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Stereopsis

Stereopsis. Reconstruction. Only need to match features across epipolar lines. Geometric Reconstruction. Pinhole Camera Model. Basic Stereo Derivations. Derive expression for Z as a function of x 1 , x 2 , f and B. Basic Stereo Derivations. Basic Stereo Derivations.

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Stereopsis

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  1. Stereopsis

  2. Reconstruction Only need to match features across epipolarlines

  3. Geometric Reconstruction

  4. Pinhole Camera Model

  5. Basic Stereo Derivations Derive expression for Z as a function of x1, x2, f and B

  6. Basic Stereo Derivations

  7. Basic Stereo Derivations Define the disparity:

  8. Stereo image rectification

  9. Stereo image rectification • 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, 1999.

  10. Image Rectification • Common Image Plane • Parallel Epipolar Lines • Search Correspondenceson scan line

  11. p p’ O’ O Reconstruction

  12. Equation 1 Equation 2 (From equations 1 and 2) Reconstruction

  13. Reconstruction up to a Scale Factor • Assume that intrinsic parameters of both cameras are known • Essential Matrix is known up to a scale factor (for example, estimated from the 8 point algorithm).

  14. Reconstruction up to a Scale Factor

  15. Reconstruction up to a Scale Factor Let It can be proved that

  16. Reconstruction up to a Scale Factor We have two choices of t,(t+and t-) because of sign ambiguity and two choices of E, (E+ and E-). This gives us four pairs of translation vectors and rotation matrices.

  17. Reconstruction up to a Scale Factor Given and • Construct the vectors w, and compute R • Reconstruct the Z and Z’ for each point • If the signs of Z and Z’ of the reconstructed points are • both negative for some point, change the sign ofand go to step 2. • different for some point, change the sign of each entryof and go to step 1. • both positive for all points, exit.

  18. Finding Correspondences

  19. Stereo matching algorithms • Match Pixels in Conjugate Epipolar Lines • Assume brightness constancy • This is a tough problem • Numerous approaches • dynamic programming [Baker 81,Ohta 85] • smoothness functionals • more images (trinocular, N-ocular) [Okutomi 93] • graph cuts [Boykov 00] • A good survey and evaluation: http://www.middlebury.edu/stereo/

  20. For each epipolar line For each pixel in the left image • Improvement: match windows • This should look familar... • Can use Lukas-Kanade or discrete search (latter more common) Your basic stereo algorithm • compare with every pixel on same epipolar line in right image • pick pixel with minimum match cost

  21. Correspondence using Discrete Search

  22. Sum of Squared Differences (SSD)

  23. Image Normalization

  24. Foreshortening

  25. Problems with window matching

  26. Stereo results • Data from University of Tsukuba • Similar results on other images without ground truth Scene Ground truth

  27. Results with window correlation Window-based matching (best window size) Ground truth

  28. Results with better method • State of the art method • Boykov et al., Fast Approximate Energy Minimization via Graph Cuts, • International Conference on Computer Vision, September 1999. Ground truth

  29. Final Exam Thursday, April 24, 200319:00-21:45

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