Exercise questions for Signal Processing and Analysis
Lecture 1 and 2:
1. How is a Linear Time Independent system defined, explain its properties? 2. What is meant by a “causal” system and how can you find out from a systems impulse
response if it is “causal”? 3. An LTI system is computing its output signal ���� from its input ���� and according to
the following equation, ���� � ���� ∗ ���� � ∑ ��� �� ∙ ���� ���
a) What input signal should be used if you by simulation want to find out the impulse
response ����? b) Draw the signal flow graph (SFG) for an FIR filter and for K = 3. c) Can this FIR filter become unstable? Motivate preferably by equations.
4. This is the system function for a filter, ���� ����
����.���������.��
a) Develop the difference equation for this filter b) Draw a diagram showing the corresponding Signal Flow Graph for its simplest
direct implementation form I c) Define the filter coefficients d) Draw a diagram showing SFG for direct form IIt e) What type of filter is this, FIR or IIR? f) What order K has this filter? g) Draw a zero-pole diagram. Is this filter stable?
5. Graphs below are showing the response of an FIR filter having linear phase characteristic.
Compute the group delay, meaning how much is a signal delayed when passing through
the filter?
Lecture 3:
6. Graphs below are showing filter output signals after simulation where the unity impulse
has been used as input. What kind of conclusions can you make about those two graphs?
7. Graphs below is showing simulation output for a signal, time plot to the left and a
histogram of signal values to the right. Signal is represented using 32 bits floating point
arithmetic. Minimum and maximum signal values during simulation was -0.3961 and
0.3726.
• Select a proper fixed point representation of this signal, assuming that you have in
total 16 bits for sign, integer and fractional bits. Motivate your answer.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Time [ms]
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Filter output
0 1 2 3 4 5 6 7 8 9
Time [ms]
-8
-6
-4
-2
0
2
4
6
8
Filter output
10112
0 1 2 3 4 5 6 7 8 9
Time [ms]
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Filter output
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Signal values
0
50
100
150
200
250
300
Counts
Lecture 4 and 5:
8. Figure 1 depicts a diagram for the amplitude characteristic of a 2D linear filter. What kind
of filter is this, High Pass, Low Pass or Band Pass ? What do you expect to be the visual
effect on an image if this filter where applied on it?
Figure 1. Amplitude characteristic for a 2D filter.
9. The three pictures above show the amplitude transfer function for a 2D Butterworth filter.
The amplitude characteristics is illustrated as a mesh plot, an intensity image and as a
radial plot for different orders n of the filter.
a) What class of filter is this, Low Pass, High Pass, Band Pass or Band Stop filter?
b) Which one of the pictures labeled A to E is filtered using the smallest value for r as
defined in the amplitude characteristics above.
r
r
10. Explain shortly what kind of image processing operations is necessary for high quality
downscaling of an image?
11. Figure 2 shows a picture of the silhouette of a screw taken at back lightening. The
silhouette is highlighted at subpixel precision by image processing. Suggest a method for
how this image processing can be done.
Figure 2. Screw thread.
12. Assume a gray-level image f(r,c) and its smoothened correspondence g(r,c). The region of
interest is R. Then the dynamic thresholding of brighter objects on a dark background can
be defined as, { }diffgcrgcrfRcrS ≥−∈= ),(),(|),(
Where gdiff is a fixed constant. Pictures A and B both have bright spots on a darker
background. If compared with using simple global thresholding, which one of the pictures
A or B will require the use of dynamic thresholding in order to successfully segment the
bright spots from the background? Motivate your answer shortly. If you answer with a
long and not precise story, your credits will be reduced.
Pic. A Pic. B
Original
image
A
B C
D E
13. Image (2) was processed by Histogram equalisation to create image (1).
a) Which one of the histograms A and B correspond to image (1) and (2)?
Explain and motivate
b) How is the graylevel transformation function computed for Histogram equalisation?
Image (1)
Image (2)
Histogram A Histogram B
0
2000
4000
6000
8000
10000
0 50 100 150 200 250
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 50 100 150 200 250
Lecture 6: 14. The following equation defines a morphological operation, { }∅≠∩= ABxOP x)(|
).
a) What is the name of this operation?
b) Which one of the following pictures below is the correct graphical illustration of the
effect of that operation for a binary image A and a structural element B?
Illustration A)
Illustration B)
Illustration C)
Illustration D)
15. Two image points (x1,y1), (x2,y2), lying
on a single line are shown. The
corresponding lines in a parameter space
are also shown. This transformation can
be utilized in the Hough transform to find
lines in an image. Explain shortly how
this detection of lines works for the Hough transform and how the line parameters for that
line can be measured.
A
16. Figure 3 depicts an image object A and a structural element B used for the morphological
operation Dilate. Dilate is defined as, { }∅≠∩=⊕ ABxBA x)(|)
. M)
is the reflection of
a region M and xM is the translation of region M by a vector x. Draw a nice picture and
show how BA ⊕ will look like.
Figure 3. Image object A and the structural element B used for morphological operations.
17. Explain how Template matching is working and suggest also a method how to cope with
the increasing execution times for Template matching as the resolution of the image is
increased.
18. The following drawing shows a square shaped region A of pixels belonging to one single
image component in a binary image. Region B is a circular shaped structural element
having the diameter 2 and with its origin at the center, indicated with a dark spot.
A morphological operation OP is defined as, { }ABxOP x ⊆= )(| .
a) What is the name of this operation as known in all reference litterature?
b) Make a simple sketch having the right proportions showing the visual effect on region
A after applying this morphological operation OP using structural element B. Also make
an indication in your drawing on what the size of the processed region will be. I want just
the sketch as an answer, nothing else.
A
B 3
2
19. The following binary picture to the left shows vertical and horizontal lines having a width
of 5 pixels. Distances between lines are at least 30 pixels. Consider lines as belonging to
region A. The drawing to the right shows a structuring element B.
A morphological operation OP1 is defined as, { }ABxOP x ⊆= )(|1
.
Another morphological operation OP2 is defined as, { }∅≠∩= ABxOP x)(|2
)
a) What are the names for operations OP1 and OP2 ?
b) Apply firstly OP1 on region A and then apply OP2 such that C = OP2(OP1(A)).
Make a drawing and show how region C will look like.
20. An Edge Histogram Descriptor EHD is computed on the two pictures shown below.
Estimate and illustrate EHD for the two pictures A and B. Explain what the diagrams show
and why they look like they do.
Picture A Picture B
B 10 pixels
1 pixel
Lecture 7 and 8:
21. Explain shortly how a minimum distance classifier works. What kind of priori statistics is
computed for the trainings sets?
22. Explain the KNN classifier and how it works, preferably as pseudo code describing the
classification. How is the KNN classifier trained?
23. Explain how the K-means clustering works, preferably as pseudo code describing the
clustering. To which one of the following groups of artificial intelligence does the K-
means belong to: (1) Supervised learning, (2) Unsupervised learning? Motivate!
24. A linear classifier is described by a hyperplane (or hyperline if 2D) according to the
following equation, �� ∙ �� + ! � 0. The following parameters are given for the classifier:
• �� � �9,3� and ! � 18
The three input data vectors, ��(to ��*should be classified by the classifier above. Do this
manual calculation and find out which group of two vectors out three is belonging to the same
class?
• ��( � �2,10� , ��, � �0,5� , ��* � �2,20� Finally, prepare a graph that illustrates the relevant feature space, the separating hyperline,
and the three data vectors, ��(to ��*.
Lecture 9: 25. Draw a picture and explain how a sheet of light laser can be used together with an area
scan sensor for acquisition of a 3D-surface and based on triangulation techniques. Just
explain the measurement principle how it works.
26. Figure 4 depicts a schematic setup for stereo imaging based on two image sensors and an
object W at distance Z given by 12 xx
BZ
−−=
λλ . The object W is projected onto the image
sensors 1 and 2 at position (x1 ,y1) and (x2 ,y2) respectively. Explain what kind of image
processing is necessary in order to measure the distance Z from the two sensors to the
object W. Relate your explanation to the given expression for Z.
Figure 4. Stereo imaging.
27. The position of a laser line projected onto an image detector versus height of object is
shown in Figure 5. One curve is representing measured values used for calibration and
second curve shows a computed transfer function. These curves comes from a setup for
laser scanning used to capture a 3D surface. It shows almost a perfect linear relation
between pixels and height. From measurements and it was shown that the standard
deviation of computed position of laser line was 0.2 pixels.
a) Explain shortly what property of captured images is limiting precision of laser line
position to 0.2 pixels?
b) What is the precision of height measurement that this scanner can achieve?
pixels
0 10 20 30 40 50 60
hei
ght
[mm
]
3.4
3.6
3.8
4
4.2
Measured slope =0.014406
Computed slope =0.014209
Calibration reference level =3.3139 mm
Deviation in slopes =0.0089986 mm
Comparison of computed and measured levels
Measured Heights
Computed Heights
Figure 5. Height versus position on image detector.
28. The intensity profile of an imaged laser line is shown in Figure 6. When Center Of
Gravity (COG) is computed to find position of laser line in one of the spatial dimensions,
a threshold can be used.
a) Explain and motivate why this threshold is used for a laser scanner.
Figure 6. Gray level versus pixels for an imaged laser line.
29. A laser scanner is using a step size of 0.5 mm. What is the highest frequency along the
scanning dimension that can be resolved?
30. A laser scanner is using a telecentric lens having an optical amplification of 0,25. Pixel
size of image detector is 10 µm.
What is the highest frequency along the laser line that can be resolved?
0 50 100 150 200 2500
50
100
150
200
250
[Pixels]
Gray level
Threshold
Lecture 10: 31. The homogeneous camera coordinates ./ and the homogeneous world coordinates �/ are
approximately linearly dependent according to calibration matrix A,
⋅
=
⋅=
144434241
34333231
24232221
14131211
4
3
2
1
Z
Y
X
aaaa
aaaa
aaaa
aaaa
c
c
c
c
WAC
h
h
h
h
hh
a) Describe briefly the practical camera calibration procedure to acquire matrix A.
You are not required to define the mathematical computation, just how the
calibration is done.
b) Lens distortion can be model according to,
01 � �1 + 234� + 2�4
5 + 2�46� ∙ 07 + 81
where 81 � �229�7�7 + 25�4� + 2�7
�� 29�4� + 2�7
�� + 225�7�7�:
What impact can the lens distorsion model have on the residual of the camera
calibration?
c) What physical property is the parameter r modelling?
32. What is meant by camera intrinsic and extrinsic parameters?
33. What is meant by an “affine” transformation?
34. A rotation around z-axis by angle ; is defined by the geometric affine transformation <=.
−
=
1000
0100
00cossin
00sincos
θθ
θθ
ZR
Y
Z
X
θ
. (X,Y,Z)
A translation T by vector ���, >�, ��� is defined as,
−
−
−
=
1000
100
010
001
0
0
0
Z
Y
X
T
a) Compute a transformation matrix that is a combination of firstly a rotation <=and
then a translation T.
b) Compute a transformation matrix that is a combination of firstly a translation T
and then a rotation <=.
35. The following plot shows synthetically generated point cloud data.
Z-dim
ension
The following left and right side graphs show two different transformations of previous point
cloud,
Which one of graphs A or B are showing transformation <= ∙ 8and which one is showing T?
Motivate your answer.
Z-dim
ension
Z-dim
ension
A) B)
Formulas: Z-transforms,
Operation ?�@� A�B�
Single sided Z-transform for
causal systems ����
C���� ∙ D�7E
7��FG4|D| I <�
Right shift ��� J� D�K��D� C���� ∙ D��K
K
��3
Left shift ��� J� DK��D� C ���� ∙ DK��
K�3
���
Group delay, LM ≡ 1(OMPQ�R�S1R
Convolution, ���� � ���� ∗ ���� ≡ ∑ ��� �� ∙ ����E��� � ∑ ���� ∙ ��� ��E���