Geometric Camera Calibration
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1 Geoetrc Caera Calbraton EECS Fall 24! Foundatons of Coputer Vson!! Instructor: Jason Corso (jjcorso)! web.eecs.uch.edu/~jjcorso/t/598f4!! Readngs: F.; SZ 6. (FL 4.6; extra notes)! Date: 9/7/4!! Materals on these sldes have coe fro any sources n addton to yself; I a nfntely grateful to these, especally Greg Hager, Slvo Savarese, and Steve Setz.!
2 lan 2 Revew erspectve rojecton! Geoetrc Caera Calbraton! Indrect caera calbraton! Solve for projecton atrx then the paraeters! Drect caera calbraton! Mult-planes ethod! Exaple wth the Matlab Toolbox! Catadoptrc Sensng! Dfferent slde-deck. (See Chrs Geyer s CVR 2 Tutoral)! Other calbraton ethods not covered! Vanshng ponts-based ethod (see SZ)! Self-calbraton!
3 rojecton equaton The projecton atrx odels the cuulatve effect of all paraeters Useful to decopose nto a seres of operatons x ΠX * * * * * * * * * * * * Z Y X s sy sx ' ' x x x x x x c y c x y fs x fs T I R Π projecton ntrnscs rotaton translaton dentty atrx Caera paraeters A caera s descrbed by several paraeters Translaton T of the optcal center fro the orgn of world coords Rotaton R of the age plane focal length, prncple pont (x c, y c ), pxel sze (s x, s y ) blue paraeters are called extrnscs, red are ntrnscs The defntons of these paraeters are not copletely standardzed especally ntrnscs vares fro one book to another Source: S Setz sldes.!
4 rojectve Caera 4 O c focal length: Source: S Savarese sldes.!
5 rojectve Caera: The Noralzed Iage lane 5 The noralzed age plane s parallel to the physcal retna (e.g., ccd) but located at unt dstance ( ) fro the pnhole.! Iage Source: Forsyth and once Book.!
6 rojectve Caera: The Noralzed Iage lane 6 hyscal pxels n the retna (e.g. ccd) ay not be square, so we have two addtonal scale paraeters.! Unts:! s a dstance expressed n eters! A pxel wll have densons where and are n! Can replace dependent pxel paraeters!
7 rojectve Caera 7 focal length: age center-pont: Source: S Savarese sldes.!
8 rojectve Caera 8 focal length: age center-pont: non-square pxels: skew angle: Source: S Savarese sldes.!
9 z y x v u s ' o o β α K has 5 degrees of freedo! Source: S Savarese sldes.! rojectve Caera 9 focal length: age center-pont: non-square pxels: skew angle:
10 ʹ z y x v u cot o o sn θ β θ α α K has 5 degrees of freedo! Source: S Savarese sldes.! rojectve Caera focal length: age center-pont: non-square pxels: skew angle:
11 rojectve Caera j w k w O w w focal length: age center-pont: non-square pxels: skew angle: rotaton, translaton: Source: S Savarese sldes.!
12 rojectve Caera 2 j w k w O w w Internal paraeters External paraeters Source: S Savarese sldes.! focal length: age center-pont: non-square pxels: skew angle: rotaton, translaton:
13 ropertes of nhole erspectve rojecton Dstant objects appear saller! onts project to ponts! Lnes project to lnes! Vanshng ont! Angles are not preserved.! arallel lnes eet!! Source: S. Savarese sldes.!
14 rojectve Caera 4 ʹ K[ R T] w M w Internal paraeters External paraeters Source: S Savarese sldes.!
15 rojectve Caera 5 Internal paraeters External paraeters
16 v u cot K o o sn θ β θ α α T T 2 T R r r r z y x t t t T 4 rojectve Caera 6
17 Estate ntrnsc and extrnsc paraeters fro or ultple ages Change notaton: w p v u cot K o o sn θ β θ α α T T 2 T R r r r z y x t t t T 4 Source: S Savarese sldes.! Goal of Calbraton 7
18 The Calbraton roble 8 Calbraton rg j C n wth known postons n [O w, w,j w,k w ] p, p n known postons n the age Goal: copute ntrnsc and extrnsc paraeters Source: S Savarese sldes.!
19 The Calbraton roble 9 Calbraton rg age! j C n wth known postons n [O w, w,j w,k w ] p, p n known postons n the age Goal: copute ntrnsc and extrnsc paraeters Source: S Savarese sldes.!
20 The Calbraton roble 2 Calbraton rg age! j C How any correspondences do we need? M has unknown We need equatons 6 correspondences would do t Source: S Savarese sldes.!
21 The Calbraton roble 2 Calbraton rg age! j C In practce, usng ore than 6 correspondences enables ore robust results Source: S Savarese sldes.!
22 j C M v u p 2 M 2 n pxels Source: S Savarese sldes.! The Calbraton roble 22
23 u 2 ) ( v ) ( u 2 v v u 2 ) ( v 2 ) ( u Source: S Savarese sldes.! The Calbraton roble 2
24 ) ( 2 v ) ( u ) ( 2 v ) ( u ) ( 2 n n n v ) ( n n n u Source: S Savarese sldes.! The Calbraton roble 24
25 B B B B B A A A A A What s AB? B A B A B A B A B A B A B A B A AB Source: S Savarese sldes.! Block Matrx Multplcaton 25
26 The Calbraton roble u ( ) + v ( ) + 2 known unknown 26 u v n n ( n ) + n ( n ) + 2 n Hoogenous lnear syste x4 2n x 2 def 4x T T 2 T 2x Source: S Savarese sldes.!
27 Hoogeneous M x N Lnear Systes Mnuber of equatons 2n Nnuber of unknown 27! Rectangular syste (M>N) s always a soluton To fnd non-zero soluton Mnze 2 under the constrant 2 Source: S Savarese sldes.!
28 The Calbraton roble 28 How do we solve ths hoogenous lnear syste? Usng DLT (Drect Lnear Transforaton) algorth va SVD decoposton! Source: S Savarese sldes.!
29 Egenvalues and Egenvectors 29 Egendecoposton A λ λ SΛ S S S. 2 λ N Egenvectors of A are coluns of S [ ] S v v N Source: S Savarese sldes.!
30 Sngular Value Decoposton A U Σ V Σ U, V orthogonal atrx σ σ 2. σ N σ λ σ sngular value λ egenvalue of A t A Source: S Savarese sldes.!
31 ropertes of SVD Recall the sngular values of a atrx are related to ts rank.! Recall that Ax can have a nonzero x as soluton only f A s sngular.! Fnally, note that the atrx V of the SVD s an orthogonal bass for the doan of A; n partcular the zero sngular values are the bass vectors for the null space.! uttng all ths together, we see that A ust have rank 7 (n ths partcular case) and thus x ust be a vector n ths subspace.! Clearly, x s defned only up to scale.! Source: G Hager sldes.!
32 DLT algorth (Drect Lnear Transforaton) 2 H Source: S Savarese sldes.! x x ʹ H x xʹ unknown A h Functon of easureents
33 The Calbraton roble U D V T 2n Last colun of V gves SVD decoposton of Why? See pag 59 of AZ M M p Source: S Savarese sldes.!
34 Clarfcaton about SVD 4 U D V T n n n n n n Thanks to at O Keefe!! Has n orthogonal coluns Orthogonal atrx Ths s one of the possble SVD decopostons Ths s typcally used for effcency The classc SVD s actually: U D V T n n n n Source: S Savarese sldes.! orthogonal Orthogonal
35 Extractng Caera araeters 5 Intrnsc Estated values Source: S Savarese sldes.!
36 Theore (Faugeras, 99) 6 Source: S Savarese sldes.!
37 Extractng Caera araeters 7 Intrnsc Estated values Source: S Savarese sldes.!
38 Extractng Caera araeters 8 Extrnsc Estated values Source: S Savarese sldes.!
39 9 Degenerate cases s cannot le on the sae plane! onts cannot le on the ntersecton curve of two quadrc surfaces Source: S Savarese sldes.!
40 4 Takng lens dstortons nto account Chroatc Aberraton! Sphercal aberraton! Radal Dstorton! Source: S Savarese sldes.!
41 Dealng wth Radal Dstorton As Well 4 Caused by perfect lenses! Devatons are ost notceable for rays that pass through the edge of the lens! No dstorton n cushon Barrel Source: S Savarese sldes.!
42 Issues wth lenses: Radal Dstorton 42 n cushon Source: S Savarese sldes.! Barrel (fsheye lens)
43 Radal Dstorton 4 Source: S Savarese sldes.!
44 Iage agnfcaton n(de)creases wth dstance fro the optcal center p v u M λ λ d v v c u b v u a d u ± p 2p κ p d λ olynoal functon Dstorton coeffcent To odel radal behavor Source: S Savarese sldes.! Radal Dstorton 44
45 v u p 2 Q q q q 2 q q q q p v u M λ λ Q v u 2 q q q q Is ths a lnear syste of equatons? No! why? Source: S Savarese sldes.! Radal Dstorton 45
46 General Calbraton roble 46 u v q q q q 2 X f () paraeter easureent f( ) s nonlnear - Newton Method - Levenberg-Marquardt Algorth Iteratve, starts fro ntal soluton May be slow f ntal soluton far fro real soluton Estated soluton ay be functon of the ntal soluton Newton requres the coputaton of J, H Source: S Savarese sldes.! Levenberg-Marquardt doesn t requre the coputaton of H
47 General Calbraton roble 47 u v q q q q 2 X f () paraeter easureent f( ) s nonlnear A possble algorth. Solve lnear part of the syste to fnd approxated soluton 2. Use ths soluton as ntal condton for the full syste. Solve full syste usng Newton or L.M. Source: S Savarese sldes.!
48 General Calbraton roble 48 u v q q q q 2 X f () paraeter easureent f( ) s nonlnear Typcal assuptons: - zero-skew, square pxel - u o, v o known center of the age - no dstorton Just estate f and R, T Source: S Savarese sldes.!
49 Can estate and 2 and gnore the radal dstorton?! v u p 2 λ d v u Hnt: slope v u 2 q q q q Source: S Savarese sldes.! Radal Dstorton 49
50 Estatng and 2! v u p 2 λ ) ( ) ( 2 u v ) ( ) ( 2 u v ) ( ) ( 2 n n n n u v Q n 2 n Tsa technque [87] v u 2 2 ) ( ) ( ) ( ) ( Source: S Savarese sldes.! Radal Dstorton 5
51 Once that and 2 are estated! v u p 2 λ s non lnear functon of 2 λ There are soe degenerate confguratons for whch and 2 cannot be coputed,, Source: S Savarese sldes.! Radal Dstorton 5
52 Drect Calbraton: The Algorth 52. Copute age center fro orthocenter! 2. Copute the Intrnsc atrx (6.8)!. Copute soluton wth SVD! 4. Copute gaa and alpha! 5. Copute R (and noralze)! 6. Copute f x and and T z! Source: G Hager sldes.!
53 Basc Equatons 5 Source: G Hager sldes.!
54 Basc Equatons 54 Source: G Hager sldes.!
55 Basc Equatons 55 one of these for each pont! Source: G Hager sldes.!
56 Basc Equatons 56 Source: G Hager sldes.!
57 ropertes of SVD Agan 57 Recall the sngular values of a atrx are related to ts rank.! Recall that Ax can have a nonzero x as soluton only f A s sngular.! Fnally, note that the atrx V of the SVD s an orthogonal bass for the doan of A; n partcular the zero sngular values are the bass vectors for the null space.! uttng all ths together, we see that A ust have rank 7 (n ths partcular case) and thus x ust be a vector n ths subspace.! Clearly, x s defned only up to scale.! Source: G Hager sldes.!
58 Basc Equatons 58 We now know R x and R y up to a sgn and gaa.! R z R x x R y!! We wll probably use another SVD to orthogonalze! ths syste (R U D V ; set D to I and ultply).! Source: G Hager sldes.!
59 Last Detals about Drect Calbraton 59 We stll need to copute the correct sgn.! note that the denonator of the orgnal equatons ust be postve (ponts ust be n front of the caeras)! Thus, the nuerator and the projecton ust dsagree n sgn.! We know everythng n nuerator and we know the projecton, hence we can deterne the sgn.! We stll need to copute T z and f x! we can forulate ths as a least squares proble on those two values usng the frst equaton.! Source: G Hager sldes.!
60 Self-Calbraton Calculate the ntrnsc paraeters solely fro pont correspondences fro ultple ages.! Statc scene and ntrnscs are assued.! No expensve apparatus.! Hghly flexble but not well-establshed.! rojectve Geoetry age of the absolute conc.! Source: G Hager sldes.!
61 Mult-lane Calbraton Hybrd ethod: hotograetrc and Self-Calbraton.! Uses a planar pattern aged ultple tes (nexpensve).! Used wdely n practce and there are any pleentatons.! Based on a group of projectve transforatons called hoographes.! be a 2d pont [u v ] and M be a d pont [x y z ].! rojecton s! Source: G Hager sldes.!
62 lanar Hoographes Frst Fundaental Theore of rojectve Geoetry:! There exsts a unque hoography that perfors a change of bass between two projectve spaces of the sae denson.! rojecton Becoes! Notce that the hoography s defned up to scale (s).! Source: G Hager sldes.!
63 Coputng the Intrnscs We know that! Fro one hoography, how any constrants on the ntrnsc paraeters can we obtan?! Extrnscs have 6 degrees of freedo.! The hoography has 8 degrees of freedo.! Thus, we should be able to obtan 2 constrants per hoography.! Use the constrants on the rotaton atrx coluns! Source: G Hager sldes.!
64 Coputng Intrnscs Rotaton Matrx s orthonoral:! Wrte the hoography n ters of ts coluns! Source: G Hager sldes.!
65 Coputng Intrnscs Derve the two constrants:! Source: G Hager sldes.!
66 Closed-For Soluton Notce s syetrc, 6 paraeters can be wrtten as a vector b.! Fro the two constrants, we have! Stack up n of these for n ages and buld a 2n*6 syste.! Solve wth SVD (yet agan).! Extrnscs fall-out of the result easly.! Source: G Hager sldes.!
67 Non-lnear Refneent Closed-for soluton nzed algebrac dstance.! Snce full-perspectve s a non-lnear odel! Can nclude dstorton paraeters (radal, tangental)! Mnze squared dstance wth a non-lnear ethod.! Source: G Hager sldes.!
68 Exaple Calbraton rocedure Caera Calbraton Toolbox for Matlab! J. Bouguet [998-2]!! 68 Source: S Savarese sldes.!
69 Exaple Calbraton rocedure 69 Source: S Savarese sldes.!
70 Exaple Calbraton rocedure 7 Source: S Savarese sldes.!
71 Exaple Calbraton rocedure 7 Source: S Savarese sldes.!
72 Exaple Calbraton rocedure 72 Source: S Savarese sldes.!
73 Exaple Calbraton rocedure 7 Source: S Savarese sldes.!
74 Exaple Calbraton rocedure 74 Source: S Savarese sldes.!
75 Exaple Calbraton rocedure 75 Source: S Savarese sldes.!
76 Next Lecture: hotoetrc and Radoetrc Aspects 76 Readng: F 2, ; SZ 2.2, 2.!
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