Multiple View Geometry

1. Multiple View Geometry Marc Pollefeys University of North Carolina at Chapel Hill Modified by Philippos Mordohai

2. Tutorial outline • Introduction • Visual 3D modeling • Acquisition of camera motion • Acquisition of scene structure • Constructing visual models • Examples and applications

3. Visual 3D models from images “Sampling” the real world • Visualization • Virtual/augmented/mixed reality, tele-presence, medicine, simulation, e-commerce, etc. • Metrology • Cultural heritage and archaeology, geology, forensics, robot navigation, etc. Convergence of computer vision, graphics and photogrammetry

4. Visual 3D models from video What can be achieved? unknown scene unknown camera Scene (static) camera unknown motion automatic modelling Visual model

5. Example: DV video  3D model accuracy ~1/500 from DV video (i.e. 140kb jpegs 576x720)

6. (Pollefeys et al. ICCV’98; … Pollefeys et al.’IJCV04)

7. Outline • Introduction • Image formation • Relating multiple views

8. Perspective projection Perspective projection Linear equations (in homogeneous coordinates)

9. m M1 C M2 Homogeneous coordinates • 2-D points represented as 3-D vectors (x y 1)T • 3-D points represented as 4-D vectors (X Y Z 1)T • Equality defined up to scale • (X Y Z 1)T ~ (WX WY WZ W)T • Useful for perspective projection  makes equations linear

10. The pinhole camera

11. Effects of perspective projection • Colinearity is invariant • Parallelism is not preserved

12. Intrinsic parameters • Camera deviates from pinhole s: skew fx ≠ fy: different magnification in x and y (cx cy): optical axis does not pierce image plane exactly at the center • Usually: rectangular pixels: square pixels: principal point known: or

13. Extrinsic parameters Scene motion Camera motion

14. Projection matrix • Includes coordinate transformation and camera intrinsic parameters

15. Projection matrix • Mapping from 2-D to 3-D is a function of internal and external parameters

16. Radial distortion • In reality, straight lines are not preserved due to lens distortion • Estimate polynomial model to correct it

17. Radial distortion

18. Outline • Introduction • Image formation • Relating multiple views

19. L2 M? M Triangulation m2 l2 C2 3D from images m1 C1 L1 • calibration • correspondences

20. Comparing image regions Compare intensities pixel-by-pixel I(x,y) I´(x,y) (Dis)similarity measures Normalized Cross Correlation • Sum of Square Differences

21. Feature matching Multi-view relation Structure and motion recovery Self-calibration Structure and motion recovery Feature extraction

22. Other cues for depth and geometry Shading Shadows, symmetry, silhouette Texture Focus

23. P P L2 L2 m1 m1 m1 C1 C1 C1 M M L1 L1 l1 l1 e1 e1 lT1 l2 e2 e2 l2 m2 m2 m2 l2 l2 Fundamental matrix (3x3 rank 2 matrix) C2 C2 C2 Epipolar geometry Underlying structure in set of matches for rigid scenes • Computable from corresponding points • Simplifies matching • Allows to detect wrong matches • Related to calibration

24. The epipoles The epipole is the projection of the focal point of one camerain another image. e21 e12 C2 C1 Image 2 Image 1

25. Two view geometry computation:linear algorithm • For every match (m,m´):

26. Benefits from having F • Given a pixel in one image, the corresponding pixel has to lie on epipolar line • Search space reduced from 2-D to 1-D

27. Two view geometry computation:finding more matches

28. Difficulties in finding corresponding pixels

29. Matching difficulties • Occlusion • Absence of sufficient features (no texture) • Smoothness vs. sensitivity • Double nail illusion

30. Two view geometry computation:more problems • Repeated structure ambiguity • Robust matcher also finds • support for wrong hypothesis • solution: detect repetition