EE5356 Digital Image Processing
Instructor: Dr. K.R. Rao
Spring 2011, Test 1
Tuesday, 22 February 2011
2:00 – 3:20 PM (1 hour and 20 minutes)
(CLOSED BOOK, CLOSED NOTES)
INSTRUCTIONS:
1. Closed books and closed notes.
2. Please show all the steps in your works.
4. You can work problems in any order.
At the end please rearrange in the correct order.
5. Please print your name and student ID.
6. No cheating, no talking.
Name_______________________________________
Student ID ___________________________________
1. For the Rayleigh density
𝑢
𝑢2
exp (− 2 ) , 𝑢 > 0
𝑝𝑢 (𝑢) = {𝜎 2
2𝜎
0
Show that the compander transformation is
𝑥
𝑓(𝑥) = 𝑐 ∫ 𝛼 1⁄3 exp (−
0
,𝑢 < 0
𝛼2
) 𝑑𝛼 − 1
6𝜎 2
where c is a normalization constant such that f(∞) = 1.
Hint:
𝑥
∫0 [𝑝𝑢 (𝑢)]1⁄3 𝑑𝑢
𝑓(𝑥) = 𝑎 { 𝑡𝐿+1
},𝑥 ≥ 0
∫0 [𝑝𝑢 (𝑢)]1⁄3 𝑑𝑢
𝑓(𝑥) = 𝑓(−𝑥), 𝑥 < 0
2. Complete the following matrices:
1
3
2
4
5
6
1
7
3
1
1:2 upsampling H & V
Based on ZOH

1:2 upsampling H & V
Based on FOH

3. Define zero order hold (ZOH) and first order hold (FOH).
4. (a) Complete the equation for Leibnitz rule:
𝑏(𝑡)
𝑏(𝑡)
𝜕
𝜕
[∫ 𝑓(𝑥, 𝑡)𝑑𝑥 ] = ∫
[ 𝑓(𝑥, 𝑡)] 𝑑𝑥 + ⋯
𝜕𝑡 𝑎(𝑡)
𝑎(𝑡) 𝜕𝑡
(b) The sketch below shows forcing a uniform quantizer on a non-uniform pdf for step size Δ,
given the number of output levels L.
[Fig. 1. Pdf of x.]
How does the step size Δ affect granular noise and overload noise? Explain clearly.
5. Explain dithering (pseudorandom noise quantization). Sketch and explain clearly.
6. Compandor. Sketch and explain the role of a compandor.
7. In an uniform quantizer for uniform PDF, if the number of output levels is doubled, what is the
increase in PSNR in dB? What is the penalty for this?
8. Define first order entropy. (Both in words & equation.)
9. A Lloyd-max quantizer with L output levels is shown and the equations for MSE are given:
Q
u
input
)2
𝑡𝐿+1
∫𝑡 (𝑢
1
𝑢̇
output
[Fig. 2. L level quantizer.]
𝑡
− 𝑢̇ 𝑃𝑢 (𝑢)𝑑𝑢 = ∑𝐿𝑖=1 ∫𝑡 𝐿+1(𝑢 − 𝑢̇ )2 𝑃𝑢 (𝑢)𝑑𝑢
)2
MSE = 𝐸[(𝑢 − 𝑢̇ ] =
1
where,
ti = input level
ri = output level
(𝑢̇ = ri, ti < ri < ti+1, i=1, 2, … , L)
Get expressions for ti, ri (closed forms). What is the meaning of ri?
10. Sketch the midrise uniform quantizer and the midtread uniform quantizer. Label the axes.
11. A bandlimited 2 D spatial signal shown in fig. 3(a) is in the 2D frequency domain. It is uniformly
sampled in the 2D spatial domain along a uniform sampling grid shown in fig. 3(b).
𝜉1
1/2
Region of
support
𝜉1
1/2
-1/2
-1/2
(a)
y
3
2
1
-5
-4
-3
-2
-1
0
1
2
3
4
x
-1
-2
-3
(b)
[Fig. 3. (a) ℱ[𝑓(𝑥, 𝑦)] = 𝐹(𝜉1 , 𝜉2 ). (b) 2 D uniform (infinite) sampling array.]
Sketch its Fourier transform.
12. What happens if a bandlimited signal is sampled below the Nyquist rate?
13. Define the Nyquist sampling theorem.
14. The Fourier transform X(w) of a signal x(t) is shown:
|X(w)|
- fB
0
fB
f
[Fig. 4. Fourier transform of band-limited signal x(t).]
x(t) is a band limited signal with bandwidth fB Hz. x(t) is sampled uniformly by multiplying with
an impulse train:
comb(t: T0)
-3 T0 -2 T0 - T0
0
T0
2 T0 3 T0
t
[Fig. 5. comb(t: T0).]
1
)]),
Sketch ℱ([𝑥(𝑡)][𝑐𝑜𝑚𝑏(𝑡, 𝑇0
assume > 2𝑓𝐵 . How can x(t) be recovered from the uniformly
𝑇0
sampled values?
15. Find the Fourier transform of a periodic impulse train. Do not derive. Also draw a clear sketch of
the Fourier transform.
The given impulse train is represented as:
-3∆x -2∆x -∆x
∞
∆x
2∆x 3∆x
[Fig. 6. Periodic impulse train.]
comb(x: ∆x) = ∑ δ(x − m ∙ ∆x)
𝑚=−∞
0
x