BENG280A, Principles of Biomedical Imaging 10/12/09 Fall Quarter 2009 HOMEWORK #3 Due in Class on Thursday 10/22/09 Readings: Sections 2.2.3; 2.2.5, 2.2.6; 2.4, 2.5, 2.6, 3.2; 3.3; 6.3 Preview Section 2.8 Problems: 1. Professor Smith shows Professor Jones the following line in his derivation of a proof: € s(y) ∗ t(y) = s(x /m) * t(x /m) where y = x /m Professor Jones claims that Professor Smith has made a mistake. Who is correct? If Dr Jones is correct, what is the correct answer? Hint: Make use of the definition of convolution. 2. Prove the following Fourier transform properties a) Rotation property (Eqns 2.90 and 2.91). b) If € g(x) is a real and even function, show that its Fourier transform is also real and even. c) Conjugation Property (Eqn 2.84) d) Conjugate symmetry (Eqn 2.85). e) Scaling Property (Eqn 2.89) f) Show that the conjugate symmetry property holds for the function € cos(2π(x + y) + 3π/4). In addition, sketch the function and its Fourier transform. 3. Use the convolution and modulation theorems to find and graph the transforms of the following functions: € sinc(2x) ∗sinc(66x) and € sinc(5x)cos(24πx)[ ]2 4. Let € g(x, y) = exp(− j2π(16x + 9y))sin(28πx). Find and sketch its 2D Fourier Transform. Hint: the function is separable. 5. Problem 6.8 6. Problem 6.13 Matlab Exercise: The purpose of this exercise is to familiarize you with the MATLAB functions used for performing 2D Fourier transforms and manipulating and displaying images. Boldfaced items should be turned in with the homework. You can always get more information about a command by typing help <name of command>, e.g., help fft2. Steps: 1. First download the file BE280Ahw1im.mat from the course website. 2. Load the image into MATLAB with the command: load BENG280Ahw1im. 3. Type whos to see the variables in your MATLAB workspace. You should see a variable named Mimage. Type size(Mimage) to see how large the image is. 4. Use the command imagesc(Mimage) to display the image – you should see a sagittal image of a head. To change the colormap display to gray-scale, type colormap(gray(256)) which will result in a display with 256 shades of gray. Print out a copy of the image. Try experimenting with different numbers for the colormap, e.g. colormap(gray(20)); 5. Compute the 2D Fourier transform of the image with the command Mf = fft2(Mimage); where the 2D transform will now be stored in the variable Mf. Remember to add the semi-colon at the end of the command, otherwise MATLAB will display all the numbers in the matrix! The command fft2 puts the zero-frequency value of the transform at the first indices of the matrix. For display it’s convenient to put the zero-frequency value in the center of the matrix. To do this, type Mf = fftshift(Mf);6. Type imagesc(abs(Mf)) to display the magnitude of the transform. It will be hard to see anything, because the dynamic range of the Fourier transform is so large. To get a sense of the dynamic range, find the minimum value of the Fourier Transform with the command min(min(abs(Mf))) and the maximum value with the command max(max(abs(Mf))). Record these minimum and maximum values. What is the ratio of the maximum to minimum value? 7. To scale the image so that you can get a better sense of what the Fourier transform looks like, you can use the imagesc command with the syntax: imagesc(abs(Mf),[cmin cmax]) where cmin will be displayed as the darkest value on the image and cmax will be displayed as the lightest value on the image. For example, typing in imagesc(abs(Mf),[200 1e6]) should give you a nice looking result. Experiment with different values of cmin and cmax. 8. You can also look at the real part, the imaginary part and the phase of the transform with the commands imagesc(real(Mf)), imagesc(imag(Mf)), and imagesc(angle(Mf)), respectively. If necessary you can use the [cmin cmax] option to scale the image properly. Print out images of the magnitude and phase and real and imaginary parts of the transform. To plot more than one image on the same Figure, you can make use of the subplot command in MATLAB. 9. Another way to look at the transform is to use the mesh command. Try mesh(abs(Mf)). It will be a little hard to see what is going on, so do the following: Define span = 128 + (-20:20); then type mesh(abs(Mf(span,span))); 10. Resolution. What happens when we zero out the outer regions of the Fourier transform? (a) Resolution reduction in the x-direction. >> res_span = 129+(-16:16); >> Mf2 = zeros(256,256); >> Mf2(:,res_span) = Mf(:,res_span); >> Mf2 = fftshift(Mf2); >> M_resx = ifft2(Mf2); >> imagesc(abs(M_resx)); % This will show reduction of resolution in the x-direction. (b) Demonstrate resolution reduction in the y-direction. Hand in code and image. (c) Demonstrate resolution reduction in the x and y directions. Hand in code and image 11. Missing data in k-space. We can also zero out the inner regions of the Fourier Transform. >>Mfzero = Mf; >>Mfzero(ky,kx) = 0; >>iMFzero= ifft2(fftshift(Mfzero)); % look at resulting image. Zero out the following: (a) kx = 129; ky = 129 (b) kx= 1:256; ky = 129+(-16:16); (c) kx = 129+(-16:16); ky = 1:256; (d) kx = 129+(-16:16); ky = 129 + (-16:16); (e) kx = 1:2:256;ky = 1:256; For each set of parameters, plot out the Fourier transform and the resulting image. Give a qualitative explanation of why the image looks the way it does. 12. Spikes in the data. You can put a spike at location (kx,ky) in Fourier space with the following commands >>Mfspike = Mf; >>spike = 100e6; >>Mfspike(ky,kx) = Mf(ky,kx) + spike; % add spike >>iMFspike = ifft2(fftshift(Mfspike)); % look at resulting image. Put spikes at: (a) kx = 129; ky = 129 – note this point corresponds to the center of k-space (i.e. kx = 0, ky = 0 in terms of the coordinates we used in class). (b) kx= 161; ky = 129 (c) kx = 161; ky = 161For each spike location, plot out the Fourier transform and the resulting image. Give a qualitative explanation for why the image looks the way it does. For example, what accounts for the direction of the
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