ME 565 HW 4 Solutions Winter Make image black and white. Compute the FFT of our image using fft2. clear all; close all; clc %%Exercise 4.

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1 ME 565 HW 4 Solutions Winter 2017 clear all; close all; clc %%Exercise 4.1 %read in the image A= imread('recorder','jpg'); Make image black and white Abw2=rgb2gray(A); [nx,ny]=size(abw2); Compute the FFT of our image using fft2 disp('doing FFT analysis for sparsity check...') Doing FFT analysis for sparsity check... At=fft2(Abw2); F = log(abs(fftshift(at))+1); %scaling for plot F = mat2gray(f); % Use mat2gray to scale the image between 0 and 1 % figure(3) % imshow(f,[]); % Display the fft coefficients % taking the fft At=fft2(Abw2); %%%what we want is to keep exactly a certain percentage of the coefficients

2 count_pic=2; trials=[10, 1]; count=nx*ny; %the total number of pixels figure(2), subplot(1,3,1), imshow(abw2) title('original image','fontsize',12) figure (3), subplot(1,3,1),surf(double(abw2)); title('original image','fontsize',12) for ii=1:length(trials) percentage=trials(ii); indceil= ceil(0.01*percentage*count); %the number of coefficients for the final image %order the coefficients from largest to smallest and keep the top x %zero out the indices that are not in this top x% %reshape At; reshaping does not change the intrinsic order of the elements %if we use linear indexes At_rshp=reshape(At,count,1); %sort descending [At_rshp_srt, topindices]= sort(abs(at_rshp),'descend'); %the indices of top x% of the Fourier Coefficients idx_lookup= topindices(1:indceil); % check(ii)=length(idx_lookup); %I make a matrix At_reduced the size of my original At and then subsitute in the %elements that correspond to the top x% of the Fourier Coefficients At_reduced=zeros(size(At)); At_reduced(idx_lookup)=At(idx_lookup); Alow=uint8(ifft2(At_reduced)); %L2 norm comparison disp('l2 norms...') image_error=abw2-alow; L2_norm_error=norm(double(image_error),2) L2_norm_fft_orig=norm(At,2) L2_norm_fft_comp=norm( At_reduced,2) norm_low=norm(double(alow),2) %computing the L2 norm of the new image figure(2), subplot(1,3,count_pic), imshow(alow); drawnow title([num2str(percentage) '% of FFT basis'],'fontsize',12) figure(3), subplot(1,3,count_pic), surf(double(alow)); drawnow title([num2str(percentage) '% of FFT basis'],'fontsize',12) count_pic=count_pic+1; end clear idx_lookup L2 norms... L2_norm_error = L2_norm_fft_orig = e+08 L2_norm_fft_comp = e+08 norm_low = e+05

3 L2 norms... L2_norm_error = e+03 L2_norm_fft_orig = e+08 L2_norm_fft_comp = e+08 norm_low = e+05

4 The L2 norm of the error between the original image and the 10% case is smaller than for the 1% case. The L2 norm of the fourier coefficients for the 10% case is also closer to the L2 norm of the fourier coefficients of the original image indicated that it is a closer match to the original. This matches our assessment if we view the 10% and 1% images and compare these to the orignal image.

5 Exercise 4-2 clear all; close all; clc vars={'rush','fs'}; load ('r2112.mat',vars{:}) disp('play the original signal...') Play the original signal... sound(rush, FS); pause clc %Compute FFT of signal N=length(rush); FFT=fft(rush, N); %Compute PSD of signal and plot it along with the spectrogram figure PSD_clean=FFT.*conj(FFT)/N; freq=(fs/n)*(1:n); L=1:floor(N/2); subplot(211) plot(freq(l), 10*log10(PSD_clean(L))) xlabel('frequency (Hz)') title('power Spectral Density -original Signal')

6 subplot(212) spectrogram (rush, 1000, 500, 1000, FS, 'yaxis') title ('Spectrogram of original signal') %Prefiltered signal analysis vars_noisy = {'rushnoisy'}; load ('r2112noisy.mat',vars_noisy{:}); %Listen to the noisy sample disp ('Play corrrupted signal...') Play corrrupted signal... sound(rushnoisy, FS); pause clc n=length(rushnoisy); FFT_noisy=fft(rushnoisy,n); %Plot PSD and spectrogram of noisy signal figure PSD_noisy=FFT_noisy.*conj(FFT_noisy)/n; freq_n=fs/n*(1:n); subplot(221)

7 plot(freq_n(l), 10*log10(PSD_noisy(L))) xlabel('frequency (Hz)') title('psd-noisy Signal') subplot(222) spectrogram (rushnoisy, 1000, 500, 1000, FS, 'yaxis') title ('Spectrogram of noisy signal') %Filtering and reconstuction of signal FFT_filtered=FFT_noisy; FFT_filtered(n/4:n)=0; %removing last 3/4 of Fourier components clean_signal =2*real(ifft(FFT_filtered)); disp('playing filtered signal...') Playing filtered signal... sound(clean_signal, FS) pause clc disp('compare to original...') Compare to original... sound(rush, FS) pause clc n_cleansig=length(clean_signal); FFT_cleansig=fft(clean_signal, n_cleansig); PSD_cleansig=FFT_cleansig.*conj(FFT_cleansig)/n_cleansig; freq_clean=fs/(n_cleansig)*(1:n_cleansig); subplot(223) plot(freq_clean(l),10*log10(psd_cleansig(l))) xlabel('frequency (Hz)') title('psd-filtered signal') subplot(224) spectrogram (clean_signal, 1000, 500, 1000, FS, 'yaxis') title ('Spectrogram of cleaned signal')

8 The above shows that we have successfully removed the high frequency contamination (as well as the high frequency components above Hz in the original signal)