ECE 6364 Spr 2009 HW 4 Due 2/23
Problem 1.
Consider processing a blurred image B
F to form an unsharp-masked image F
ˆ via unsharp masking
B2B
ˆFFF ∇γ+= . Where blurred image B
F is formed by blurring with a 1st order neighborhood blur mask
having
δ
41−in the middle pixel and
δ
in the other 4 pixels of the mask, solve the minimization in the
handout
∑−
∑
+−
γ
+
≠
−=
+
−=
2
)0,0(),(
2
2
2
2
2]0),([])0,0([
min
nm
mnnmMKM
imize for 0>K.
to verify that the optimal
γ
is given by 556169
11028)15(
2
2
+−
+−+−
=
δδ
δδδ
γ
k (i.e. set the derivative with respect to
γ
to
0, solve for
γ
).
Problem 2.
In a 9x9 pixel window with the (0,0)th pixel in the center, label each pixel with the minimum order
neighborhood to which it belongs with respect to the center pixel of the 9x9 window.
Problem 3.
Show that the continuous LaPlacian operator ),(
2yxf∇ is rotationally invariant; ; i.e.
y
yxf
x
yxf
y
yxf
x
yxf
2
2
2
2
2
2
2
2),(),(
'
),(
'
),(
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
Problem 4.
Which of the following matrices are unitary?
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
=
121
303
222
12
1
0
A ⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−=
100
011
011
2
1
1
A
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
−−
−−
−−
=
5
1
5
3
5
3
5
11111 5
3
5
1
5
1
5
31111
2
1
2
A ⎥
⎦
⎤
⎢
⎣
⎡−
=31
13
10
1
3
A