Announcements. Image Formation: Outline. The course. Image Formation and Cameras (cont.)

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1 nnouncements Imge Formton nd Cmers (cont.) ssgnment : Cmer & Lenses, gd Trnsformtons, nd Homogrph wll be posted lter tod. CSE 5 Lecture 5 CS5, Fll CS5, Fll CS5, Fll The course rt : The phscs of mgng rt : Erl vson rt 3: econstructon rt 4: ecognton CS5, Fll Imge Formton: Outlne Fctors n producng mges rojecton erspectveorthogrphc rojecton Vnshng ponts rojectve Geometr gd Trnsformton nd SO(3) Lenses Sensors Qunttonesoluton Illumnton eflectnce nd dometr nhole Cmer: erspectve projecton bstrct cmer model - bo wth smll hole n t Geometrc propertes of projecton 3-D ponts mp to ponts 3-D lnes mp to lnes lnes mp to whole mge or hlf-plne olgons mp to polgons Importnt pont to note: ngles & dstnces not preserved, nor re nequltes of ngles & dstnces. Degenerte cses: lne through focl pont project to pont plne through focl pont projects to lne CS5, Fll Forsth&once CS5, Fll

2 Equton of erspectve rojecton Dgresson Crtesn coordntes: We hve, b smlr trngles, tht (,, ) -> (f, f, f ) Estblshng n mge plne coordnte sstem t C lgned wth nd j, we get (,,) ( f, f ) rojectve Geometr nd Homogenous Coordntes CS5, Fll CS5, Fll Wht s the ntersecton of two lnes n plne? Do two lnes n the plne lws ntersect t pont? ont No, rllel lnes don t meet t pont. CS5, Fll CS5, Fll Cn the perspectve mge of two prllel lnes meet t pont? ES rojectve geometr provdes n elegnt mens for hndlng these dfferent stutons n unfed w nd homogenous coordntes re w to represent enttes (ponts & lnes) n projectve spces. CS5, Fll CS5, Fll

3 rojectve Geometr oms of rojectve lne. Ever two dstnct ponts defne lne. Ever two dstnct lnes defne pont (ntersect t pont) 3. There ests three ponts,,,c such tht C does not le on the lne defned b nd. Dfferent thn Euclden (ffne) geometr rojectve plne s bgger thn ffne plne ncludes lne t nfnt rojectve lne ffne Lne t = lne + Infnt Homogenous coordntes w to represent ponts n projectve spce Use three numbers to represent pont on projectve plne Wh? The projectve plne hs to be bgger thn the Crtesn plne. How: dd n etr coordnte e.g., (,) -> (,,) Impose equvlence relton (,,) *(,,) such tht ( not ).e., (,,) (,, ) ont t nfnt ero for lst coordnte e.g., (,,) Wh do ths? ossble to represent ponts t nfnt Where prllel lnes ntersect Where prllel plnes ntersect ossble to wrte the cton of perspectve cmer s mtr CS5, Fll CS5, Fll Homogenous coordntes w to represent ponts n projectve spce Use three numbers to represent pont on projectve plne dd n etr coordnte e.g., (,) -> (,,) Impose equvlence relton (,,) *(,,) such tht ( not ).e., (,,) (,, ) (,,) (,) Converson Euclden -> Homogenous -> Euclden In -D Euclden -> Homogenous: (, ) -> (,,) Homogenous -> Euclden: (,, ) -> (, ) In 3-D Euclden -> Homogenous: (,, ) -> (,,,) Homogenous -> Euclden: (,,, w) -> (w, w, w) (,,) (,) CS5, Fll CS5, Fll onts t nfnt Lnes n rojectve spce ont t nfnt ero for lst coordnte (,,) nd equvlence relton (,,) *(,,) No correspondng Euclden pont (,,) (,,). Lne n Euclden plne. lne through orgn n homogenous coordntes 3. lne s represented b ts norml N 4. Equton for plne s N. (,,) = or M. (,,) = where M = λn N rojectve lne ffne Lne t = lne + Infnt CS5, Fll CS5, Fll 3

4 The equton of projecton End of the Dgresson Crtesn coordntes: (,,) ( f, f ) Homogenous Coordntes nd Cmer mtr U V W f T CS5, Fll CS5, Fll rllel lnes meet n the mge Vnshng pont Vnshng ponts H VL V Imge plne Formed b lne through O rllel to the gven lne(s) sngle lne cn hve vnshng pont Dfferent drectons correspond dfferent vnshng ponts V V CS5, Fll CS5, Fll V 3 Vnshng onts Vnshng ont In the projectve plne, prllel lnes meet t pont t nfnt. The vnshng pont s the perspectve projecton of tht pont t nfnt, resultng from multplcton b the cmer mtr. CS5, Fll CS5, Fll 4

5 5 CS5, Fll ffne Cmer Model Te perspectve projecton equton, nd perform Tlor seres epnson bout some pont (,, ). Drop terms tht re hgher order thn lner. esultng epresson s ffne cmer model pproprte n Neghborhood bout (,, ) CS5, Fll p b v u ewrte ffne cmer model n terms of Homogenous Coordntes w v u ffne cmer model CS5, Fll v u Orthogrphc projecton Strtng wth ffne cmer mode Te Tlor seres bout (,, ) pont on optcl s (,, ) CS5, Fll The projecton mtr for scled orthogrphc projecton T W V U rllel lnes project to prllel lnes tos of dstnces re preserved under orthogrphc projecton CS5, Fll Wht f cmer coordnte sstem dffers from object coordnte sstem {c} {W} CS5, Fll Euclden Coordnte Sstems O O O O j j...

6 Coordnte Chnges: ure Trnsltons otton Mtr CS5, Fll O O O O = + O.. j. CS5, Fll j. j. j j... j. T T j T j Coordnte Chnges: ure ottons Coordnte Chnges: gd Trnsformtons O j j CS5, Fll CS5, Fll O convenent notton CS5, Fll O onts: Ledng superscrpt ndctes the coordnte sstem tht the coordntes re wth respect to Subscrpt n dentfer otton Mtrces Lower left (Gong from ths sstem) Upper left (Gong to ths sstem) To dd vectors, coordnte sstems must gree To rotte vector, ponts coordnte sstem must gree wth lower left of rotton mtr CS5, Fll Some ponts bout SO(n) SO(n) = { nn : T = I, det() = } SO(): rotton mtrces n plne SO(3): rotton mtrces n 3-spce 3 Forms Group under mtr product operton: Identt Inverse ssoctve Closure Closed (fnte ntersecton of closed sets) ounded,j [-, +] Does not form vector spce. Mnfold of dmenson n(n-) Dm(SO()) = Dm(SO(3)) = 3 6

7 SO(3) rmetertons of SO(3) 3-D mnfold, so between 3 prmeters nd n+ prmeters (Whtne s Embeddng Thm.) oll-tch-w Euler ngles s ngle (odrgues formul) Cle s formul Mtr Eponentl Quternons (four prmeters + one constrnt) gd Trnsformtons s Mppngs: otton bout the s = rot(,θ) CS5, Fll CS5, Fll otton: Homogenous Coordntes bout s ' ' ' = cos θ sn θ -sn θ cos θ rot(,θ) θ p' p bout s: bout s: ' ' ' ' ' ' otton cos θ = sn θ cos θ = -sn θ -sn θ cos θ sn θ cos θ CS5, Fll CS5, Fll oll-tch-w rot( ˆ, ) rot( ˆ, j ) rot( ˆ, ) otton bout (,, ), unt vector on n rbtrr s (odrgues Formul) otte(, θ) θ Euler ngles ' ' ' = (-c)+c (-c)+s (-c)-s (-c)-s (-c)+c (-c)-s (-c)+s (-c)-s (-c)+c rot( ˆ'', ) rot( ˆ', j ) rot( ˆ, ) where c = cos θ & s = sn θ CS5, Fll CS5, Fll 7

8 Quternons q = (,) q s quternon (generlton of mgnr numbers) s ts rel prt 3 s ts mgnr prt. Opertons on quternons: Sum of quternons: (, ) + ( b, ) (( +b ),(+ ) Multplcton b sclr: (, ) (, ) Quternon product: (, ) ( b, ) (( b ), ( + b + )) Conjugte: q = (,) q (, -) Unt Quternons nd ottons Let denote the rotton of ngle bout the unt vector u. Defne unt quternon q = (cos, sn u). Note q = (.e., q les on unt sphere for n u nd. Then for n vector, = mgnr(q * q ) where *= (, ) q nd q defne the sme rotton mtr. If q = (, ( b, c, d ) T) s unt quternon, the correspondng rotton mtr s: Norm: q q q = q q = + CS5, Fll CS5, Fll loc Mtr Multplcton Wht s? Homogeneous epresentton of gd Trnsformtons T O O T rojectve trnsformton 3 3 lner trnsformton of homogenous coordntes onts mp to ponts, lnes mp to lnes u u u CS5, Fll CS5, Fll Centrl rojecton lnr Homogrph Fgure borrowed from Hrtle nd ssermn Multple Vew Geometr n computer vson Fgure borrowed from Hrtle nd ssermn Multple Vew Geometr n computer vson CS5, Fll CS5, Fll 8

9 lnr Homogrph: ure otton pplcton: norms Fgure borrowed from Hrtle nd ssermn Multple Vew Geometr n computer vson CS5, Fll CS5, Fll CS5, Fll Cmer prmeters Issue World unts (e.g., cm), cmer unts (pels) cmer m not be t the orgn, loong down the -s etrnsc prmeters one unt n cmer coordntes m not be the sme s one unt n world coordntes ntrnsc prmeters - focl length, prncpl pont, spect rto, ngle between es, etc. U Trnsformton Trnsformton V representng representng W ntrnsc prmeters etrnsc prmeters T : gd trnsformton CS5, Fll Cmer Clbrton, estmte ntrnsc nd etrnsc cmer prmeters See Tet boo for how to do t. Cmer Clbrton Toolbo for Mtlb (ouguet) 9