Computer Engineering Department
TED University
Fall 2020 - CMPE 362/562 Digital Image Processing
Midterm
Due Date: November 29, 2020 at 23:59
Note: This work is to be done individually. The answers can be either handwritten or typed
on word document. If you chose to submit handwritten results: give clean answers, and
before submission, make sure that your pdf is readable. Do not mix the pages and submit
questions in given order. For each question give clear explanations. Do not just write the
answers.
Q1 (15 points)
Given the 4x4 image shown below, compute the histogram equalized image (assume that the
gray-levels are in the range [0,7]). Show all your steps.
4224
2047
6226
2324
Computer Engineering Department
TED University
Q2 (15 points)
A 3 bits/pixel image of size 5x5 is given below. Find the following: (a) the output of a 3x3
averaging filter at the pixel (3,3) (b) the output of a 3x3 median filter at the pixel (3,3), and (c)
the outputs of the Sobel filters (shown below) at the pixel (3,3). Show your work.
Image:
y=1 y=2 y=3 y=4 y=5
-------------------------------------------------------x=1|
3
7
6
2
0
|
x=1|
|
x=2|
2
5
6
2
1
|
x=1|
|
x=3|
4
7
2
0
4
|
x=1|
|
x=4|
3
5
6
3
1
|
x=1|
|
x=5|
5
7
5
1
2
|
x=1|
|
--------------------------------------------------------
Sobel filters:
Computer Engineering Department
TED University
Q3 (15 points)
Consider the continuous function 𝑓(𝑥) = sin(𝜋𝑥) + 4 ∗ sin(6𝜋𝑥) + 2 ∗ sin(8𝜋𝑥).
a. What is the highest frequency of 𝑓(𝑥)?
b. What is the Nyquist rate?
c. At what rate would you sample 𝑓 (𝑥 ) so that perfect recovery of the function from its
samples is possible?
Computer Engineering Department
TED University
Q4 (20 points)
Consider a one dimensional discrete function 𝑓(𝑥) with samples 𝑓 (0) = 2, 𝑓 (1) = 2, 𝑓 (2) = 4,
𝑓 (3) = 2. What are 𝐹(0) and 𝐹(3) (the Fourier transform of 𝑓(𝑥) at the frequencies 𝑢 = 0 and
𝑢 = 3)?
Computer Engineering Department
TED University
Q5 (20 points)
Show the matching between the spatial domain images and the Fourier magnitude images where
F(0,0) is located at the center. Explain your reasoning.
Spatial domain images:
A
B
D
E
C
Fourier magnitude images:
1
2
4
5
3
Computer Engineering Department
TED University
Q6 (15 points)
Consider that we want to estimate the intensity value of the following image at the coordinate
x = 3.2 and y = 2.7.
Image:
y=1 y=2 y=3 y=4 y=5
-------------------------------------------------------x=1|
3
7
6
2
0
|
x=1|
|
x=2|
2
5
6
2
1
|
x=1|
|
x=3|
4
7
2
0
4
|
x=1|
|
x=4|
3
5
6
3
1
|
x=1|
|
x=5|
5
7
5
1
2
|
x=1|
|
-------------------------------------------------------a. What is the interpolated value if we use nearest neighbor interpolation?
b. What is the linear system of equations to be solved if we want to compute the
interpolated value using bilinear interpolation?
Show your work.