Epipolar Geometry: Essential Matrix

1. Epipolar Geometry: Essential Matrix Various slides from previous courses by: D.A. Forsyth (Berkeley / UIUC), I. Kokkinos (Ecole Centrale / UCL). S. Lazebnik (UNC / UIUC), S. Seitz (MSR / Facebook), J. Hays (Brown / Georgia Tech), A. Berg (Stony Brook / UNC), D. Samaras (Stony Brook) . J. M. Frahm (UNC), V. Ordonez (UVA), Steve Seitz (UW).

2. Last Class • Stereo Vision – Dense Stereo • More on Epipolar Geometry

3. Today’s Class • More on Epipolar Geometry • Essential Matrix • Fundamental Matrix

4. Estimating depth with stereo Stereo: shape from “motion” between two views We’ll need to consider: Info on camera pose (“calibration”) Image point correspondences scene point image plane optical center

5. Key idea: Epipolar constraint X X X x x’ x’ x’ Potential matches for x have to lie on the corresponding line l’. Potential matches for x’ have to lie on the corresponding line l.

6. Epipolar geometry: notation X x x’ • Baseline – line connecting the two camera centers • Epipoles • = intersections of baseline with image planes • = projections of the other camera center • Epipolar Plane – plane containing baseline (1D family)

7. Epipolar geometry: notation X x x’ • Baseline – line connecting the two camera centers • Epipoles • = intersections of baseline with image planes • = projections of the other camera center • Epipolar Plane – plane containing baseline (1D family) • Epipolar Lines - intersections of epipolar plane with image planes (always come in corresponding pairs)

8. Epipolar Geometry: Another example Credit: William Hoff, Colorado School of Mines

9. Example: Converging cameras

10. Geometry for a simple stereo system First, assuming parallel optical axes, known camera parameters (i.e., calibrated cameras):

11. Simplest Case: Parallel images • Image planes of cameras are parallel to each other and to the baseline • Camera centers are at same height • Focal lengths are the same • Then epipolar lines fall along the horizontal scan lines of the images

12. Geometry for a simple stereo system Similar triangles (pl, P, pr) and (Ol, P, Or): disparity Assume parallel optical axes, known camera parameters (i.e., calibrated cameras). What is expression for Z?

13. Depth from disparity image I´(x´,y´) image I(x,y) Disparity map D(x,y) (x´,y´)=(x+D(x,y), y) So if we could find the corresponding points in two images, we could estimate relative depth…

14. Correspondence search Left Right • Slide a window along the right scanline and compare contents of that window with the reference window in the left image • Matching cost: SSD or normalized correlation scanline Matching cost disparity

15. Correspondence search Left Right scanline SSD

16. Correspondence search Left Right scanline Norm. corr

17. Basic stereo matching algorithm • If necessary, rectify the two stereo images to transform epipolar lines into scanlines • For each pixel x in the first image • Find corresponding epipolarscanline in the right image • Examine all pixels on the scanline and pick the best match x’ • Compute disparity x–x’ and set depth(x) = B*f/(x–x’)

18. Failures of correspondence search Occlusions, repetition Textureless surfaces Non-Lambertian surfaces, specularities

19. camera • projector Active stereo with structured light • Project “structured” light patterns onto the object • Simplifies the correspondence problem • Allows us to use only one camera • L. Zhang, B. Curless, and S. M. Seitz. Rapid Shape Acquisition Using Color Structured Light and Multi-pass Dynamic Programming.3DPVT 2002

20. Kinect and Iphone X Solution • Add Texture!

21. Kinect: Structured infrared light • http://bbzippo.wordpress.com/2010/11/28/kinect-in-infrared/

22. iPhone X

23. Basic stereo matching algorithm • If necessary, rectify the two stereo images to transform epipolar lines into scanlines • For each pixel x in the first image • Find corresponding epipolarscanline in the right image • Examine all pixels on the scanline and pick the best match x’ • Compute disparity x–x’ and set depth(x) = B*f/(x–x’)

24. Effect of window size • Smaller window • More detail • More noise • Larger window • Smoother disparity maps • Less detail W = 3 W = 20

25. Results with window search Data Window-based matching Ground truth

26. Better methods exist... Ground truth Graph cuts • Y. Boykov, O. Veksler, and R. Zabih, Fast Approximate Energy Minimization via Graph Cuts, PAMI 2001 • For the latest and greatest: http://www.middlebury.edu/stereo/

27. When cameras are not aligned:Stereo image rectification • Reproject image planes onto a common • plane parallel to the line between optical centers • Pixel motion is horizontal after this transformation • Two homographies (3x3 transform), one for each input image reprojection • C. Loop and Z. Zhang. Computing Rectifying Homographies for Stereo Vision. IEEE Conf. Computer Vision and Pattern Recognition, 1999.

28. Rectification example

29. Back to the General Problem Credit: William Hoff, Colorado School of Mines

30. Back to the General Problem Credit: William Hoff, Colorado School of Mines

31. Back to the General Problem Credit: William Hoff, Colorado School of Mines

32. Back to the General Problem Credit: William Hoff, Colorado School of Mines

33. Back to the General Problem Credit: William Hoff, Colorado School of Mines

34. Back to the General Problem Credit: William Hoff, Colorado School of Mines

35. Back to the General Problem Credit: William Hoff, Colorado School of Mines

36. Brief Digression: Cross Product as Matrix Multiplication • Dot Product as matrix multiplication: Easy • Cross Product as matrix multiplication:

37. Back to the General Problem Credit: William Hoff, Colorado School of Mines

38. The Essential Matrix

39. The Essential Matrix Essential Matrix (Longuet-Higgins, 1981) Credit: William Hoff, Colorado School of Mines

40. Epipolar constraint: Calibrated case X • Intrinsic and extrinsic parameters of the cameras are known, world coordinate system is set to that of the first camera • Then the projection matrices are given by K[I | 0] andK’[R | t] • We can multiply the projection matrices (and the image points) by the inverse of the calibration matrices to get normalized image coordinates: x x’

41. Epipolar constraint: Calibrated case X = (x,1)T x x’ = Rx+t t R The vectors Rx, t, and x’ are coplanar

42. Epipolar constraint: Calibrated case X x x’ = Rx+t Recall: The vectors Rx, t, and x’ are coplanar

43. Epipolar constraint: Calibrated case X x x’ = Rx+t Essential Matrix (Longuet-Higgins, 1981) The vectors Rx, t, and x’ are coplanar

44. Epipolar constraint: Calibrated case X • E x is the epipolar line associated with x(l' = E x) • Recall: a line is given by ax + by + c = 0 or x x’

45. Epipolar constraint: Calibrated case X • E x is the epipolar line associated with x(l' = E x) • ETx'is the epipolar line associated with x'(l = ETx') • E e= 0 and ETe' = 0 • Eis singular (rank two) • E has five degrees of freedom x x’

46. Epipolar constraint: Uncalibrated case X • The calibration matrices K and K’ of the two cameras are unknown • We can write the epipolar constraint in terms of unknown normalized coordinates: x x’

47. Epipolar constraint: Uncalibrated case X x x’ Fundamental Matrix (Faugeras and Luong, 1992)

48. Epipolar constraint: Uncalibrated case X x x’ • F x is the epipolar line associated with x(l' = F x) • FTx' is the epipolar line associated with x' (l = FTx') • F e= 0 and FTe' = 0 • Fis singular (rank two) • F has seven degrees of freedom

49. Questions?