1. Multi-View Geometry

2. Review: Pinhole Camera

3. Review: Perspective Projection

4. Review: Perspective Projection • Points go to Points • Lines go to Lines • Planes go to whole image or Half-planes • Polygons go to Polygons

5. Perspective cues

6. Perspective cues

7. Perspective cues

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

9. Review: Extrinsic Parameters Y M Image plane Y C Z v X X Z Focal plane m u By Rigid Body Transformation:

10. Recovering 3D from images • What cues in the image provide 3D information?

11. Visual cues • Shading Merle Norman Cosmetics, Los Angeles

12. Visual cues • Shading • Texture The Visual Cliff, by William Vandivert, 1960

13. Visual cues • Shading • Texture • Focus From The Art of Photography, Canon

14. Visual cues • Shading • Texture • Focus • Motion

15. Visual cues • Shading • Texture • Focus • Motion • Shape From X • X = shading, texture, focus, motion, ...

16. 3D World Points • Camera Centers • Camera Orientations Multi-View Geometry Relates

17. 3D World Points • Camera Centers • Camera Intrinsic Parameters • Image Points Multi-View Geometry Relates • Camera Orientations

18. Stereo scene point image plane optical center

19. Stereo • Basic Principle: Triangulation • Gives reconstruction as intersection of two rays • Requires • calibration • point correspondence

20. Stereo Constraints p’ ? p Given p in left image, where can the corresponding point p’in right image be?

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

22. Epipolar Constraint

23. P p p’ O’ O From Geometry to Algebra

24. P p p’ O’ O From Geometry to Algebra

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

26. Review: Matrix Form of Cross Product

27. Review: Matrix Form of Cross Product

28. Matrix Form

29. The Essential Matrix