Image warping and stitching May 5 th, 2015

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1 Image warpng and sttchng Ma 5 th, 2015 Yong Jae Lee UC Davs PS2 due net Frda Announcements 2 Last tme Interactve segmentaton Feature-based algnment 2D transformatons Affne ft RANSAC 3 1

2 Algnment problem In algnment, we wll ft the parameters of some transformaton accordng to a set of matchng feature pars ( correspondences ). ' T Slde credt: Adapted b Dev Parkh from Krsten Grauman 4 Man questons T Algnment: Gven two mages, what s the transformaton between them? T Warpng: Gven a source mage and a transformaton, what does the transformed output look lke? 5 Motvaton for feature-based algnment: Recognton 6 Fgures from Davd Lowe 2

3 Motvaton for feature-based algnment: Medcal mage regstraton 7 Motvaton for feature-based algnment: Image mosacs 8 Image from Parametrc (global) warpng Eamples of parametrc warps: translaton rotaton aspect affne perspectve 9 Source: Alosha Efros 3

4 Parametrc (global) warpng p = (,) T p = (, ) Transformaton T s a coordnate-changng machne: p = T(p) What does t mean that T s global? Is the same for an pont p can be descrbed b just a few numbers (parameters) Let s represent T as a matr: p = Mp ' M ' 10 Source: Alosha Efros Homogeneous coordnates To convert to homogeneous coordnates: homogeneous mage coordnates Convertng from homogeneous coordnates 11 Fttng an affne transformaton Assumng we know the correspondences, how do we get the transformaton? (, ), ) ( m1 m3 m2 t1 m 4 t2 12 4

5 5 2D Affne Transformatons Affne transformatons are combnatons of Lnear transformatons, and Translatons Parallel lnes reman parallel w f e d c b a w ' ' ' 13 Projectve Transformatons Projectve transformatons: Affne transformatons, and Projectve warps Parallel lnes do not necessarl reman parallel w h g f e d c b a w ' ' ' 14 Fttng an affne transformaton Assumng we know the correspondences, how do we get the transformaton? ), ( ), ( t t m m m m t t m m m m

6 RANSAC loop: RANSAC: General form 1. Randoml select a seed group of ponts on whch to base transformaton estmate (e.g., a group of matches) 2. Compute transformaton from seed group 3. Fnd nlers to ths transformaton 4. If the number of nlers s suffcentl large, re-compute estmate of transformaton on all of the nlers Keep the transformaton wth the largest number of nlers 16 RANSAC eample: Translaton Putatve matches 17 Source: Rck Szelsk RANSAC eample: Translaton Select one match, count nlers 18 6

7 RANSAC eample: Translaton Select one match, count nlers 19 RANSAC eample: Translaton Fnd average translaton vector 20 RANSAC pros and cons Pros Smple and general Applcable to man dfferent problems Often works well n practce Cons Lots of parameters to tune Doesn t work well for low nler ratos (too man teratons, or can fal completel) Can t alwas get a good ntalzaton of the model based on the mnmum number of samples 21 Lana Lazebnk 7

8 Toda Image mosacs Fttng a 2D transformaton Homograph 2D mage warpng Computng an mage mosac 22 HP frames commercals Qk 23 Mosacs... mage from S. Setz Obtan a wder angle vew b combnng multple mages. 24 8

9 Panoramc Photos are old Sdne, 1875 Berut, late 1800 s Slde credt: James Has How to sttch together a panorama (a.k.a. mosac)? Basc Procedure Take a sequence of mages from the same poston Rotate the camera about ts optcal center Compute transformaton between second mage and frst Transform the second mage to overlap wth the frst Blend the two together to create a mosac (If there are more mages, repeat) but wat, wh should ths work at all? What about the 3D geometr of the scene? Wh aren t we usng t? 26 Source: Steve Setz Pnhole camera Pnhole camera s a smple model to appromate magng process, perspectve projecton. Image plane Vrtual mage pnhole If we treat pnhole as a pont, onl one ra from an gven pont can enter the camera. Fg from Forsth and Ponce 27 9

10 Mosacs: generatng snthetc vews real camera snthetc camera Can generate an snthetc camera vew as long as t has the same center of projecton! 28 Source: Alosha Efros Mosacs... mage from S. Setz Obtan a wder angle vew b combnng multple mages. 29 Image reprojecton mosac PP The mosac has a natural nterpretaton n 3D The mages are reprojected onto a common plane The mosac s formed on ths plane Mosac s a snthetc wde-angle camera 30 Source: Steve Setz 10

11 Image reprojecton Basc queston How to relate two mages from the same camera center? how to map a pel from PP1 to PP2 Answer Cast a ra through each pel n PP1 Draw the pel where that ra ntersects PP2 Observaton: Rather than thnkng of ths as a 3D reprojecton, thnk of t as a 2D mage warp from one mage to another. PP1 PP2 31 Source: Alosha Efros Image reprojecton: Homograph A projectve transform s a mappng between an two PPs wth the same center of projecton rectangle should map to arbtrar quadrlateral parallel lnes aren t preserved but must preserve straght lnes called Homograph PP2 w' * * w' * * w * * p H * * * 1 p PP1 32 Source: Alosha Efros The projectve plane Wh do we need homogeneous coordnates? represent ponts at nfnt, homographes, perspectve projecton, mult-vew relatonshps What s the geometrc ntuton? a pont n the mage s a ra n projectve space - (0,0,0) -z (,,1) (s,s,s) mage plane Each pont (,) on the plane s represented b a ra (s,s,s) all ponts on the ra are equvalent: (,, 1) (s, s, s) 11

12 Homograph 1, 1 1, 1 2, 2 2, 2,, n n n n To compute the homograph gven pars of correspondng ponts n the mages, we need to set up an equaton where the parameters of H are the unknowns 34 Solvng for homographes Upto a scale factor. Constrant Frobenus norm of H to be 1. Problem to be solved: p = Hp w' h00 h01 w' h h w h20 h21 h02 h 12 h22 1 mn Ah b 2 s.t. h 1 where vector of unknowns h = [h 00,h 01,h 02,h 10,h 11,h 12,h 20,h 21,h 22 ] T 2 Adapted from Dev Parkh 35 Solvng for homographes w ' h w ' h w h h h h h02 h 12 h

13 Solvng for homographes A h 0 2n 9 9 2n Defnes a least squares problem: Snce h s onl defned up to scale, solve for unt vector ĥ Soluton: ĥ = egenvector of A T A wth smallest egenvalue Works wth 4 or more ponts (. e., h 2 1), Homograph w w w, w, To appl a gven homograph H Compute p = Hp (regular matr multpl) Convert p from homogeneous to mage coordnates w' * w' * w * p * * * * * * 1 38 H p Toda RANSAC for robust fttng Lnes, translaton Image mosacs Fttng a 2D transformaton Homograph 2D mage warpng Computng an mage mosac 39 13

14 Image warpng T(,) f(,) g(, ) Gven a coordnate transform and a source mage f(,), how do we compute a transformed mage g(, ) = f(t(,))? Slde from Alosha Efros, CMU 40 Forward warpng T(,) f(,) g(, ) Send each pel f(,) to ts correspondng locaton (, ) = T(,) n the second mage Q: what f pel lands between two pels? Slde from Alosha Efros, CMU 41 Forward warpng T(,) f(,) g(, ) Send each pel f(,) to ts correspondng locaton (, ) = T(,) n the second mage Q: what f pel lands between two pels? A: dstrbute color among neghborng pels (, ) Known as splattng Slde from Alosha Efros, CMU 42 14

15 Inverse warpng T -1 (,) f(,) g(, ) Get each pel g(, ) from ts correspondng locaton (,) =T -1 (, ) n the frst mage Q: what f pel comes from between two pels? Slde from Alosha Efros, CMU 43 Inverse warpng T -1 (,) f(,) g(, ) Get each pel g(, ) from ts correspondng locaton (,) =T -1 (, ) n the frst mage Q: what f pel comes from between two pels? A: Interpolate color value from neghbors nearest neghbor, blnear >> help nterp2 44 Slde from Alosha Efros, CMU Blnear nterpolaton Samplng at f(,): Slde from Alosha Efros, CMU 45 15

16 Recap: How to sttch together a panorama (a.k.a. mosac)? Basc Procedure Take a sequence of mages from the same poston Rotate the camera about ts optcal center Compute transformaton (homograph) between second mage and frst usng correspondng ponts. Transform the second mage to overlap wth the frst. Blend the two together to create a mosac. (If there are more mages, repeat) 46 Source: Steve Setz Image warpng wth homographes mage plane n front black area where no pel maps to mage plane below 47 Source: Steve Setz Image rectfcaton p p 48 16

17 Analsng patterns and shapes What s the shape of the b/w floor pattern? The floor (enlarged) Automatcall Slde from Antono Crmns 49 rectfed floor Analsng patterns and shapes Automatc rectfcaton From Martn Kemp The Scence of Art (manual reconstructon) Slde from Antono Crmns 50 Analsng patterns and shapes What s the (complcated) shape of the floor pattern? Automatcall rectfed floor St. Luc Altarpece, D. Venezano Slde from Crmns 51 17

18 Analsng patterns and shapes Automatc rectfcaton From Martn Kemp, The Scence of Art (manual reconstructon) Slde from Crmns 52 Analzng patterns and shapes The Ambassadors b Hans Holben the Younger, 1533 Julan Beever: Manual Homographes 18

19 Changng camera center Does t stll work? snthetc PP PP1 PP2 55 Source: Alosha Efros Recall: same camera center real camera snthetc camera Can generate snthetc camera vew as long as t has the same center of projecton. 56 Source: Alosha Efros Or: Planar scene (or far awa) PP3 PP1 PP2 PP3 s a projecton plane of both centers of projecton, so we are OK! Ths s how bg aeral photographs are made 57 Source: Alosha Efros 19

20 58 RANSAC for estmatng homograph RANSAC loop: 1. Select four feature pars (at random) 2. Compute homograph H (eact) 3. Compute nlers where SSD(p, Hp )< ε 4. Keep largest set of nlers 5. Re-compute least-squares H estmate on all of the nlers 59 Slde credt: Steve Setz Robust feature-based algnment 60 Source: L. Lazebnk 20

21 Robust feature-based algnment Etract features 61 Source: L. Lazebnk Robust feature-based algnment Etract features Compute putatve matches 62 Source: L. Lazebnk Robust feature-based algnment Etract features Compute putatve matches Loop: Hpothesze transformaton T (small group of putatve matches that are related b T) 63 Source: L. Lazebnk 21

22 Robust feature-based algnment Etract features Compute putatve matches Loop: Hpothesze transformaton T (small group of putatve matches that are related b T) Verf transformaton (search for other 64 matches consstent wth T) Source: L. Lazebnk Robust feature-based algnment Etract features Compute putatve matches Loop: Hpothesze transformaton T (small group of putatve matches that are related b T) Verf transformaton (search for other 65 matches consstent wth T) Source: L. Lazebnk Summar: algnment & warpng Wrte 2d transformatons as matr-vector multplcaton (ncludng translaton when we use homogeneous coordnates) Fttng transformatons: solve for unknown parameters gven correspondng ponts from two vews (affne, projectve (homograph)). Perform mage warpng (forward, nverse) Mosacs: uses homograph and mage warpng to merge vews taken from same center of projecton

23 Net tme: whch features should we match? 67 Questons? 68 23

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