Multi-linear Systems and Invariant Theory. in the Context of Computer Vision and Graphics. Class 5: Self Calibration. CS329 Stanford University

Transcription

1 Mlti-linar Systms and Invariant hory in th ontt of omtr Vision and Grahics lass 5: Slf alibration S39 Stanford Univrsity Amnon Shasha lass 5

2 Matrial W Will ovr oday h basic qations and conting argmnts h absolt conic and its imag. ra s qations Rcovring intrnal aramtrs. lass 5

3 lass 5 3 h Basic Eqations and onting Argmnts ; W V U R Rcall, 3D->D from Eclidan world fram to imag Lt, b th intrnal aramtrs of camra, and choos canonical fram in which RI and for first camra. P ; P t R Z Y X P W V U R Z Y X + world fram to first camra fram

4 lass 5 4 h Basic Eqations and onting Argmnts µ WW I ; µ W Z Y X whr W mas from th rojctiv fram to Eclidan n W Z Y X Z (8 nknown aramtrs)

5 h Basic Eqations and onting Argmnts ( X, Y, Z,) ar th oints on th lan at infinity (in Ec fram) (,,,) is th lan at infinity π W is th lan at infinity in Proj fram (rcall: if W mas oints to oints (Ec -> Proj), thn th dal W mas lans to lans) lass 5 5

6 lass 5 6 h Basic Eqations and onting Argmnts n n W π n

7 lass 5 7 h Basic Eqations and onting Argmnts µ Projctiv fram µ WW µ W n µ W t R n t R

8 lass 5 8 n t R h Basic Eqations and onting Argmnts R n + R n + n sinc thn, R bt rovids 5 (non-linar) constraints! I RR

9 lass 5 9 R I RR ) ( ) ( Sinc th right-hand sid is symmtric and to scal, w hav 5 constraints. h Basic Eqations and onting Argmnts

10 h Basic Eqations and onting Argmnts Lts do som conting: Lt # 5 b th nmbr of intrnal aramtrs m b th nmbr of viws 5 ( m ) (# ) + ( m )(# ) + 3 lass 5

11 h Basic Eqations and onting Argmnts 5 ( m ) (# ) + ( m )(# ) + 3 # 5 not nogh masrmnts (!) # 4 5( m ) 7 + 4( m ) m 8 # 3 5( m ) 6 + 3( m ) m 4 # 5( m ) 4 + ( m ) m... 5( m ) 8 m 3 (fid intrnal arams) lass 5

12 h rmaindr of this lctr is abot a gomtric insight of ( ) ( ) lass 5

13 h Absolt onic whr rrsnts a conic in D (,, 3,) ar th oints on th lan at infinity (in Ec fram) (,,,) is th lan at infinity 3 3 is conic on th lan at infinity whn + + I 3 lass 5 is th absolt conic (imaginary circl) Ω 3

14 lass 5 4 Plan at infinity is rsrvd ndr affin transformations: v A W bcas 3 3 W Ω is rsrvd ndr similarity transformation (R,t to scal) 3 3 A W if and A ) ( A A thn A A bt, I so in ordr that I w mst hav: A A I A is orthogonal h Absolt onic

15 h Imag of th Absolt onic Imag of oints at infinity: lt A R R t 3 R 3 if is a conic on th lan at infinity thn A A is th rojctd conic onto th imag sinc I, thn ( R ) ( R ) th imag of Ω is lass 5 5

16 h Imag of th Dal Absolt onic l is tangnt to th conic at l l l l l is th imag of th dal absolt conic h basic qation: ( ) ( ) Bcoms: ( ) ( ) Why 8 aramtrs? 5 for th conic, 3 for th lan lass 5 6

17 lass 5 7 Gomtric Intrrtation of d d d λ d dirction of otical ray + λ d h angl btwn two otical rays d,d ) cos( d d d d θ givn on can masr angls

18 lass 5 8 ra s Eqations Gnral ida: liminat n from th basic qation. l l l l i i i i l l i i l i t i i i i t i i i, t t ar dgnrat (rank ) conics

19 Not: ra s Eqations t l l + l l is a dgnrat conic t iff l or l Lt π b th homograhy indcd by th lan of th conic t (slid 4) t π π t t lass 5 9

20 lass 5 π π ra s Eqations Rcall: π F F F π In or cas and th conic is Ω F F F F Likwis: F F

21 lass 5 Dtrmining givn π Rcall: π th location of th lan at inifinity in th rojctiv coordinat fram. W wish to rrsnt th homograhy indcd by π Lt π P b a oint on th lan at infinity. µ P + µ P ) ( ) ( µ

22 lass 5 ) ( ) ( Dtrmining givn Rcall: (slid 6) Not: this cold b drivd from first rincils as wll: π l l tangnts lins to th imag of th absolt conic l l l l l l

23 Dtrmining givn π Assm fid intrnal aramtrs dt( ) Not: dt( ) R R R Provids 4 indndnt linar constraints on Why 4 and not 5? lass 5 3 w nd 3 viws (sinc has 5 nknowns)

24 Why 4 onstraints? dt( ) R and R ar similar matrics, i.., hav th sam ignvals Lt w b th ais of rotation, i.., Rw w has an ignval, with ignvctor v w ( R v v R ( v ) v ) lass 5 4

25 Why 4 onstraints? v v if is a soltion to thn + λvv is also a soltion + λ λ ( ( vv ) + v )( v ) + λvv W nd on mor camra motion (with a diffrnt ais of rotation). lass 5 5

26 lass 5 6 ra s Eqations (rvisitd) ra s qations: Start with th basic qation: Mltily th trms by F F on both sids: F