Geometric Transformations and Multiple Views

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1 CS 2770: Computer Vson Geometrc Transformatons and Multple Vews Prof. Adrana Kovaska Unverst of Pttsburg Februar 8, 208

2 W multple vews? Structure and dept are nerentl ambguous from sngle vews. Multple vews elp us to perceve 3d sape and dept. Krsten Grauman, mages from Svetlana Lazebnk

3 Algnment problem We prevousl dscussed ow to matc features across mages, of te same or dfferent objects Now let s focus on te case of two mages of te same object (e.g. and ) Wat transformaton relates and? In algnment, we wll ft te parameters of some transformaton accordng to a set of matcng feature pars ( correspondences ). T Adapted from Krsten Grauman and Derek Hoem

4 Motvaton: Image mosacs Krsten Grauman Image from ttp://grapcs.cs.cmu.edu/courses/5-463/200_fall/

5 Frst, wat are te correspondences? Mn dst = matc? Compare content n local patces, fnd best matces. Scan wt template formed from a pont n, and compute e.g. Eucldean dstance between pel ntenstes n te patc Compare SIFT features Adapted from Krsten Grauman

6 Second, wat are te transformatons? Eamples of transformatons: translate rotate cange aspect rato squs cange perspectve Adapted from Alosa Efros

7 Parametrc (global) warpng T p = (,) p = (, ) Transformaton T s a coordnate-cangng macne: p = T(p) Wat does t mean tat T s global? It s te same for an pont p It can be descrbed b just a few numbers (parameters) Let s represent T as a matr: p = Mp Alosa Efros M

8 Scalng Scalng a coordnate means multplng eac of ts components b a scalar Unform scalng means ts scalar s te same for all components: (2, ) 2 (4, 2) Adapted from Alosa Efros

9 Scalng Non-unform scalng: dfferent scalars per component (2, ) X 2, Y 0.5 (4, 0.5) Adapted from Alosa Efros

10 Scalng Scalng operaton: Or, n matr form: b a q p n m scalng matr S Adapted from Alosa Efros b a 0 0

11 2D Lnear transformatons a c b d Onl lnear 2D transformatons can be represented wt a 22 matr. Lnear transformatons are combnatons of Scale, Rotaton, Sear, and Mrror Alosa Efros

12 2D Rotate around (0,0)? (see dden slde) * cos * sn * sn * cos cos sn sn cos 2D Sear? s s * * s s 2D Scalng? s s * * s s 0 0 Modfed from Alosa Efros Fg. from ttps:// Wat transforms can we wrte w/ 22 matr?

13 Wat transforms can we wrte w/ 22 matr? 2D Mrror about Y as? 0 0 2D Mrror over (0,0)? 0 0 2D Translaton? t t CAN T DO! Alosa Efros

14 Homogeneous coordnates To convert to omogeneous coordnates: omogeneous mage coordnates Convertng from omogeneous coordnates Krsten Grauman

15 t t Translaton t t t t t = 2 t = Homogeneous Coordnates Adapted from Alosa Efros (2, ) (4, 2)

16 2D affne transformatons Affne transformatons are combnatons of Lnear transformatons, and Translatons Maps lnes to lnes, parallel lnes reman parallel w f e d c b a w 0 0 Adapted from Alosa Efros

17 Fttng an affne transformaton Assumng we know te correspondences, ow do we get te transformaton? ), ( ), ( t t m m m m t t m m m m Alosa Efros

18 Fttng an affne transformaton How man matces (correspondence pars) do we need to solve for te transformaton parameters? Once we ave solved for te parameters, ow do we compute gven? t t m m m m ), ( new new Adapted from Krsten Grauman ), ( new new

19 Detour: Kepont matcng for searc A Quer A 2 A 3 In database. Fnd a set of dstnctve keponts 2. Defne a regon around eac kepont (wndow) f A f B 3. Compute a local descrptor from te regon 4. Matc descrptors d( f A, f B ) T Adapted from K. Grauman, B. Lebe

20 Detour: solvng for translaton wt outlers A A 2 A 3 B B 2 B 3 Gven matced ponts n {A} and {B}, estmate te translaton of te object A A B B t t Derek Hoem

21 Detour: solvng for translaton wt outlers A A 5 B 4 A 2 A B 3 (t, t ) A 4 B 2 B 3 B 5 Problem: outlers RANSAC soluton. Sample a set of matcng ponts ( par) 2. Solve for transformaton parameters 3. Repeat N tmes B B A A t t Adapted from Derek Hoem

22 Detour: solvng for translaton wt outlers B 4 A B 5 B 6 A 2 A B 3 (t, t ) A 4 B 2 B 3 A 5 A 6 Problem: outlers, multple objects Houg transform soluton. Intalze a grd of parameter values 2. Eac matced par casts a vote for consstent values 3. Fnd te parameters wt te most votes B B A A t t Adapted from Derek Hoem

23 Projectve transformatons Projectve transformatons: Affne transformatons, and Projectve warps Parallel lnes do not necessarl reman parallel w g f e d c b a w Krsten Grauman

24 Projectve transformatons A projectve transformaton s a mappng between an two projectve planes wt te same center of projecton Also called Homograp PP2 PP * * * * * * * * * w w w H p p Adapted from Alosa Efros

25 Image mosacs: Goals... mage from S. Setz Obtan a wder angle vew b combnng multple mages. Krsten Grauman

26 Image mosacs: Camera setup Two mages wt camera rotaton but no translaton (0, 0) (0, 0) (50, 70) (50, 20) Adapted from Derek Hoem Camera Center

27 Image mosacs: Man 2D vews, one 3D object Steve Setz mosac plane Te mosac as a natural nterpretaton n 3D Te mages are reprojected onto a common plane Te mosac s formed on ts plane Mosac s a sntetc wde-angle camera

28 How to sttc togeter panorama (mosac)? Basc Procedure Take a sequence of mages from te same poston Rotate te camera about ts optcal center Compute te omograp (transformaton) between frst and second mage Transform te second mage to overlap wt te frst Blend te two togeter to create a mosac (If tere are more mages, repeat) Adapted from Steve Setz

29 Computng te omograp,, 2, 2, 2 2,, n n n n To compute te omograp gven pars of correspondng ponts n te mages, we need to set up an equaton were te parameters of H are te unknowns Krsten Grauman

30 Computng te omograp Assume we ave four matced ponts: How do we compute omograp H? w w w p H Adapted from Derek Hoem, Krsten Grauman p =Hp A g f e d c b a w w w Dervaton: ttp:// Can set scale factor 9 =. So, tere are 8 unknowns. Need at least 8 eqs, but te more te better p DEMO

31 How to sttc togeter panorama (mosac)? Basc Procedure Take a sequence of mages from te same poston Rotate te camera about ts optcal center Compute te omograp (transformaton) between frst and second mage Transform te second mage to overlap wt te frst Blend te two togeter to create a mosac (If tere are more mages, repeat) Adapted from Steve Setz

32 * * * * * * * * * w w w H p p w w w w,,, To appl a gven omograp H Compute p = Hp (regular matr multpl) Convert p from omogeneous to mage coordnates Modfed from Krsten Grauman Transformng te second mage Image 2 canvas Image

33 Transformng te second mage Image Image 2 canvas H(,) f(,) g(, ) Forward warpng: Send eac pel f(,) to ts correspondng locaton (, ) = H(,) n te rgt mage Modfed from Alosa Efros

34 Transformng te second mage H(,) f(,) g(, ) Forward warpng: Send eac pel f(,) to ts correspondng locaton (, ) = H(,) n te rgt mage Q: wat f pel lands between two pels? A: dstrbute color among negborng pels (, ) Alosa Efros

35 Transformng te second mage Image Image 2 canvas H - (,) f(,) g(, ) Inverse warpng: Get eac pel g(, ) from ts correspondng locaton (,) = H - (, ) n te left mage Modfed from Alosa Efros

36 Transformng te second mage H - (,) f(,) g(, ) Inverse warpng: Get eac pel g(, ) from ts correspondng locaton (,) = H - (, ) n te left mage Q: wat f pel comes from between two pels? A: nterpolate color value from negbors Alosa Efros

37 Homograp eample: Image rectfcaton p p To unwarp (rectf) an mage solve for omograp H gven p and p : p =Hp Derek Hoem

38 Summar of affne/projectve transforms Wrte 2d transformatons as matr-vector multplcaton (ncludng translaton wen we use omogeneous coordnates) Fttng transformatons: solve for unknown parameters gven correspondng ponts from two vews lnear, affne, projectve (omograp) Mosacs: uses omograp and mage warpng to merge vews taken from same center of projecton Perform mage warpng (forward, nverse) Adapted from Krsten Grauman

39 Net: Stereo vson Homograp: Same camera center, but camera rotates Stereo vson: Camera center s not te same (we ave multple cameras) Eppolar geometr Relates cameras from two postons/cameras Stereo dept estmaton Recover dept from dspartes between two mages Adapted from Derek Hoem

40 Stereo potograp and stereo vewers Take two pctures of te same subject from two slgtl dfferent vewponts and dspla so tat eac ee sees onl one of te mages. Invented b Sr Carles Weatstone, 838 Image from fser-prce.com Krsten Grauman

41 Dept from stereo for computers Two cameras, smultaneous vews Sngle movng camera and statc scene Krsten Grauman

42 Dept from stereo Goal: recover dept b fndng mage coordnate tat corresponds to X X z f f C Baselne C B Derek Hoem

43 Dept from stereo Goal: recover dept b fndng mage coordnate tat corresponds to Sub-Problems. Calbraton: How do we recover te relaton of te cameras (f not alread known)? 2. Correspondence: How do we searc for te matcng pont? X Derek Hoem

44 Geometr for a smple stereo sstem Assume parallel optcal aes, known camera parameters (.e., calbrated cameras). Wat s epresson for Z? Smlar trangles (p l, P, p r ) and (O l, P, O r ): T l Z f r T Z Dept s nversel proportonal to dspart. dept dspart Z f T r l Adapted from Krsten Grauman

45 Dept from dspart We ave two mages from dfferent cameras. If we could fnd te correspondng ponts n two mages, we could estmate relatve dept How do we matc a pont n te frst mage to a pont n te second effcentl? mage I(,) Dspart map D(,) mage I (, ) Krsten Grauman

46 Stereo correspondence constrants Gven p n left mage, were can correspondng pont p be? Krsten Grauman

47 Eppolar constrant world pont Geometr of two vews constrans were te correspondng pel for some mage pont n te frst vew must occur n te second vew. It must be on te lne were () te plane connectng te world pont and optcal centers, and (2) te mage plane, ntersect. Potental matces for p ave to le on te correspondng lne l. Potental matces for p ave to le on te correspondng lne l. Adapted from Krsten Grauman, Derek Hoem

48 Eppolar geometr: notaton P p p Adapted from Derek Hoem Baselne lne connectng te two camera centers Eppoles = ntersectons of baselne wt mage planes = projectons of te oter camera center Eppolar Plane plane contanng baselne Eppolar Lnes - ntersectons of eppolar plane wt mage planes (alwas come n correspondng pars)

49 Eppolar constrant Te eppolar constrant s useful because t reduces te correspondence problem to a D searc along an eppolar lne. Krsten Grauman, mage from Andrew Zsserman

50 Stereo geometr, wt calbrated cameras If te stereo rg s calbrated, we know ow to rotate and translate camera reference frame to get to camera reference frame 2 Rotaton: 33 matr R; translaton: 3 vector T. X RX T (See dden sldes for ow we get to te net slde.) Adapted from Krsten Grauman

51 X X Essental matr T RX 0 [T ] RX 0 Let E [T ] R XEX X T EX 0 E s called te essental matr, and t relates correspondng mage ponts between bot cameras, gven te rotaton and translaton. Before we sad: If we observe a pont n one mage, ts poston n oter mage s constraned to le on lne defned b above. It turns out tat: E T s te eppolar lne l troug n te second mage, correspondng to. E s te eppolar lne l troug n te frst mage, correspondng to. Adapted from Krsten Grauman, Derek Hoem

52 Essental matr eample: parallel cameras R I p [,, f ] T E [ d,0,0] [ T ]R d 0 d 0 p [,, f ] p Ep 0 For te parallel cameras, mage of an pont must le on same orzontal lne n eac mage plane. Krsten Grauman

53 mage I(,) Dspart map D(,) mage I (, ) (, )=(+D(,),) Adapted from Krsten Grauman

54 Basc stereo matcng algortm For eac pel n te frst mage Fnd correspondng eppolar scanlne n te rgt mage Searc along eppolar lne and pck te best matc : slde a wndow along te rgt scanlne and compute Eucldean dstance between contents of tat wndow wt te reference wndow n te left mage; take te wndow correspondng to te mnmum as te matc Compute dspart - and set dept() = f*t/(- ) Adapted from Derek Hoem

55 Results wt wndow searc Data Left mage Rgt mage Wndow-based matcng Predcted dept Ground trut Ground trut Derek Hoem

56 Summar of stereo vson Eppolar geometr Eppoles are ntersecton of baselne wt mage planes Matcng pont n second mage s on a lne passng troug ts eppole Eppolar constrant lmts were ponts from one vew wll be maged n te oter, wc makes searc for correspondences qucker Essental matr E maps from a pont n one mage to a lne (ts eppolar lne) n te oter Stereo dept estmaton Fnd correspondng ponts along eppolar scanlne Estmate dspart (dept s nverse to dspart) Adapted from Krsten Grauman and Derek Hoem

57 Projectve structure from moton Gven: m mages of n fed 3D ponts j = P X j, =,, m, j =,, n Problem: estmate m projecton matrces P and n 3D ponts X j from te mn correspondng 2D ponts j X j j 3j P 2j Svetlana Lazebnk P 2 P 3

58 Poto toursm Noa Snavel, Steven M. Setz, Rcard Szelsk, "Poto toursm: Eplorng poto collectons n 3D," SIGGRAPH 2006 ttp://pototour.cs.wasngton.edu/

59 3D from multple mages Sameer Agarwala, Noa Snavel, Ian Smon, Steven M. Setz, Rcard Szelsk, "Buldng Rome n a Da," ICCV 2009

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