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# CS 663 Assignment 3 solution

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1. Consider the two images in the homework folder ‘barbara256.png’ and ‘kodak24.png’. Add zero-mean
Gaussian noise with standard deviation σ = 5 to both of them. Implement a mean shift based filter
and show the outputs of the mean shift filter on both images for the following parameter configurations:
(σs = 2, σr = 2); (σs = 0.1, σr = 0.1); (σs = 3, σr = 15). Comment on your results in your report. Repeat
when the image is corrupted with zero-mean Gaussian noise of σ = 10 (with the same bilaterial filter parameters). Comment on your results in your report. Include all image ouputs as well as noisy images in the
report. [20 points]

2. Consider the barbara256.png image from the homework folder. Implement the following in MATLAB: (a)
an ideal low pass filter with cutoff frequency D ∈ {40, 80}, (b) a Gaussian low pass filter with σ ∈ {40, 80}.
Show the effect of these on the image, and display all filtered images in your report. Display the frequency
response (in log absolute Fourier format) of all filters in your report as well. Comment on the differences in
the outputs. Also display the log absolute Fourier transform of the original and filtered images. Comment
on the differences in the outputs. Make sure you perform appropriate zero-padding while doing the filtering!
[20 points]

3. Prove the convolution theorem for 2D Discrete fourier transforms. [10 points]
4. Consider a 201×201 image whose pixels are all black except for the central row (i.e. row index 101 beginning
from 1 to 201) in which all pixels have the value 255. Derive the Fourier transform of this image analytically,
and also plot the logarithm of its Fourier magnitude using fft2 and fftshift in MATLAB. Use appropriate
colorbars. [8+2=10 points]
5. If a function f(x, y) is real, prove that its Discrete Fourier transform F(u, v) satisfies F

(u, v) = F(−u, −v).
If f(x, y) is real and even, prove that F(u, v) is also real and even. The function f(x, y) is an even function
if f(x, y) = f(−x, −y). [15 points]

6. If F is the continuous Fourier operator, prove that F(F(F(F(f(t))))) = f(t). Hint: Prove that F(F(f(t))) =
f(−t) and proceed further from there. [15 points]
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Figure 1: Figures required for the last question. Fourier domain (first figure) and spatial domain (second figure)
representations of various filters.
7. Provide an explanation for the presence of strong spikes in the center of the filters in the second sub-figure
Of Fig. 1. Note that the fourier transform magnitudes of these filters are plotted in the first figure. [10
points]
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