Problem Set 4.
1. Form a low-pass spatial filter that averages the 4-neighbors of a
point but excludes the point itself. Assuming an NxN image: (a) Find the equivalent filter H(u,v)
in the frequency domain. (b) Show that this is a low-pass filter.
g(x,y)=(1/4)[f(x,y+1) + f(x+1,y) + f(f(x-1,y) + f(x,y-1)]
![](images/Image206.gif)
Because of the cosine terms, this filter has a gain of +1 at the center or
zero frequency, a gain of zero at u = v = N/4, and a gain of -1 at u = v = N/2.
Of course, N/2 is the maximum valid frequency for an N x N image. Since the gain
decreases with increasing frequency, the filter is a low pass filter.
2. Using the image shown below (icextl.tif), use sharpening to enhance
the pictures. What can you do about the uneven illumination? Can you
improve the picture?
![](images/Image207.gif)
The illumination can be made more uniform either by applying a large mean
filter to a copy of the picture until the detail is all gone and the resulting
image used as a background, a fraction of which (0.8 ?) can be subtracted from
the original. If too much background is subtracted, the picture will be very
dark. After subtracting background, try contrast stretching and/or histogram
equalization. Finally, see if applying a high-frequency boost will bring up the
crystal edges a bit.