Multiple View Geometry: Exercise Sheet 1
Prof. Dr. Florian Bernard, Florian Hofherr, Tarun Yenamandra
Computer Vision Group, TU Munich
Link Zoom Room , Password: 307238
Exercise: April 21st, 2021
Part I: Theory
1. Show for each of the following sets (1) whether they are linearly independent, (2) whether they
spanR3 and (3) whether they form a basis ofR3:
(a) B1 =
11
1
,
01
1
,
00
1
(b) B2 =
21
0
,
11
0
(c) B3 =
21
0
,
31
0
,
00
1
,
10
1
2. Which of the following sets forms a group (with matrix-multiplication)? Prove or disprove!
(a) G1 :=
{
A ∈ Rn×n| det(A) 6= 0 ∧A> = A
}
(b) G2 := {A ∈ Rn×n| det(A) = −1}
(c) G3 := {A ∈ Rn×n| det(A) > 0}
3. Prove or disprove: There exist vectors v1, …,v5 ∈ R3 \ {0}, which are pairwise orthogonal,
i.e.
∀i, j = 1, …, 5 : i 6= j =⇒ 〈vi,vj〉 = 0
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https://tum-conf.zoom.us/s/62772800235?pwd=SUpZN2QrV0JpeXJyR2R1TWx5cHEwdz09
Part II: Practical Exercises
1. Basic image processing
(a) Download lena.png, provided in ex01.zip.
(b) Load the image into the workspace.
(c) Determine the size of the image and show the image.
(d) Convert the image to gray scale and determine the maximum and the minimum value of
the image.
(e) Apply a gaussian smoothing filter and save the output image.
(f) Show 1) the original image, 2) the gray scale image and 3) the filtered image in one figure
and give the figures appropriate titles.
(g) Compare the gray scale image and the filtered image for different values of the smoothing.
2. Basic operations
(a) Let A =
(
2 2 0
0 8 3
)
and b =
(
5
15
)
. Solve Ax = b for x.
(b) Define a matrix B equal to A.
(c) Change the second element in the first row of A to 4.
(d) Compute the following:
c = 0;
for i ∈ {−4, 0, 4}
c = c+ i ∗A> ∗ b
end
print c
(e) Compare A .* B and A’ * B and explain the difference.
3. Write a function approxequal(x, y, �) checking if two vectors x and y are almost equal, i.e.
if
∀i : |xi − yi| ≤ � .
The output should be logical 1 or 0.
If the input consists of two matrices, your function should compare the columns of the matrices
if they are almost equal. In this case, the output should be a vector with logical values 1 or 0.
4. Write a function addprimes(s, e) returning the sum of all prime numbers between and in-
cluding s and e.
Use the Matlab-function isprime.
2