= x. I (x,y ) Example: Translation. Operations depend on pixel s Coordinates. Context free. Independent of pixel values. I(x,y) Forward mapping:

Transcription

1 Gomric Transormaion Oraions dnd on il s Coordinas. Con r. Indndn o il valus. (, ) ' (, ) ' I (, ) I ' ( (, ), ( ) ), (,) (, ) I(,) I (, ) Eaml: Translaion (, ) (, ) (, ) I(, ) I ' Forward Maing Forward maing: Sourc (, ) (, ) Targ Forward Maing (,) I(,) (, ) I (, ) Problms wih orward maing du o samling: Hols (som arg ils ar no oulad) Ovrlas (som arg ils assignd w colors)

2 Forward Maing ( ), Sourc Targ ( ), Invrs Maing Invrs maing: (, ) (, ) 4 Sourc Targ Invrs Maing ( ),. 7 Sourc? Targ ( ), Each arg il assignd a singl color. Color Inrolaion is ruird. Eaml: Scaling along X Inrolaion Forward maing: Sourc (,) ; Targ (,) Wha hans whn a maing uncion calculas a racional il locaion? Invrs maing: Sourc / ; Targ Inrolaion: gnras a nw il b analzing h surrounding ils.

3 Inrolaion Good inrolaion chnius am o ind an oimal balanc bwn hr undsirabl ariacs: aliasing, blurring, and dg halos. 4 scaling Nars Nighbor Inrolaion Th assign valu is akn rom h il closs o h gnrad locaion: I (, ) I round (, ) Advanag: Fas Disadvanag: Jaggd rsuls Aliasing nar dgs ( { }, round{ (, ) }) aliasing blurring halos 9 Original Imag Original Imag Nars N. Inrolaion Nars N. Inrolaion

4 Bilinar Inrolaion Linar Inrolaion Th assign valu is a wighd sum o h our nars ils. Each wigh is roorional o h disanc rom ach ising il. v v w v w w w v v v v w w Isolaing v in h abov uaion: ( α) w v αv v whr α w w Bilinar Inrolaion Bilinar Inrolaion NW V N NE NW V N NE SW S SE ( ) ( ) ( ) S SE SW N NE NW V N S SW S Th bilinar inrolaion is h bs i low-dgr olnomial o h orm: v(, ) SE Th il s boundaris ar C coninuous (coninuous valus across boundaris). i, j i a ij j 4

5 Bilinar aml 5 z5 z7 v 5 z z.5.5 Nars N. Inrolaion Bilinar Inrolaion Bicubic Inrolaion Th assign valu is a wighd sum o h 44 nars ils: v(, ) i, j i a ij j Nars N. Inrolaion Bilinar Inrolaion 5

6 How can w ind h righ coicins? Dno h il valus V {,..} Th unknown coicins ar a ij {i,j..} ij i, j v i a W hav a linar ssm o 6 uaions wih 6 coicins. j or, {.. },, [, ] s N.N Bilinar Bicubic Th il s boundaris ar C coninuous (coninuous drivaivs across boundaris). N.N Bilinar 6

7 Aling h Transormaion T % ransormaion mari [r,c] siz(img) % cra arra o dsinaion, coordinas [X,Y]mshgrid(:c,:r); % calcula sourc coordinas sourccoor inv(t) * [X(:) Y(:) ] ; % calcula nars nighbor inrolaion Xs round(sourccoor(,:)); Ys round(sourccoor(,:)); indind(xs< Xs>r); %ou o rang ils Xs(ind); Ys(ind); indind(ys< Ys>c); %ou o rang ils Xs(ind); Ys(ind); Bicubic % calcula nw imag nwimag img((xs-).*rys); nwimag(ind); nwimag(ind); nwimag rsha(nwimag,r,c); Ts o linar D ransormaions Rigid (Euclidan) ransormaion: Translaion Roaion (disanc rsrving). Similari ransormaion: Translaion Roaion Uniorm Scal (angl rsrving). Ain ransormaion: Translaion Roaion Scal Shar (aralllism rsrving). Projciv ransormaion Cross-raio rsrving All abov ransormaions ar grous whr Rigid Similari Ain Projciv Ts o linar D ransormaions All abov ransormaions ar grous whr Rigid Similari Ain Projciv 7

8 Mari Noaion Evr locaion (,) is rad as a column vcor: Coordina ransormaion is obaind b muliling wih a mari? a c b a b d c d Mari Noaion - Scal Scal(a,b): (,) (a,b) a I a or b ar ngaiv, w g rlcion. Invrs: S - (a,b)s(/a,/b) a a b b ' b' Mari Noaion - Shar Shar(a,b): (,) (a,b) b a a b a.5, b Mari Noaion - Roaion Roa(θ): (,) (cosθsinθ, -sinθ cosθ) cosθ sinθ Invrs: R - (θ)r T (θ)r(-θ) sinθ cosθ sinθ cosθ sinθ cosθ 8

9 9 Mari Noaion - Translaion Mari Noaion - Translaion Translaion(a,b): Canno rrsn ranslaion using marics. Invrs: b a b a Homognous Coordinas Homognous Coordinas Homognous Coordinas is a maing rom R n o R n : No: (,,) all corrsond o h sam nonhomognous oin (,). E.g. (,,) (6,9,) (4,6,). Invrs maing: ( ) ( ),,,, W Y W X W Y X ),, ( ),, ( ), ( W Y X Homognous Coordinas Homognous Coordinas (,) (,,) (,,) Som D Transormaions Som D Transormaions Translaion : Ain ransormaion: Projciv ransormaion: W Y X d c b a W Y X d c b a W Y X Ain

10 Hirarch o Linar D Transormaions Hirarch o Linar D Transormaions Global Transormaions Imag Rciicaion Global Transormaions Imag Rciicaion Global Transormaions Global Waring Global Transormaions Global Waring A Global Transormaions Global Waring Global Transormaions Global Waring oins i oins i mach A Y

11 Global Transormaions Global Waring Global Transormaions Global Waring oins i oins i mach A Y Invrs Maing: Global Transormaions Global Waring Global Transormaions Global Waring Solv or A - in rms o h las man suar. i.. ind A - which minimizs: i i i A Global Transormaions Global Waring Global Transormaions Global Waring soluion: ( ) ) ( T XX T X X inv Y inv A Global Transormaions Global Waring Global Transormaions Global Waring Alrnaiv rrsnaion: d c b a d c b a A Rarrang:

12 Global Transormaions Global Waring Global Transormaions Global Waring d cb a Global Transormaions Global Waring Global Transormaions Global Waring d cb a Global Transormaions Global Waring Global Transormaions Global Waring inv d c b a ' ( ) T T Q Q Q Q inv ) '( soluion: Global Transormaions Global Waring Global Transormaions Global Waring Wha abou Projciv Transormaions? h g d c b a Homogni mus b rsrvd! ' ; ' h g d c h g b a

13 Global Transormaions Global Waring Wha abou Projciv Transormaions? Global Transormaions Imag Rciicaion So who ARE w? a b ' g h ; c d ' g h ( ' ' ) And similarl or a b c d g h Local Transormaions Imag Waring Local Transormaions Imag Waring Ara o inlunc Ara o inlunc s d s d s sourc oin d dsinaion oin Dmo al

14 Local Transormaions Imag Waring Imag Morhing (Imag Mamorhosis) Ara o inlunc s s sourc oin d dsinaion oin d dsinaion d s s sourc d Dmo bw Cross Dissolv (il oraions) Waring Cross Dissolv War sourc imag owards inrmdia imag. War dsinaion imag owards inrmdia imag. Cross-dissolv h wo imags b aking h wighd avrag a ach il. Sourc Imag Dsinaion Imag sourc im cross dissolv I ( ) ( ) [,] S T Cross-dissolv war dissolv dsinaion waring imags 4

15 war Imag Mamorhosis war Cross-dissolv Cross-dissolv L S,T b h sourc and h arg imags L G() b h ransormaion rom S owards T, whr G()I (h idni) L [,] h im s o b snhsizd Algorihm:. War S owards T:. War T oward S:. Cross dissolv: S T ( ) G( ){ S} I ( ) G( ( ) ) { T} ( ) ( ) S( ) T( ) sours S()G( ){S} Faur Basd Morhing Morh on sha ino anohr sha Us local aurs o din h gomric waring I()(-) S() (T()) T()G((-) ) - {T} arg 5

16 Q Q Q Q P P P P Q Q Q Q P P P P 6

17 Q Q Q Q P P P P On Sgmn Waring Q u v β R α Sourc Imag Ds Imag α [,] is h rlaiv osiion along h sgmn (P,Q ). β is h acual rndicular disanc o h sgmn. (u,v ) is h local coordinas o h sgmn (P,Q ): u is a uni vcor aralll o Q -P v is h uni vcor rndicular o Q -P ( Q P) u u Q P v u u P Q u v β α R P Q u v β α Th oin R is mad ino (α,β) : ( R P ) u α ; β Q P whr R P ( R P ) v R' P' α Q' P' u' βv' 7

18 Q β u v R α P Q u v β α Sourc Imag Ds Imag Invrs Maing: R( α, β ) P α Q P u βv whr (u,v) is h local coordinas o h sgmn (P,Q): ( Q P) u u Q P v u u R P Mulil Sgmn Waring Q R β R P Q β P Q R β β P P Q In mulil sgmn waring h oin R is inluncd b mulil sgmns. Th inlunc srngh o ach sgmns is roorional o: Sgmn lngh Th disanc rom h oin R Th inlunc o ach sgmns is: Qi P i W i aβ i Th valu [,] conrols h inlunc o h lin lngh. Th valu a is a small numbr avoiding division b zro. Th valu b drmins how h rlaiv wigh diminish as h β incrass Th inal maing is: R k k W R k W k b k Eaml: Eaml imags rom: h:// For mor dails s: Thaddus Bir & Shawn Nl / Faur-Basd Imag Mamorhosis Siggrah '9 h:// 8

19 Anohr Eaml: Msh Waring From: h:// 7 -Pass Msh Waring Algorihm Th irs ass wars h rows o h imag: -Pass Msh Waring Algorihm Thn, ach row o h imag is ward individuall b linarl inrolaing ach sgmn bwn h -coordinas dind b h sourc msh o h siz o h corrsonding sgmn dind b h -coordinas o h dsinaion Msh. For ach column o h msh drmin h -coordinas a which h msh column crosss ach imag row. Th scond ass rorms h ac sam rocdur on h columns o h imag b inrolaing h -coordinas o h mshs. 9

20 VidMorh Fun Morh