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

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