Midterm 1 and Solutions

Questions:

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05/04/2016

TUESDAY

ECE 599 DIGITAL IMAGE PROCESSING

1. MIDTERM EXAM

Duration: 120min

Rules and notes: Closed book, no cheat-sheet. You can use your calculator. Please attempt all questions.

Grading
Q1	Q2	Q3	Q4	Q5	Q6	Total
10	15	20	15	20	20	100

Q1) Putting the corresponding numbers into the boxes below, order the following electromagnetic waves in descending order with respect to their frequencies:

1. X-rays,
2. GSM signals (cell phones),
3. Infrared,
4. Ultraviolet,
5. Red,
6. Black,
7. Blue,
8. Gamma rays,
9. Microwave,
10. FM radio

Q2) Affine matrix of rotation is given in Fig. 1. Rotate the image in Fig. 2. by 60° and find new coordinates of the pixels. (Note: new coordinates do not need to be an integer.)

sin(60)=0.87, cos(60)=0.5,

cos() sin() 0 Tr   = -sin() cos() 0 0 0 1 Figure 1

7 8 9 10 Figure 2

Q3) In an 2 bit image (That is, pixels can take values between 0 and 3), number of occurrences of each of pixel values (histogram) are given as:

n0 (Black) = 0,

n1 = 5,

n2 = 80,

n3 = 15

a) Find histogram equalization transformation function of this image. Show your work.

b) Apply the transformation function and find the equalized histogram. Plot it.

Q4) In Fig. 3, a 3x3 box filter in spatial domain (not in frequency domain) is given. We can use box filters to soften the image. If we want even softer image, we apply it again. Find a filter which gives the same effect with applying the box filter in Fig. 3 for two times. (Hint: size of your filter might be bigger than the original box filter.)

1/9	1/9	1/9
1/9	1/9	1/9
1/9	1/9	1/9
Figure 3. A box filter

Q5) Magnitude plot of frequency domain representation of function f(t) is given in Fig. 4. The following questions are related to Fig. 4.

Figure 4. Magnitude of F(𝞵), which is Fourier transform of f(t).

1. What would you see in frequency domain (draw) if you sample the signal f(t) in
1. 250,
2. 200,
3. 150 and
4. 100 Hz.
2. After sampling it, in which cases is it possible to perfectly recover the original signal?
3. For each case where there is no aliasing, How do you recover the original signal after sampling it ? (what kind of filter would you apply to it in freq. domain? (Box?) What are its cutoff frequencies?)
4. For each case where there is aliasing, before sampling the signal, What would you do to prevent aliasing ? (Give numerical values of your filter.)

Q6) Roughly sketch magnitude of DFT of the image in Fig. 5. It is an image of an equilateral triangle with a thick bottom edge. Explain your drawing. (Hint: DFT of a rotated image rotates in same degree.)

Figure 5. An image

Good Luck!

Solutions