
838:The
267:, a
251:and
80:and
54:pose
32:pose
2977:doi
2798:.
1491:.
1420:).
3055::
2983:.
2973:22
2971:.
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1126:.
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570:.
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2892:.
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2801:(
2706:0
2703:=
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2686:2
2682:h
2673:T
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2664:h
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2579:(
2571:)
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2530:(
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2238:3
2234:h
2226:2
2222:h
2214:1
2210:h
2203:[
2198:=
2195:J
2190:H
2185:=
2176:2
2172:x
2162:2
2158:h
2154:j
2151:+
2146:1
2142:h
2138:=
2133:]
2127:0
2120:j
2113:1
2107:[
2100:]
2092:3
2088:h
2080:2
2076:h
2068:1
2064:h
2057:[
2052:=
2049:I
2044:H
2039:=
2030:1
2026:x
1998:0
1995:=
1990:2
1986:x
1977:T
1972:2
1968:x
1947:0
1944:=
1939:1
1935:x
1926:T
1921:1
1917:x
1795:T
1788:]
1782:0
1777:j
1769:1
1763:[
1757:=
1754:J
1751:,
1748:I
1725:x
1679:x
1656:H
1623:T
1603:R
1583:n
1563:K
1543:n
1519:H
1498:)
1494:(
1404:R
1384:T
1379:T
1375:R
1368:=
1365:T
1360:1
1353:R
1346:=
1343:C
1323:C
1303:T
1283:T
1255:T
1252:,
1249:R
1226:4
1220:4
1214:]
1208:1
1201:3
1195:1
1191:0
1181:1
1175:3
1171:T
1163:3
1157:3
1153:R
1146:[
1100:0
1096:v
1073:0
1069:u
1024:f
1002:y
998:m
975:x
971:m
948:y
944:m
937:f
934:=
929:y
902:x
898:m
891:f
888:=
883:x
846:K
821:]
815:1
810:0
805:0
796:0
792:v
784:y
774:0
765:0
761:u
748:x
737:[
732:=
729:K
701:w
679:w
675:z
671:,
666:w
662:y
658:,
653:w
649:x
628:T
624:R
604:K
584:v
581:,
578:u
556:]
550:T
545:R
539:[
533:K
530:=
527:M
502:]
496:1
487:w
483:z
473:w
469:y
459:w
455:x
448:[
443:M
440:=
435:]
429:1
420:w
416:z
406:w
402:y
392:w
388:x
381:[
374:]
368:T
363:R
357:[
351:K
348:=
343:]
337:w
330:v
327:w
320:u
317:w
311:[
278:M
225:T
221:]
217:1
209:w
205:z
196:w
192:y
183:w
179:x
175:[
149:T
145:]
141:1
135:v
129:u
126:[
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