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Multi-View Geometry (Cont.)

Multi-View Geometry (Cont.). Stereo Constraints (Review). p’. ?. p. Given p in left image, where can the corresponding point p’ in right image be?. Epipolar Line. p’. Y 2. X 2. Z 2. O 2. Epipole. Stereo Constraints (Review). M. Image plane. Y 1. p. O 1. Z 1. X 1. Focal plane.

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Multi-View Geometry (Cont.)

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  1. Multi-View Geometry (Cont.)

  2. Stereo Constraints (Review) p’ ? p Given p in left image, where can the corresponding point p’in right image be?

  3. Epipolar Line p’ Y2 X2 Z2 O2 Epipole Stereo Constraints (Review) M Image plane Y1 p O1 Z1 X1 Focal plane

  4. Epipolar Constraint (Review)

  5. P p p’ O’ O From Geometry to Algebra (Review)

  6. P p p’ O’ O From Geometry to Algebra (Review)

  7. Linear Constraint:Should be able to express as matrix multiplication.

  8. The Essential Matrix (Review)

  9. The Essential Matrix • Based on the Relative Geometry of the Cameras • Assumes Cameras are calibrated (i.e., intrinsic parameters are known) • Relates image of point in one camera to a second camera (points in camera coordinate system). • Is defined up to scale • 5 independent parameters

  10. The Essential Matrix Similarly p is the epipolar line corresponding to p in theright camera

  11. The Essential Matrix Similarly, Essential Matrix is singular with rank 2 e’

  12. Small Motions and Epipolar Constraint

  13. Velocity Vector Translational Component of Velocity Angular Velocity Motion Models (Review) 3D Rigid Motion

  14. Small Motions Velocity Vector Translational Component of Velocity Angular Velocity

  15. Translating Camera Focus of expansion (FOE): Under pure translation, the motionfield at every point in the image points toward the focus ofexpansion

  16. FOE for Translating Camera

  17. FOE from Basic Equations of Motion q p v O

  18. What if Camera Calibration is not known

  19. Review: Intrinsic Camera Parameters Y M Image plane C Z v X Focal plane m u P

  20. Fundamental Matrix If u and u’ are corresponding image coordinates then we have

  21. Fundamental Matrix Fundamental Matrix is singular with rank 2 In principal F has 7 parameters up to scale and can be estimatedfrom 7 point correspondences Direct Simpler Method requires 8 correspondences

  22. Estimating Fundamental Matrix The 8-point algorithm Each point correspondence can be expressed as a linear equation

  23. The 8-point Algorithm

  24. Shape from Stereo

  25. Pinhole Camera Model

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

  27. Basic Stereo Derivations

  28. Basic Stereo Derivations Define the disparity:

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