Compressed Sensing Inspired Image Reconstruction from Overlapped Projections

Algorithm 2

Image Reconstruction from overlapped projections (IROP).

Initialization:

    (i)  𝐝𝐟=0;

    (ii)  𝐠= A good initial guess, such as a blurry image from a low-resolution CT scan;

    (iii)  𝑁_src=2;

    (iv)  𝑁_data=𝑁_src⋅𝑁_bin⋅𝑁_rot;

    (v)  𝑁_linear= A number of the system linearization steps;

    (vi)𝑁_itr=A number of the POCS-gradient or SART-gradient iterations for each

      linear approximation;

    (vii)   𝑁_grad=A number of the gradient descent search steps;

    (viii)  𝑎= A step size for the gradient descent search, say 0.2;

    (ix)  Obtain projection data using Algorithm 1 or from a real scan;

Reconstruction:

   Linearization updating loop for 𝐼=1∶𝑁_linear (or until ‖𝐩−exp(𝑀₁𝐠)−exp(𝑀₂𝐠)‖^𝑞_𝑝≤𝜀)

       (i) Compute 𝐌_new;

     POCS-gradient (or SART-gradient) reconstruction loop for 𝐽=1∶𝑁_itr

       For 𝑃𝑂𝐶𝑆_itr=1∶𝑁_data

         (a) Project 𝐝𝐟 onto the constraint equation:

            𝐝𝐟=𝐝𝐟−𝐌_𝑛𝑒𝑤𝑝_new−𝐌_new𝐝𝐟‖𝐌_new‖²;

       End

       For GRAD_itr=1∶𝑁_grad

         (b) Compute the TV with respect to 𝐝𝐟;

         (c) Perform the gradient descent search;

       End

     End

       (ii) Update the current guess 𝐠=𝐠+𝐝𝐟;

    End