BIM 252/EEC 205 Computational Imaging Winter 2022

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BIM 252/EEC 205 Computational Imaging Winter 2022

Homework #1 

1. Radon transform and central slice theorem.

(a) An alternative way to write the Radon transform is using the delta function:

(*).

Verify that this is equivalent to .

(Hint: change coordinates from  to .)

(b) Use the Radon transform in equation (*) to verify the central slice theorem.

(Hint: what is the Fourier transform of a shifted delta function?)

(c) Find the Radon transform of a rectangular function

 (d) First find the 2D Fourier transform  of the rectangular function in (c) and take its profile along  to get  Second find the 1D Fourier transform  of the projection of f(x,y) at angle . Verify

2. Backprojection and the point response function h(x,y)

(a) Show that . (you can assume ).

(b) Use the -function definition of  (see problem 1(a)) to show that

      where . (Recall .)

(c) Show that . (Hint: write  and use part (a).)

3. Radon transform in 3D

(a) Derive the radon transform of a centered ball of radius

(b) Find an expression for the filtered projections, i.e., . (Notice the singularities at )

(c) Verify that the filtered backprojection gives the correct answer for .

(d) Now consider a point outside the ball, (0,0,z), where z>R, find the range of angle , for which the projection  backprojects a non-zero value onto point (0,0,z). Note that the summation of all backprojections at this point is actually zero.

4. Image reconstruction:

Implement the backprojection filtering method and compare with the filtered backprojection (FBP) method.

Use these codes to generate phantom images and projections:

img = phantom(256);  % create the phantom image

pp = radon(img,[0:179]);  % 180 angels of parallel beam projections

FBP reconstruction using the iradon.m function in MATLAB and compute the mean squared error (MSE)

fbp_recon = iradon(pp,[0:179],'linear','ram-lak',1,256);

figure, imagesc(fp), axis image

fbp_mse = mean( (fbp_recon(:) – img(:)).^2);

Now implement a backprojection filtering method. The backprojection can be performed using the iradon.m function with filter option ‘none’, i.e.

Bp_img = iradon(pp, [0:179],'linear','none',1, OUTPUT_SIZE);

Then perform the 2D cone filtering operation using Fourier transform (fft2.m and ifft2.m):

After that, crop the image back to 256x256 so that it is aligned with the original phantom image in order to compute the MSE.

Please try your reconstruction code with OUTPUT_SIZE = 256, 384, 512, and use tic, toc commands to measure the computational time. Compare the results with those of FBP method.

Please fill out the following table and submit it with your reconstructed images.

5. Image quality vs. angular sample

First, use the matlab functions radon.m to generate parallel beam projections with N projection angles uniformly distributed between 0 and 180 degree, and then reconstruct the image from the projections using iradon.m. Compute the mean squared error (MSE) between the reconstructed image and original phantom image and plot the MSE for N=10:10:180.

The phantom image is the same as that used in problem 4.

Submit your matlab code and the MSE plot.