1. 1 Three-view geometry 3-view constraint along F Minimal algebraic sol The content described in these slides is not required in the final exam!

2. 2 Where are we? 1.1-view geometry  P matrix 2.2-view geometry  P, P’  F matrix 3.3-view geometry  P, P’, P’’  T tensor The understanding of the tensor T is not required for our class!!!!

3. 3 u O u’ O u” How about points?

4. 4 Transferring points in 3-view It’s about re-projection or transfer from the first two views into the third one. This tensorial equation gives 9 scalar equations, 4 of which are linearly independent.

5. 5 Minimal data for algebraic sol. of 3 views Invariants of 6 pts and projective reconstruction from 3 uncalibrated images, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 17, no. 1, 1995 Given 6 corresponding points in 3 uncalibrated images, we can compute the three projection matrices P, P’, and P’’ (similar to 7 points in 2 uncalibrated images, get P and P’ (via F)). It is important to know this result, even though we will not derive it!

6. 6 N-view geometry No algebraically independent geometric constraints for a set of more than 3 views.

7. 7 Bundle adjustment: practical and optimal method of 3D reconstruction for multiple views Similar equation to calibration, but 1. It is for all images indexed by k, and 2. are unknowns, xi,yi, zi, and ti are also unknowns!!! 3. A very large optimisation problem

8. 8 Automatic computation of a projective reconstruction for a sequence projective reconstruction: 2-view, 3-view, N-view obtaining correspondences over N-views

9. 9 Pairwise matches: compute point matches between view pairs using robust F estimation Putative correspondences: over three views from two view matches RANSAC (6-pt algo) robust estimation of three view geometry, P, P’ and P’’ Generate additional matches From 2 images to 3 images:

10. 10 Compute all 2-view reconstructions for consecutive views Compute all 3-view reconstruction for consecutive views Extend to sequence by hierarchical merging (projective) bundle-adjustment (autocalibration) (euclidean bundle-adjustment)

11. 11 Some real examples of reconstruction.