Vision 3D artificielle - Final exam (duration: 2h30)
P. Monasse, M. Aubry, and R. Marlet November 10th, 2015
You can choose to answer in French or English, at your convenience.
1 SVD
The two questions are independent.
1. LetE be a 3×3 real matrix. Show that:
(det(E) = 0
2EETE= tr(EET)E ⇔E’s singular values are (σ, σ,0) with σ≥0.
2. In the following, we use the Frobenius normkAk2 = tr(ATA). LetA be ann×nreal matrix of coefficientsaij.
(a) Show that kAk2=P
ija2ij.
(b) Show that this defines a matrix norm, indeed.
(c) Show that ifU is orthogonal,kU Ak=kAk.
(d) Show that
arg min
A0,det(A0)=0
kA0−Ak
can be computed simply through the SVD of A. Explain how and justify it.
2 Fundamental matrix
1. When trying to compute the fundamental matrix F with automatically matched feature points and RANSAC algorithm on the images of Fig- ure 1, we find a solution, but clicking points on the box do not show lines corresponding to matching points. Explain why.
1
Q
P
Figure 1: Top: original image pair and detected feature points. Bottom: two clicked points, one on the central left part of the ground (P) and one on the box (Q) with their computed epipolar lines. The one associated toQis not correct.
2. LetH be the homography relating a planar part of the scene in a stereo pair. Show that we can write the fundamental matrixF = [e]×H−1 with esome 3-vector.
3. Show that the data of two matched points outside the planar part of the scene are sufficient to determinee.
4. Propose an algorithm (in pseudo-code), variant of RANSAC, that may be able to handle correctly the problem of Figure 1.
3 Multi-view constraints
Suppose that a 3D point is seen in three views through pinhole cameras, at positionsx0= x10 x20 x30T
, x1,x2. We can write the trifocal constraints
[x1]× X
i
xi0Ti
!
[x2]×= 0
whereTi (i= 1,2,3) are 3×3 matrices. Show that among these 9 constraints on the coefficients ofTi, at most 4 are linearly independent equations.
2
4 Disparities estimation with graph-cut techniques
Given two rectified images, consider the problem of assigning a disparity to each pixel on each image line, in the set{dmin, . . . , dmax}. We consider the following strategy:
Initially consider all pixels as unassigned;
ford←dmin tod←dmax do
Build a graph whose cut best separates pixels into {D≤d, D>d}where D≤d are pixels with disparityd0≤dandD>d are pixels with
disparityd0 > d;
end
forall thepixelpdo
Assign topthe disparity max
d∈{dmin,...,dmax} p∈D≤d\D>d
d;
end
Questions: What are the advantages and disadvantages of such a strategy compared to the other techniques presented in the course, in terms of:
1. problem size,
2. well-definedness of the algorithm, 3. quality of the solution being modeled, 4. optimality of the solution reached.
3
