ARIZONA STATE UNIVERSITY EEE 508 Spring 2015 Problem Set

ARIZONA STATE UNIVERSITY
SCHOOL OF ELECTRICAL, COMPUTER, AND ENERGY ENGINEERING
EEE 508
Spring 2015
Problem Set #2
Assigned: 5 February 2015
Due Date: 19 February 2015
Reading: Finish reading Chapters 1 and 2 in the OpenCV book.
Read Chapters 3, 4, and 6 in the OpenCV book.
PROBLEM 2.1:
(4 points) The 2 × 2 DFT is defined as
X(k1 , k2 ) =
NX
1 −1 N
2 −1
X
−j
x(n1 , n2 )e
2πk1 n1
N1
−j
e
2πk2 n2
N2
n1 =0 n2 =0
where N1 = N2 = 2.
(a) This transform may also be written in the matrix form Y = T T XT where X,Y , and T are
the input, output, and transform matrices. Determine T .
(b) The 2 × 2 DFT can also be expressed by y = T T x, where x is an input vector formed by
unwrapping x(n1 , n2 ).
For example, if x(0, 0) = a, x(0, 1) = b, x(1, 0) = c, and x(1, 1) = d, then




x=
a
b
c
d



;

y is a 4-element column vector (corresponding to the unwrapped output) and T is a 4 × 4
matrix. Determine T.
(c) With x defined as in (b), determine T if y = T T x is the 4-point DFT of the 4-element vector
x.
PROBLEM 2.2:
(3 points) Consider the N1 × N2 transform X(K1 , K2 ) of the N1 × N2 image x(n1 , n2 ):
s
X(K1 , K2 ) =
2
N1
s
N1 −1 NX
2 −1
2 X
πn2 K2
2πn1 K1
cos
x(n1 , n2 ) cos
N2 n =0 n =0
N1
N2
1
2
where 0 ≤ K1 ≤ N1 − 1 and 0 ≤ K2 ≤ N2 − 1.
(a) (1 point) Express X(K1 , K2 ) in matrix form when N1 = N2 = 2.
(b) (1 point) Express X(K1 , K2 ) in vector form when N1 = N2 = 2.
(c) (1 point) Is X(K1 , K2 ) a unitary transform? Why? (Justify your answer)
PROBLEM 2.3:
(3 points) The analysis section of a 2-D separable filterbank system is shown below.
x(n1 , n2 )
- H (ω )
0 1
-
- H (ω )
1 1
-
- H (ω )
0 2
-
?2
- H (ω )
1 1
-
?2
- y (n , n )
0 1 2
?2
?2
- H (ω )
0 2
-
?2
- y (n , n )
1 1 2
- y (n , n )
2 1 2
H0 (ω) is an ideal lowpass filter with cutoff π/2; H1 (ω) is an ideal highpass filter with cutoff π/2.
Consider the following 2-D frequency bands of X(ω1 , ω2 ) (where X(ω1 , ω2 ) = DFT{x(n1 , n2 )}).
ω2 6
π
M
N
O
P
I
J
K
L
F
G
H
B
C
D
3π/4
π/2
E
.
π/4
A
0
π/4
π/2
3π/4
π
- ω
1
(a) (1 point) Indicate the frequency band or the group of frequency bands corresponding to
subband image yo (n1 , n2 ).
(b) (1 point) Indicate the frequency band or the group of frequency bands corresponding to
subband image y1 (n1 , n2 ).
(c) (1 point) Indicate the frequency band or the group of frequency bands corresponding to
subband image y2 (n1 , n2 ).