Problem 1: Exercises on Low-pass and High-pass Filters in the Frequency Domain. Some common steps to transform an image into frequency domain: I = double(I); % to save accuracy I = fft2(I); I = fftshift(I); % to shift the low frequency components into center.

a) Design your own Gaussian low-pass filter with the standard deviation of 40 at both directions in the frequency domain. Obtain the filtered image by filtering the original image Sample with the designed Gaussian filter. Display the original image, the Gaussian low-pass filter (treat it as an image), and the filtered image in figure 1 with the appropriate titles. (hint: Frequency domain filter has the same size with image, e.g., 300x300. You need to use the formula in lecture to construct this filter image, the D0 is the distance to the center of the image (X/2, Y/2))

b) Design your own butterworth high-pass filter of order 2 (n=2) with a cutoff frequency of 80 in the frequency domain. Obtain the filtered image by filtering the original image Sample with the designed butterworth filter. Display the original image, the butterworth filter (treat it as an image), and the filtered image in figure 2 with the appropriate titles. (hint: the filter should have the same size as the image, the high-pass filter can be treated as 1-lowpass filter).

c) Summarize the effects of using the Gaussian low-pass filter and the butterworth high-pass filter by using display command.

d) Close all figures and all variables in the workspace.

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