ECE 6364 Spring 2009 HW 11 Due 4/27
Problem 1.
Given an image blurred by camera motion
[]
)()(
2
)( 2
0Ttutu
ta
tx −−=
0)(
0=ty
Find the transfer function ),( vuH for the blurring function.
Hint:
∫
=xdxC
0
2)cos()(
γγ
is known as the Fresnel Cosine integral and ∫
=xdxS
0
2)sin()(
γγ
as the Fresnel Sine integral.
There is no known closed form solution.
)(xC and )(xS must be evaluated via numerical integration. )(xC and )(xS versus x are available in table form.
When you use )(xC or )(xS in a program (e.g. image restoration in the presence of motion blur) and you want the program to work for
different values of x, you simply load the table into a matrix and index the entry that applies from within the program.
In deriving a problem solution, one simply replaces ∫
xd
0
2)cos(
γγ
with )(xC and continues towards the answer, knowing that )(xC is
a well-defined quantity, known to all.
Problem 2.
You want to compute the linear convolution of two 1-D images ][**][ fedcba using DFTs. You load the two 1-D images into
zero-padded vectors as
[]
cba ][00 and
[]
0][0 fed , compute the DFTs of the two vectors, multiply the two DFTs
together term-by-term, and then take the inverse DFT. Your result is the vector
[
]
9.52.40.32.21.1 . Determine the numerical
result of the linear convolution ][**][ fedcba , marking the 0th pixel with a box.
Problem 3.
Let
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
987
6]5[4
321
H be a blurring mask, and ⎥
⎦
⎤
⎢
⎣
⎡
=dc
ba][
X be a 2x2 image. Where
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
tsrq
ponm
lkji
hgfe
][
** XH , we are only
interested in the central 2x2 portion ⎥
⎦
⎤
⎢
⎣
⎡
on
kj][ of the linear convolution that is the same size as the image X.
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