1. Three-view geometry
3-view constraint along F
Minimal algebraic sol

2. Where are we? 1-view geometry  P matrix
2-view geometry  P, P’  F matrix
3-view geometry  P, P’, P’’  T tensor

3. Lines vs. points
Lines (line segments, straight lines) are natural mathematical object
Popular image features:
(Canny-like) edge, link, chains, polygonal approximation
Robust and global feature
1D features vs. 2D features (points)
‘1D’ also in terms of geometric constraint

4. ‘natural object’ for 3 views
Impossible for 2-views
‘nicely’ described by ‘incidence’ in 3-views

6. Basic ‘incidence’ constraint for lines
Three planes intersect into a common line!

7. Rank 2 implies a linear dependency, and
linear combination gives line transfer equation
By noting
In tensor notation for T1, T2 and T3:
2 scalar constraints, line transfer

8. Some tensorial notations
on pts, lines and conics:
Transforms contravariantly
Co-variantly to preserve incidence
Co-variantly
NB: co-,contra-variance is w.r.t. the basis trans.
Transpose is of no importance, il accommodates row/column vectors

9. Upper indices for contravariant indices---points
Lower indices for covariant indices---lines
Incidence:

12. How about points? u” u u’ O O

13. Transferring points in 3-view
It’s about re-projection or transfer from the first two views into the third one.
Why two t? in fact, it is the redundancy,
By equating two t, we get the fundamental matrix!
We have only 4 trilinear equations as each transfer gives two equations for u’’, then with 2 t, we have 4
But tensor notation gives us 9 equations, but only 4 of them linearly independent,
Look at free indexes, not contracted, b, d and k, that gives 9 scalar equations.

14. Properties of T T for 3-view similar to F for 2-view
T both for points and lines
Points and lines could be mixed up
Rank of each matrix Ti and rank of nullspace of Ti?
How many d.o.f.?
#algebraic constraints

15. Extraction of geometry from T
Matrices (I,0), (A,a), (B,b) and T
O=(0,0,0,1), a=e’, and b=e’’
Kernel ci of Ti^T, di of Ti
Kernel e21 of (c1,c2,c3)
Kernel e31 of (d1,d2,d3)
From pt-line-line incidence

16. Get pair-wise: Get P-matrix from F: CF. 2-view geometry
Each Ti e gives a vector, so we have a matrix of 3 vectors
A little bit carful about P’’

17. Estimation of T Mixture of points and lines
2 for lines and 4 for points
Linear equations in the entries of T from trilinearity
Some minimal data configuration
Linear solution with data normalisation
It’s still difficult, why? Count properly the d.o.f.
A short story about T

18. Minimal data for algebraic sol. of 3 views
Cf. Invariants of 6 pts and projective reconstruction from 3 uncalibrated images.

19. Comparative study 3-view 2-view 18 d.o.f. = 3*11-15 7 d.o.f. = 2*11-15
8 constraints
6 pts --- cubic
7 pts --- linear
T, 3*3*3-1=26
2-view
7 d.o.f. = 2*11-15
1 constraint
7 pts --- cubic
8 pts --- linear
F, 8 param

20. N-view geometry

21. Algebraic or linear methods are for the initial solution of both camera and structure geoemtry.
We always need an optimisation tool to obtain the final results

22. Bundle adjustment Similar equation to calibration, but
1. It is for all images indexed by k , and
2. xi,yi, zi, and ti are also unknowns
3. A very large optimisation problem

23. Cost function definition: robustness Local parameterisation
Projective, euclidean
Gauge freedom
Optimisation method
Gauss-Newton method
Sparse matrix method

24. Block reduction for sparse matrices:

25. Quasi-dense reconstruction
Pair-wise quasi-dense correspondence
Re-sampling quasi-dense
Robust 2-view geometry estimation
Robust 3-view geometry estimation
Merging subsequences into long sequences by projective bundle adjustment
Auto-calibration to get metric update
Euclidean bundle adjustment
Surface models?