HW 6, EECS 203A
Problem 5.16, Digital Image Processing, 2nd edition by Gonzalez, Woods.
Nasser Abbasi, UCI. Fall 2004
Question
Consider a linear position-invariant image degradation system with impulse
response
Suppose that the input to the system is an image consisting of a line of
infinitesimal width located at
and modeled by
where
is the impulse. Assuming no noise, find the output image
Solution
In general,
where
is the noise. Hence since
,
we have
Now, I can solve this using spatial domain (convolution), or solve in Fourier
transform domain, then inverse transform to get
.
I'll try the direct spatial approach:
where
are the dimensions of the image
Since
then
only when
,
that we get a non-zero value for the the output image. At all other values for
Hence
Since
then Substitute
we get
How to evaluate these sums?
Since we are told that the input image is of infinitesimal width, then this
means we can consider the sum
to have only one point. i.e.
so
To continue, best I could do is to look up the tables for integrals, and use the results for
where
is the error function defined as
Hence, using these, we get
Since
are constants, then
is a constant, call it
Then
where
new
constant
Notice that
is a function of
as well, since constant
value depends on
via the equation given above for
I