Chair of Computer Vision and Artificial Intelligence
Department of Informatics
Technical University of Munich
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Signature
Note:
• Cross your Registration number(with leading zero). It will be evaluated automatically.
• Do not sign the above signature field. A signature field will be provided as part of the first
problem.
Computer Vision II: Multiple View Geometry
Exam: IN2228 / Midterm Date: Thursday 16th July, 2020
Examiner: Florian Bernard Time: 14:00 – 14:20
Working instructions
• This exam consists of 8 pages with a total of 6 problems.
Please make sure now that you received a complete copy of the exam.
• The total amount of achievable credits in this exam is 25 credits.
• Detaching pages from the exam is prohibited.
• Allowed resources:
– this is an open book graded exercise
• This graded exercise uses the TUMexam platform which offers a student manual on their webpage1.
• The boxes on the sides of the subproblems (those with numbers) are used for correction and ticking them is
prohibited. A ticked box that is not part of a multiple choice question may result in zero points for the problem.
• In the multiple choice questions there is always exactly one correct answer. A correct answer (comprising of
exactly one tick at the correct position) will give the indicated credits, whereas a wrong answer will result in 0
credits.
Mark correct answers with a cross ×
To undo a cross, completely fill out the answer option �
To re-mark an option, use a human-readable marking �
• Remark: We aimed to provide a large variety of the type of problems that may occur in the final/re-take exam.
To obtain the grade bonus it is not necessary to attain all credits. Do not get discouraged if the time is
not sufficient to solve all problems. Start with the problems that you can solve easily and then progress to the
harder ones.
Left room from to / Early submission at
1https://tumexam.de/static/handreichung_submissions_students.pdf
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https://tumexam.de/static/handreichung_submissions_students.pdf
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Problem 1 Personal Information (0 credits)
a) Please sign the following by entering your full name: “I hereby assure that I solve and submit this exam myself
under my own name by only using the allowed tools listed on the first page”.
b) Please enter your matriculation number with leading zero.
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Problem 2 Linear Algebra (5 credits)
The solution to problem a) should be given in MATLAB syntax, e.g.,
A = [a11, a12, a13, a14; a21, a22, a23, a24; a31, a32, a33, a34]
a) Consider the linear spaces
U = span
10
0
,
00
1
,
V = span
1
0
−1
0
,
0
2
0
−1
.
Find a matrix A ∈ R3×4 such that U and V are equal to the range and kernel of A . A =
b) Let B ∈ Rn×k , C ∈ Rm×k , D ∈ Rm×n and consider the matrix
X = (rank (C>DB) + 1)−1 BC>D .
For given n, k , m, write down an expression for the minimal rank of X and the maximal rank of X , as well as a
brief justification.
Minimal rank of X :
Maximal rank of X :
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Problem 3 MATLAB (7 credits)
a) Consider the matrix A and vector b given by
A =
4 69 2
7 2
b =
18
6
.
Write a MATLAB one-line expression for x that for the specific A , b gives a near-identical result to x_ref = A \ b,
i.e.
assert(norm(x-x_ref) < 1e-12)
does not throw an error. Your expression may use only the following MATLAB symbols/functions A, b, *, +, -,
inv, ’, rank, zeros, eye, ones, magic and parenthesis. x =
b) Let w be a vector in R3 which is represented in MATLAB as a column vector w s.t. size(w) gives [3, 1] . Let R
be a rotation matrix in SO(3) which is represented in MATLAB by R. Given are a large number of vectors as the
rows of the matrix X in R100 000×3 which is represented in MATLAB by X. Given are programA
and programB
which are called like Y_A = programA(w, R, X); Y_B = programB(w, R, X).
Give an explanation of what is computed, i.e what is the mathematical relationship between YA , YB and the inputs
w, R, X . What is the relationship between Y_A and Y_B? What do the rows of Y_A and Y_B contain?
Which of the two programs is significantly faster? Explain your choice.2
Write a one-line expression which computes the same result even faster, i.e. at least a factor of three in median
run-time over 10 runs.3 You are given w_hat which is the 3× 3 matrix corresponding to ŵ. Your expression may
use only the following MATLAB symbols/functions R, X, w, w_hat, *, +, -, inv, ’, rank, zeros, eye, ones,
magic and parenthesis. Y_C =
2Assume execution on a standard desktop PC with an i5 CPU and without GPU. You do not have to use this fact in your explanation.
3This subproblem gives few points in comparison to the rest of the graded exercise.
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Problem 4 Image Formation (4 credits)
A 3D point P = (2, 0, 4)> is observed by a camera, which has its optical center at C = (0,−1, 0)> and no rotation
(R = Id3×3). The intrinsic parameter matrix K is given by
K =
500 0 3200 400 240
0 0 1
.
Calculate the pixel-position of the projected point in the image and tick the correct answer.
a)
u = 430
u = 260
u = 620
u = 750
u = 770
u = 950
u = 200
u = 570
b)
v = 340
v = 950
v = 650
v = 860
v = 350
v = 810
v = 490
v = 670
c) Given that the camera is nearly perfect what is the size of the image?
height = 800 px, width = 1000 px
height = 480 px, width = 640 px
height = 240 px, width = 320 px
height = 400 px, width = 500 px
d) Briefly justify your answer from c). 0
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Problem 5 The Lucas-Kanade Method (4 credits)
Given is the following image where the black, dashed boxes are annotations used within this exercise.
The image is the first image of an image pair used within the Lucas-Kanade algorithm. Assume that the second
image is taken 0.2 seconds later. Furthermore assume that the camera motion is slow and that the translation and
rotation can be in any direction. The red ball in A is falling. The Lucas-Kanade algorithm is used to estimate the
optical flow for the two images. For the marked image regions A – D decide and explain if the algorithm will yield a
good result. The neighborhood W (x) is set to be the same as the annotation boxes. Use precise, technical terms
from the lecture for your explanations.
a) Image region A:
b) Image region B:
c) Image region C:
d) Image region D:
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Problem 6 Fundamental Matrix (5 credits)
Let F ∈ R3×3 be the fundamental matrix for the cameras C1 and C2. Let K be the intrinsic camera matrix for C1, C2
and let T be the translation between the two camera centers as seen in the coordinate system of camera C2. Let e2
be the epipole in the second image.
This exercise will prove that e>2 F = 0 holds. Below each step there will be a solution box in which you should give a
brief explanation of why the step is correct. Please briefly reference the appropriate material from the lecture or
the tutorials. Basic mathematical facts do not need to be referenced.
a)
e>2 F = (
1
λ1
KT )>(K−>T̂RK−1)
b)
=
1
λ1
T>K>K−>T̂RK−1
c)
=
1
λ1
T>T̂RK−1
d)
= −
1
λ1
(T̂T )>RK−1
e)
= 0
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