Multiple View Geometry: Exercise Sheet 7
Prof. Dr. Florian Bernard, Florian Hofherr, Tarun Yenamandra
Computer Vision Group, TU Munich
Link Zoom Room , Password: 307238
Exercise: June 9th, 2021
Part I: Theory
1. Coimages of Points and Lines
Suppose p1, p2 ∈ R3 are two points on the line L ⊂ R3. Let x1, x2 ∈ R3 be the images of the
points p1, p2 in homogeneous coordinates, respectively, and let l ∈ R3 be a vector that spans the
coimage of the line L. All vectors are given in the image coordinate system.
Furthermore suppose L1, L2 ⊂ R3 are two lines intersecting in the point p ∈ R3. Let x ∈ R3
be the image of the point p in homogeneous coordinates and let l1, l2 ∈ R3 be vectors that span
the coimages of the lines L1, L2, respectively.
Draw a picture and convince yourself of the following relationships:
(a) Show that
l ∼ x̂1×2, x ∼ l̂1l2,
(b) Show that there exist r, s, u, v ∈ R3 such that,
l1 ∼ x̂u, l2 ∼ x̂v, x1 ∼ l̂r, x2 ∼ l̂s
where ∼ means equivalence in the sense of homogeneous coordinates.
2. Rank Constraints
Let x1, x2 ∈ R3 be two image points in homogeneous coordinates with projection matrices
Π1,Π2 ∈ R3×4. Show that the rank constraint
rank
(
x̂1Π1
x̂2Π2
)
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ensures that x1 and x2 are images (projections) of the same three-dimensional point X .
3. Projection and Essential Matrix
Suppose two projection matrices Π = [R, T ] and Π′ = [R′, T ′] ∈ R3×4 are related by a
common transformation H of the form
H =
[
I 0
v> v4
]
∈ R4×4 where v =
v1v2
v3
.
That is, [R, T ]H ∼ [R′, T ′] are equal up to scale.
Show that Π and Π′ give the same essential matrices (E = T̂R and E′ = T̂ ′R′) up to a scale
factor.
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https://tum-conf.zoom.us/s/62772800235?pwd=SUpZN2QrV0JpeXJyR2R1TWx5cHEwdz09
Part II: Practical Exercises
Epipolar lines
1. Download the package ex07.zip from the website. Extract the images batinria0.pgm
and batinria1.pgm. Their corresponding camera calibration matrices can be found in the
file calibration.txt.
2. Show the two images with matlab and select a point in the first image. You can use the command
[x,y]=ginput(n) to retrieve the image coordinates of a mouse click.
3. Think about where the corresponding epipolar line l2 in the second image could be.
4. Now compute the epipolar line l2 = Fx1 in the second image corresponding to the point x1 in
the first image. To this end you will need to compute the fundamental matrix F between the
two images. Use the calibration data from the file calibration.txt.
Remark: Note that l2 does not directly encode the epipolar line itself. Rather, l1 is the coim-
age of the epipolar plane in the second coordinate system from which the epipolar line can be
computed. This representation is chosen due to the easy formula shown above.
5. Test your program for different points x1. What do you observe?
6. Bonus: Determine the best matching point on the epipolar line via normalized cross correlation.
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