Stereo Vision (Correspondences)

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1 Stereo Vision (Correspondences) EECS Fall 2014! Foundations of Computer Vision!! Instructor: Jason Corso (jjcorso)! web.eecs.umich.edu/~jjcorso/t/598f14!! Readings: FP 7; SZ 11; TV 7! Date: 10/27/14!! Materials on these slides have come from many sources in addition to myself; individual slides reference specific sources. The primary source for these slides is Silvio Savarese.!

2 Plan 2 Stereo vision! Rectification! Correspondence problem! Active stereo vision systems!

3 Stereo Vision P! p! p! O 1! O 2! Goal: estimate the position of P given the observation of! P from two view points! Assumptions: known camera parameters and position (K, R, T)!

4 Stereo Vision P! p! p! O 1! O 2! Subgoals:! - Solve the correspondence problem! - Use corresponding observations to triangulate!

5 The Correspondence Problem P! p! p! O 1! O 2! Given a point in 3d, discover corresponding observations in left and right images!

6 Triangulation P! p! p! O 1! O 2! Intersecting the two lines of sight gives rise to P!

7 Epipolar Geometry P! p! p! e 1! e 2! Epipolar Plane! Epipoles e 1, e 2! Baseline! Epipolar Lines! O 1! O 2! = intersections of baseline with image planes! = projections of the other camera center!!

8 The Epipolar Constraint p T E p = P! p 1! l 1! l 2! e 1! e 2! p 2! O 1! O 2! E p 2 is the epipolar line associated with p 2 (l 1 = E p 2 )! E T p 1 is the epipolar line associated with p 1 (l 2 = E T p 1 )! E e 2 = 0 and E T e 1 = 0! E is 3x3 matrix; 5 DOF! E is singular (rank two)!!

9 Parallel Image Planes P! p! p! O! O! When views are parallel both correspondence and triangulation become much easier!!

10 Parallel Image Planes P! v! v! e 2! p! p! e 2! y! u! u! z! K! K! O x!! O! Parallel epipolar lines! Epipoles at infinity! v = v! Rectification: making two images parallel!

11 Parallel Image Planes P! v! v! e 2! p! p! e 2! y! u! u! z! K! K! O x!! O! K 1 =K 2 = known! x parallel to O 1 O 2! E =[ t ] R

12 b a b a ] [ = = z y x x y x z y z b b b a a a a a a Cross product as matrix multiplication 12

13 Parallel Image Planes P! v! v! e 2! p! p! e 2! y! u! u! z! K! K! O x!! O! K 1 =K 2 = known! x parallel to O 1 O 2! E = [ t = ] R 0 0 T 0 T 0 v = v?!

14 Parallel Image Planes P! v! v! e 2! p! p! e 2! y! u! u! K! K! O x!! O! uʹ 1 0 Tv ʹ ( u v 1) 0 0 T vʹ = 0 ( u v 1) T = 0 Tv = Tv T z! 0 0 p T E pʹ = 0 ʹ

15 Making Image Planes Parallel P! H! p! p! O! O! GOAL: Estimate the perspective transformation H!! that makes the images parallel!

16 Making the Image Planes Parallel The Projective Transformation H! x i xʹ i x ʹ = H x i i Now we don t have the destination image L!

17 Making the Image Planes Parallel P! H! p! p! O! O! GOAL: Estimate the perspective transformation H that makes images parallel! Impose v =v! This leaves degrees of freedom for determining H! If an inappropriate H is chosen, severe projective distortions on image take place! We impose a number of restrictions while computing H!

18 Making the Image Planes Parallel P K[ I 0] P P! P Kʹ [ R T] P p! p! e! e! O! R,T! O! 0. Compute epipoles! e = e e 1] T = T [ 1 2 K R T eʹ = Kʹ T

19 Making the Image Planes Parallel P! p! p! e! e! O! O! 1. Map e to the x-axis at location [1,0,1] T (normalization)! e = [e T 1 e2 1] [ 1 0 1] T H = R T 1 H H

20 Making the Image Planes Parallel P! p! p! e! 2. Send epipole to infinity:! O! O! H 2 1 = e = [ 1 0 1] T [ 1 0 0] T Minimizes the distortion in a! neighborhood (approximates id. mapping)!

21 Making the Image Planes Parallel P! H = H 2 H 1! e! p! p! e! O! O! 3. Define: H = H 2 H 1! 4. Align epipolar lines!

22 Projective Transformation of a Line H = A v t b l! H T l

23 Making the Image Planes Parallel P! H = H 2 H 1! e! p! l lʹ p! e! O! O! 3. Define: H = H 2 H 1! Hʹ T lʹ = H T l 4. Align epipolar lines! [HZ] Chapters: 11 (sec )! These are called matched pair of transformation!

24 Making the Image Planes Parallel H! Courtesy figure S. Lazebnik!

25 Why is Rectification Useful? Makes the correspondence problem easier! Makes triangulation easy!

26 Application: View Morphing S. M. Seitz and C. R. Dyer, Proc. SIGGRAPH 96, 1996, 21-30!

27 Morphing Without Using Geometry

28 Rectification

29

30 Stereo Vision P! p! p! O 1! O 2! Subgoals:! - Solve the correspondence problem! - Use corresponding observations to triangulate!

31 Computing Depth P! v! v! e 2! p! p! e 2! y! u! u! z! K! K! O x!! O!

32 Computing Depth P z x x f f O Baseline B! O x xʹ = B f z = disparity! Note: Disparity is inversely proportional to depth!

33 Computing Depth X 33 Z! u 2! u 1! c 2! x x 2! 1! c 1! f! T! O 2! O 1!

34 Disparity Maps x! x! Stereo pair! Disparity map / depth map! Disparity map with occlusions!

35 Stereo Vision P! p! p! O 1! O 2! Subgoals:! - Solve the correspondence problem! - Use corresponding observations to triangulate!

36 The Correspondence Problem P! p! p! O 1! O 2! Given a point in 3d, discover corresponding observations! in left and right images [also called binocular fusion problem]!

37 The Correspondence Problem A Cooperative Model (Marr and Poggio, 1976)! Correlation Methods (1970--)! Multi-Scale Edge Matching (Marr, Poggio and Grimson, )!! [FP] Chapters: 8!!

38 Correlation Methods (1970--) p! p! 210! 195! 150! 210!

39 Correlation Methods (1970--) p! Pick up a window around p(u,v)!

40 Correlation Methods (1970--) Pick up a window around p(u,v)! Build vector W! Slide the window along v line in image 2 and compute w! Keep sliding until w w is maximized.!

41 Correlation Methods (1970--) Normalized Correlation; minimize:! (w (w w)(wʹ w)(wʹ w ʹ ) w ʹ )

42 Left! Right! scanline! Credit slide S. Lazebnik! Norm. corr!

43 Correlation Methods Smaller window! - More detail! - More noise! Larger window! - Smoother disparity maps! - Less prone to noise! Window size = 3 Window size = 20 Credit slide S. Lazebnik!

44 Issues Fore shortening effect! O! O! Occlusions! O! O!

45 Issues It is desirable to have small T/z ratio!! Small error in measurements implies large error in estimating depth! O! O!

46 Issues Homogeneous regions! Hard to match pixels in these regions!

47 Issues Repetitive patterns!

48 The Correspondence Problem is Difficult - Occlusions! - Fore shortening! - Baseline trade-off! - Homogeneous regions! - Repetitive patterns! Apply non-local constraints to help enforce the correspondences!

49 Results with window search Data! Ground truth! Window-based matching! Credit slide S. Lazebnik!

50 Improving correspondence: Non-local constraints Uniqueness! For any point in one image, there should be at most one matching point in the other image!

51 Improving correspondence: Non-local constraints Uniqueness! For any point in one image, there should be at most one matching point in the other image! Ordering! Corresponding points should be in the same order in both views! Not always! in presence! of occlusions!! courtesy slide to J. Ponce!

52 Dynamic Programming (Baker and Binford, 1981)! [Uses ordering constraint]! Nodes = matched feature points (e.g., edge points).! Arcs = matched intervals along the epipolar lines.! Arc cost = discrepancy between intervals.!! Find the minimum-cost path going monotonically! down and right from the top-left corner of the! graph to its bottom-right corner.!! courtesy slide to J. Ponce!

53 Dynamic Programming (Baker and Binford, 1981)! courtesy slide to J. Ponce!

54 Dynamic Programming (Ohta and Kanade, 1985)! Reprinted from Stereo by Intra- and Intet-Scanline Search, by Y. Ohta and T. Kanade, IEEE Trans. on Pattern Analysis and Machine! Intelligence, 7(2): (1985) IEEE.!

55 Improving correspondence: Non-local constraints Uniqueness! For any point in one image, there should be at most one matching point in the other image! Ordering! Corresponding points should be in the same order in both views! Smoothness! Disparity is typically a smooth function of x (expect in occluding boundaries)!

56 Smoothness

57 Stereo Matching as Energy Minimization Y. Boykov, O. Veksler, and R. Zabih, Fast Approximate Energy Minimization via Graph Cuts, PAMI 01! I 1 I 2 D W 1 (i ) W 2 (i+d(i )) D(i ) E = α Edata ( I1, I2, D) + β Esmooth ( D) E ( W ( i) W ( i D( i ) 2 + data = 1 2 )) i ( D( i) D( j ) Esmooth = ρ ) neighbors i, j Energy functions of this form can be minimized using graph cuts! Credit slide S. Lazebnik!

58 Stereo Matching as Energy Minimization Y. Boykov, O. Veksler, and R. Zabih, Fast Approximate Energy Minimization via Graph Cuts, PAMI 01! Ground truth! Window-based! matching! Graph cuts!

59 Stereo SDK! stereo vision software development kit.! A. Criminisi, A. Blake and D. Robertson!

60 Application: Foreground/Background Segmentation V. Kolmogorov, A. Criminisi, A. Blake, G. Cross and C. Rother.! Bi-layer segmentation of binocular stereo video CVPR 2005!

61 Application: 3D Urban Scene Modeling 3D Urban Scene Modeling Integrating Recognition and Reconstruction,! N. Cornelis, B. Leibe, K. Cornelis, L. Van Gool, IJCV 08.! Link to movie.!

62 Active Stereo: Point P! p! projector! O 2! Replace one of the two cameras by a projector! - Single camera! - Projector geometry calibrated! - What s the advantage of having the projector?! Correspondence problem solved!

63 Active Stereo: Stripe projector! O! - Projector and camera are parallel! - Correspondence problem solved!!

64 Laser scanning Digital Michelangelo Project Optical triangulation! Project a single stripe of laser light! Scan it across the surface of the object! This is a very precise version of structured light scanning! Source: S. Seitz

65 Laser scanning The Digital Michelangelo Project, Levoy et al.! Source: S. Seitz

66 Active Stereo: Shadows J. Bouguet & P. Perona, 99! Light source! O! - 1 camera,1 light source! - very cheap setup! - calibrated the light source!

68 Active Stereo: Color-Coded Stripes L. Zhang, B. Curless, and S. M. Seitz 2002! S. Rusinkiewicz & Levoy 2002! projector! O! - Dense reconstruction! - Correspondence problem again! - Get around it by using color codes!

69 L. Zhang, B. Curless, and S. M. Seitz. Rapid Shape Acquisition Using Color Structured Light and Multi-pass Dynamic Programming. 3DPVT 2002!

70 L. Zhang, B. Curless, and S. M. Seitz. Rapid Shape Acquisition Using Color Structured Light and Multi-pass Dynamic Programming. 3DPVT 2002! Rapid shape acquisition: Projector + stereo cameras!

71 Next Lecture: Affine Structure from Motion 71 Readings: FP 8.1-2; SZ 7!

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