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Exercise No. 1 – Again the Essentials

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Mathematical Foundations of Computer Vision

Michael Breuß Released:26.01.2012

Assigned to:Tutorial at 02.02.2012

Assignment 8 – The Final

TM

Sheet

Exercise No. 1 – Again the Essentials

Let the essential matrix

E =

0.76 4.32 −2.4

−4.32 1.76 1.8

0 0 0

 (1)

be given. A singular value decomposition for E is

E =

0.8 0.6 0

−0.6 0.8 0

0 0 1

5 0 0 0 5 0 0 0 0

0.64 0.48 −0.6

−0.6 0.8 0 0.48 0.36 0.8

 (2)

Compute the epipoles in both images. Give them in homogeneous coordinates. (20pts)

Exercise No. 2 – Homogeneous and projective

Consider homogeneous transformations H (expressed as 3×3-matrix) in the (projective) image plane.

DetermineHsuch that the points with homogeneous coordinates

~a= (0,0,1)>, ~b= (1,0,1)>, ~c= (1,1,1)>, d~= (0,1,1)> (3) are mapped to the points with homogeneous coordinates

~a0= (6,5,1)>, ~b0 = (4,3,1)>, ~c0 = (6,4.5,1)>, d~0 = (10,8,1)> (4)

respectively. (10pts)

Hints: 1. Be aware to work properly with homogeneous coordinates. 2. As usual for projective transforma- tions, the matrix H is determined only up to scale! 3. At the appropriate point, you may use a computer.

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