Image Rectification (Stereo) (New book: 7.2.1, old book: 11.1)

Transcription

1 Image Rectification (Stereo) (New book: 7.2.1, old book: 11.1) Guido Gerig CS 6320 Spring 2013 Credits: Prof. Mubarak Shah, Course notes modified from: Lecture 25

2 Example: converging cameras courtesy of Andrew Zisserman

3 Epipolar Lines in Converging Cameras Epipolar lines all intersect at epipoles.

4 Stereo image rectification In practice, it is convenient if image scanlines are the epipolar lines.

5 Image Rectification If the two cameras are aligned to be coplanar, the search is simplified to one dimension - a horizontal line parallel to the baseline between the cameras

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10 Image Reprojection reproject image planes onto common plane parallel to line between optical centers a homography (3x3 transform) applied to both input images pixel motion is horizontal after this transformation C. Loop and Z. Zhang. Computing Rectifying Homographies for Stereo Vision. IEEE Conf. Computer Vision and Pattern Recognition, Stereo image rectification

11 Stereo image rectification: example Source: Alyosha Efros

12 Stereo image rectification: example Source: Alyosha Efros

13 Stereo image rectification: example Source: Alyosha Efros

14 Image Rectification O O

15 Image Rectification e O O e

16 Image Rectification P e O O e

17 Image Rectification P e p O O p e

18 Image Rectification P e p l O O l p e

19 Image Rectification P e p l O O l p e

20 Image Rectification P e p l O O l p e p p

21 Image Rectification P e p l O O l p e l p p l

22 Image Rectification P Common Image Plane Parallel Epipolar Lines Search Correspondences on scan line Epipoles e p l O O l p e l p p l

23 Image Rectification

24 Image Rectification All epipolar lines are parallel in the rectified image plane.

25 Image Rectification

26 Essential matrix example: parallel cameras R T E [ d,0,0] I [T x]r d 0 d 0 Grauman

27 Essential matrix example: parallel cameras R T E [ d,0,0] I [T x]r d 0 d 0 Grauman

28 Essential matrix example: parallel cameras R T E [ d,0,0] I [T x]r d 0 d 0 Grauman

29 Essential matrix example: parallel cameras R T E [ d,0,0] I [T x]r d 0 d 0 Grauman

30 Essential matrix example: parallel cameras R T E [ d,0,0] I [T x]r d 0 d 0 p Ep 0 Grauman

31 Essential matrix example: parallel cameras R T E [ d,0,0] I [T x]r d 0 d 0 p Ep 0 Grauman

32 Essential matrix example: parallel cameras R T E [ d,0,0] I [T x]r d 0 d 0 p Ep 0 Grauman

33 Essential matrix example: parallel cameras R T E [ d,0,0] I [T x]r d 0 d 0 p Ep 0 Grauman

34 Essential matrix example: parallel cameras R T E [ d,0,0] I [T x]r d 0 d 0 p Ep 0 For the parallel cameras, image of any point must lie on same horizontal line in each image plane. Grauman

35 Example I: compute the fundamental matrix for a parallel camera stereo rig K = K = α 0 u0 0 β v R=I t= tx 0 0 X Y Z f = 1 a b u0 a v0 b tx 0 tx u0 a a 0 1 b v0 b = tx b 0 tx b 0 f reduces to y = y /, i.e. raster correspondence (horizontal scan-lines)

36 X Y f Z f Geometric interpretation?

37 Image pair rectification Goal: Simplify stereo matching by warping the images Apply projective transformation so that epipolar lines correspond to horizontal scanlines e

38 Image pair rectification Goal: Simplify stereo matching by warping the images Apply projective transformation so that epipolar lines correspond to horizontal scanlines e e He map epipole e to (1,0,0)

39 Image pair rectification Goal: Simplify stereo matching by warping the images Apply projective transformation so that epipolar lines correspond to horizontal scanlines e e He map epipole e to (1,0,0) try to minimize image distortion

40 Image pair rectification Goal: Simplify stereo matching by warping the images Apply projective transformation so that epipolar lines correspond to horizontal scanlines e e He map epipole e to (1,0,0) try to minimize image distortion problem when epipole in (or close to) the image

41 Planar rectification Image Transformations: Find homographies H and H so that after transformations, F n becomes F of parallel cameras:

42 Planar rectification Image Transformations: Find homographies H and H so that after transformations, F n becomes F of parallel cameras:

43 Planar rectification Image Transformations: Find homographies H and H so that after transformations, F n becomes F of parallel cameras:

44 Planar rectification Image Transformations: Find homographies H and H so that after transformations, F n becomes F of parallel cameras: Bring two views to standard stereo setup (moves epipole to ) (not possible when in/close to image)

45 Algorithm Rectification Following Trucco & Verri book pp. 159 known T and R between cameras Rotate left camera so that epipole e l goes to infinity along horizontal axis Apply same rotation to right camera to recover geometry Rotate right camera by R -1 Adjust scale

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47 From: Trucco & Verri, Introductory Techniques for 3-D Computer Vision, pp

48 More elegant Solution Idea: Mapping epipole to infinity [1,0,0] T Factorization of matrix F=SM, where S is skew symmetric and M representing the required homography (projective transformation). Use SVD:

49 Stereo matching with general camera configuration

50 Image pair rectification

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54 Other Material /Code Epipolar Geometry, Rectification: COPIES/FUSIELLO2/rectif_cvol.html Fusiello, Trucco & Verri:Tutorial, Matlab code etc:

55 Run Example Updated Webpages: Demo for stereo reconstruction (out of date): SFM Example: Software:

56 Example: Zhengyou Zhang Fundamental matrix between the two cameras:

57 Points have been extracted using Harris corner detector, point matches via fundamental matrix F and search along epipolar lines.

58 Point matches found by a correlation technique

59 3D reconstruction represented by a pseudo stereogram

60 Additional Materials: Forward Translating Cameras

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63 Example II: compute F for a forward translating camera X Y f Z f

64 X Y Z f f first image second image

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67 e e

68 Summary: Properties of the Fundamental matrix

69 Goal: 3D from Stereo via Disparity Map image I(x,y) Disparity map D(x,y) image I (x,y ) (x,y )=(x+d(x,y),y) 71 F&P Chapter 11

70 Example: Stereo to Depth Map

71 Example: Stereo to Depth Map