Description

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]
1

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-figureOf Fig. 1. Note that the fourier transform magnitudes of these filters are plotted in the first figure. [10
points]
2