$2$ Tasks

Before beginning with the exercises, prepare the Shepp$-$Logan phantom and compute the sinogram using

either the radon transform from scikit$-$image or "line" projector from ASTRA.

Task$1$: Implement SIRT

Create the operator representing the projection matrix $W$ and implement the SIRT algorithm using matrix

notation. A single update step is given by

$b^{(i+1)}=b^{(i)}+C{W}^{T}R(p-W{b}^{(i)}),$

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with the matrices $C=\{c_{ij}\}$ and $R=\{r_{ij}\}$ being diagonal matrices containing the inverse column and row

sums, respectively:

$c_{kk}=\frac{1}{{\displaystyle \underset{j=1}{\overset{N}{}}}}{w}_{jk},Cin{R}^{MM}$

$r_{jj}=\frac{1}{{\displaystyle \underset{k=1}{\overset{M}{}}}}{w}_{jk},Rin{R}^{NN},$

with $N$ the number of projection pixels, and $M$ the number of voxels in the XCT image. The initial image

$b^{(0)}$ typically contains zeros. Reconstruct the Shepp$-$Logan phantom using SIRT after adding gaussian noise

to it $($sigma $=1,5,10\%$ of the maximum value in the image$).$

Add replacing negative values in $b^{(i)}$ with zeros at every iteration. Does the resulting reconstruction

somehow differ from the case when you set no boundaries on $b^{(i)}?$

How many iterations do you need to get a result that looks good to you?

Let the algorithms run for twice that number of iterations. What is the effect?