Image warping/morphing

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Stewart Webster
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1 Image arping/morphing Image arping Digial Visual Effecs Yung-Yu Chuang ih slides b Richard Szeliski, Seve Seiz, Tom Funkhouser and leei Efros Image formaion Sampling and quanizaion B

2 Wha is an image We can hink of an image as a funcion, f: R 2 R: f(, gives he inensi i a posiion (, defined over a recangle, ih a finie range: f: [a,b][c,d] [ [0,1] f digial image We usuall operae on digial (discree images: Sample he 2D space on a regular grid Quanize each sample (round o neares ineger If our samples are D apar, e can rie his as: f[i,j] = Quanize{ f(i D, j D } The image can no be represened as a mari of ineger values color image r(, f (, g (, b(, Image arping image filering: change range of image g( ( = h(f( h(f( f g h h(= image arping: change domain of image g( = f(h( h(=2 f h g Image arping image filering: change range of image g( ( = h(f( h(f( f g h(= h image arping: change domain of image g( = f(h( h([,]=[,/2] f h g

3 Parameric (global arping Eamples of parameric arps: ranslaion roaion aspec affine perspecive clindrical Parameric (global arping T p = (, p = (, Transformaion T is a coordinae-changing machine: p = T(p Wha does i mean ha T is global? Is he same for an poin p can be described b jus a fe numbers (parameers Represen T as a mari: i p = M*p M Scaling Scaling a coordinae means mulipling each of is componens b a scalar Uniform scaling means his scalar is he same for all componens: f g 2 2 Scaling Non-uniform scaling: differen scalars per componen: f g 2 2, 0.5

4 Scaling Scaling operaion: Or, in mari form: Wha s inverse of S? a b a 0 0 b scaling mari S 2-D Roaion This is eas o capure in mari form: cos sin sin cos R Even hough sin( and cos( are nonlinear o, is a linear combinaion of and is a linear combinaion of and Wha is he inverse ransformaion? Roaion b For roaion marices, de(r = 1 so 11 T R R 22 Marices Wha pes of ransformaions can be represened ih a 22 mari? 22 Marices Wha pes of ransformaions can be represened ih a 22 mari? 2D Ideni? D Scale around (0,0? s * s 0 s * 0 s 2D Roae around (0,0? cos * sin * sin * cos * 2D Shear? sh * sh * cos sin sin cos 1 sh sh 1

5 22 Marices Wha pes of ransformaions can be represened ih a 22 mari? 2D Mirror abou Y ais? D Mirror over (0,0? ll 2D Linear Transformaions Linear ransformaions are combinaions of Scale, Roaion, Shear, and Mirror Properies of linear ransformaions: Origin maps o origin Lines map o lines Parallel lines remain parallel Raios are preserved Closed under composiion a c b d 22 Marices Translaion Wha pes of ransformaions can no be represened ih a 22 mari? Eample of ranslaion Homogeneous Coordinaes 2D Translaion? NO! Onl linear 2D ransformaions can be represened ih a 222 mari = 2 =

6 ffine Transformaions ffine ransformaions are combinaions of Linear ransformaions, and Translaions Properies of affine ransformaions: Origin does no necessaril map o origin Lines map o lines Parallel lines remain parallel Raios are preserved Closed under composiion a Models change of basis d 0 b e 0 c f 1 Projecive Transformaions Projecive ransformaions ffine ransformaions, and Projecive arps Properies of projecive ransformaions: i Origin does no necessaril map o origin Lines map o lines Parallel lines do no necessaril remain parallel Raios are no preserved Closed under composiion a b c Models change of basis d g e h f i Image arping Given a coordinae ransform = T( and a source image I(, ho do e compue a ransformed image I ( =I(T(? Forard arping Send each piel I( o is corresponding locaion = T( in I ( T( I( I ( T( I( I (

7 Forard arping farp(i, I, T { for (=0; <I.heigh; ++ for (=0; <I.idh; ++ { (, =T(,; I (, =I(,; I( ; } } I I T Forard arping Send each piel I( o is corresponding locaion = T( in I ( Wha if piel lands beeen o piels? Will be here holes? nser: add conribuion o several piels, normalize laer (splaing h( f( g( Forard arping Inverse arping farp(i, I, T { for (=0; <I.heigh; ++ for (=0; <I.idh; ++ { (, =T(,; Splaing(I,,,I(,,kernel; I( } } I I T Ge each piel I ( from is corresponding locaion = T -1 ( in I( T -1 ( I( I (

8 Inverse arping iarp(i, I, T { for (=0; <I.heigh; ++ for (=0; <I.idh; ++ { (,=T -1 (, ; I (, =I(,; I( ; } } I T -1 I Inverse arping Ge each piel I ( from is corresponding locaion = T -1 ( in I( Wha if piel comes from beeen o piels? nser: resample color value from inerpolaed (prefilered source image f( g( Inverse arping iarp(i, I, T { for (=0; <I.heigh; ++ for (=0; <I.idh; ++ { (,=T -1 (, ; I (, =Reconsruc(I,,,kernel; } } I T -1 I Sampling band limied

9 Reconsrucion Reconsrucion Reconsrucion generaes an approimaion o he original funcion. Error is called aliasing. sampling sample value reconsrucion The reconsruced funcion is obained b inerpolaing among he samples in some manner sample posiion i Reconsrucion Compued eighed sum of piel neighborhood; oupu is eighed average of inpu, here eighs are normalized values of filer kernel k k ( q i i qi p k( q i i color=0; eighs=0; p idh for all q s dis < idh d = dis(p, q; = kernel(d; d color += *q.color; q eighs += ; p.color = color/eighs; Reconsrucion (inerpolaion Possible reconsrucion filers (kernels: neares neighbor bilinear bicubic sinc (opimal reconsrucion

10 Bilinear inerpolaion (riangle filer simple mehod for resampling images Non-parameric image arping Specif a more deailed arp funcion Splines, meshes, opical flo (per-piel moion Non-parameric image arping Non-parameric image arping Mappings implied b correspondences Inverse arping P B C B C P B B C C Barcenric coordinae? P P P

11 Barcenric coordinaes P Non-parameric image arping C B P C B C B P Barcenric coordinae C B P C B Non-parameric image arping 2 ( r e r Gaussian i X X P k K P i ( 1 radial basis funcion log( ( 2 r r r hin plae spline i K Demo hp://.colonize.com/arp/arp04-2.php W i i f l i f i id Warping is a useful operaion for mosaics, video maching, vie inerpolaion and so on.

12 Image morphing The goal is o snhesize a fluid ransformaion from one image o anoher. Image morphing Cross dissolving is a common ransiion beeen cus, bu i is no good for morphing because of he ghosing effecs. image #1 dissolving i image #2 rifacs of cross-dissolving Image morphing Wh ghosing? Morphing = arping + cross-dissolving i shape (geomeric color (phoomeric hp://.salavon.com/

13 Image morphing Morphing sequence image #1 cross-dissolving image #2 arp morphing arp Face averaging b morphing Image morphing creae a morphing sequence: for each ime 1. Creae an inermediae arping field (b inerpolaion 2. Warp boh images oards i 3. Cross-dissolve he colors in he nel arped images average faces =0 =0.33 =1

14 n ideal eample (in 2004 n ideal eample =0 morphing =0.25 =0.75 =0.5 =1 =0 middle face (=0.5 =1 Warp specificaion (mesh arping Ho can e specif he arp? 1. Specif corresponding spline conrol poins inerpolae o a complee arping funcion Warp specificaion Ho can e specif he arp 2. Specif corresponding poins inerpolae o a complee arping funcion eas o implemen, bu less epressive

15 Soluion: conver o mesh arping Warp specificaion (field arping Ho can e specif he arp? 3. Specif corresponding vecors inerpolae o a complee arping funcion The Beier & Neel lgorihm 1. Define a riangular mesh over he poins Same mesh in boh images! No e have riangle-o-riangle i l correspondences 2. Warp each riangle separael from source o desinaion Ho do e arp a riangle? 3 poins = affine arp! Jus like eure mapping Beier&Neel (SIGGRPH 1992 Single line-pair PQ o P Q : lgorihm (single line-pair For each X in he desinaion image: 1. Find he corresponding u,v 2. Find X in he source image for ha u,v 3. desinaionimage(x = sourceimage(x Eamples: ffine ransformaion

16 Muliple Lines Full lgorihm D i X i X i lengh = lengh of he line segmen, dis = disance o line segmen The influence of a, p, b. The same as he average of X i Resuling arp Comparison o mesh morphing Pros: more epressive Cons: speed and conrol

17 Warp inerpolaion nimaion Ho do e creae an inermediae arp a ime? linear inerpolaion for line end-poins Bu, a line roaing i 180 degrees ill become 0 lengh in he middle One soluion is o inerpolae line mid-poin and orienaion angle =0 =1 nimaed sequences Resuls Specif keframes and inerpolae he lines for he inbeeen frames Require a lo of eaking Michael Jackson s MTV Black or Whie

18 Muli-source morphing Muli-source morphing References Thaddeus Beier, Shan Neel, Feaure-Based Image Meamorphosis, SIGGRPH 1992, pp Delef Ruprech, Heinrich Muller, Image Warping ih Scaered Daa Inerpolaion, IEEE Compuer Graphics and pplicaions, March 1995, pp Seung-Yong Lee, Kung-Yong Cha, Sung Yong Shin, Image Meamorphosis Using Snakes and Free-Form Deformaions, SIGGRPH Seungong Lee, Wolberg, G., Sung Yong Shin, Polmorph: morphing among muliple images, IEEE Compuer Graphics and pplicaions, Vol. 18, No. 1, 1998, pp Peinsheng Gao, Thomas Sederberg, ork minimizaion approach o image morphing, The Visual Compuer, 1998, pp George Wolberg, Image morphing: a surve, The Visual Compuer, 1998, pp

Image warping Li Zhang CS559

Image warping Li Zhang CS559 Wha is an image Image arping Li Zhang S559 We can hink of an image as a funcion, f: R 2 R: f(, ) gives he inensi a posiion (, ) defined over a recangle, ih a finie range: f: [a,b][c,d] [,] f Slides solen