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(comp344)[2007](f)final~PPSpider^sol_10207.pdf
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COMP344 Digital Image Processing
Fall 2007
Final Examination
Midterm

Time allowed: 2 hours

Name
Student ID
Email




Question 1
Question 2

Question 3



Question 4
Question 5
Question 6


Total
With model answer


HK University of Science and Technology Page: 1of 12
Course Code : COMP344 Section No.: ALL Time Allowed : 2 Hour(s)
Course Title : Digital Image Processing Total Number of Pages : 12

Time allowed: 2hours
Answer all questions.
The total mark for this quiz is 100.
This is a closed-book quiz.

1. a) Compute the Fourier transform (F ()= D f(t)e.jtdt over a suitable domain D)of
.
.1 .1 t 0
the step function f(t)= de.ned on .1 t 1. (5 marks)
1 0 <t 1

HK University of Science and Technology Page: 2of 12
Course Code : COMP344 Section No.: ALL Time Allowed : 2 Hour(s)
Course Title : Digital Image Processing Total Number of Pages : 12

2
1. b) Let f be a Gaussian function f(x)= e . 2x2 . It is known that the Fourier transform of
22
f is also a Gaussian function, i.e., F(f(x)) = F ()= 2e. 12 where is the angular frequency. Now consider the 2-dimensional .lter with frequency domain representation
2 2
uv 2222
xy
F (u,v)= e .. ,
where u,v are the angular frequencies. Derive its spatial domain representation f(x, y). (10 marks)
Answer to question 1:
1. a)
. 1
.jtdt
F ()= f(t)e
.1
. 0 . 1
.jtdt + .jtdt
= .ee
.10
. 1
.jt .ejt
= .e dt
0
. 1
= .2j sin(t)dt
0
2j 2 .2cos()
= (cos() .1) =
j
1. b) Since F (u, v) can be decomposed into the multiplication of two univariate Gaussian functions, we can consider each dimension separately. Based on the given information,
.
. x
2F.1 2e. 1 22 = e 222 (1)

2 . 222 1
F.1 . x
e 22 = e 2 ( ) (2)

. 2 . x 22
F.1 22
2
e = e (multiply on both sides ) (3)
2 2
2 2 2 2
uv v
.. . u F.1 e 2x 22y 2 = F.1 e 22 F.1 e2y 2 (4)
x
2 22
x . x y . y 2
xy
22
= e e (5)
2 2
xy x 22 y 2y 2 = e . x . (6)
22
2 2) The following .gure shows the frequency domain representation of a notch .lter. Here, the dark area has a value of 0 while the white area has value 1. The centers of the two dark rectangular (which are symmetric with respect to the origin) are (u0,v0)and (.u0, .v0), respectively. The sides of the rectangle are of lengths L1 and L2, respectively. Is this a bandpass or bandreject .lter? Explain. Write down the H(u, v) of this notch .lter. (5 marks)
HK University of Science and Technology Page: 3of 12
Course Code : COMP344 Section No.: ALL Time Allowed : 2 Hour(s)
Course Title : Digital Image Processing Total Number of Pages : 12


Answer to question 2:
This is a bandreject .lter, and

. 11
. 0 u [u0 . 1 L1,u0 + L1],v [v0