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���