CIS 5800, Machine Perception, Fall 2023 Final Project Part C
It counts for 10% of the total class grade Part A and B ≃ 30%.
Due: Friday, Dec. 22, 11:59pm ET Version of December 14, 2023, GMT
2-view special case: unknown yaw and horizontal translation Assume that a two-view config- uration consists of two cameras with identical and known intrinsic parameters displaced as follows:
cosβ 0sinβ
Xr=RXl+t where R=0 1 0 , t= tcosα 0 tsinαT, (1)
−sinβ 0 cosβ
with t the magnitude of the translation, and Xr,Xl position vectors of points wrt the right and left camera coordinate system, respectively. Another way to think of this set-up is one camera moving in the XZ-plane.
1. Write the essential matrix E (defined from the epipolar constraint xTr Exl = 0)
2. How many points do I need in order to solve for the E-matrix using a homogeneous linear system?
3. What is the necessary and sufficient condition for a matrix E to represent the above configuration. We need here a set of constraints on the elements of E like E11 = 0 or E12 +E21 = 0. No need to use SVD, trigonometry suffices. As part of the sufficiency condition you will derive the unknowns of the problem α and β.
4. How many solutions α and β derived from the E-matrix satisfy the epipolar constraint?
5. Extra credit: Forget the E-matrix. Can you directly solve xTr (t × Rxl) = 0 for α and β?
6. In this configuration, the two optical axes intersect the way that the human eyes converge. Compute the coordinates (Xl,Zl) of the intersection of the optical axes wrt the left coordinate system.
7. We switch now to the continuous case where the only 3D velocities in this planar motion case are Vx,Vz,Ωy. Write the equations of the optical flow in terms of the unknowns and Z,x,y.
8. What are the coordinates of the focus of expansion?
9. Eliminate Z from the two equations of flow. Write a Z-free “continuous” epipolar constraint.
10. Extra-credit: Solve for the two unknowns.
Code Help