ECE 181b Homework 3
Image Rectification
April 20, 2006
The goal of this project is to explore some fundamental concepts of projective geometry
related to the problem of image rectification. We will rectify the image of one of the facades
of the Bren School of Environmental Science & Management at UCSB (see Figure 1) and
verify how points at infinity can be mapped to finite points via homographies.
Caveat.
To receive full credit your answers must be clearly justified. Make sure to include
the relevant steps that were required to obtain the numerical answer.
1
Calculating the Homography
A transformation from the projective space
P
2
to itself is called homography. A homography
is represented by a 3
×
3 matrix with 8 degrees of freedom (scale, as usual, does not matter):
x
y
w
=
h
1
h
4
h
7
h
2
h
5
h
8
h
3
h
6
h
9
x
y
w
To estimate the coefficients of the homography
H
we will use the approach described
in class (see the notes that can be found at
http://www.ece.ucsb.edu/
~
manj/ece181b/
homography_zuliani.pdf
). The test images can be downloaded from the course website.
For sake of convenience we will adopt the coordinate system convention displayed in Figure 2
(for a more detailed image go to
http://vision.ece.ucsb.edu/
~
zuliani/Code/lattice.
png
).
Question 1
How many point correspondences are necessary to compute the homography that
relates Image 1(a) to Image 1(b) under the constrain
that the vector
h
obtained stacking the
components of
H
one on top of the other has unit norm (i.e.
h
= 1
)? Why?
Answer 1
We need at least 4 point correspondences.
In fact each point correspondence
provides two equations (for a total of 8 constraints) and we have 9 unknowns.
The extra
degree of freedom is fixed by imposing
h
= 1
.
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 Fall '08
 Staff
 LG, Projective geometry, Bren School of Environmental Science & Management, WarpHomography

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