The Algebraic Reconstruction Technique (ART) is an iterative method for Computed Tomography (CT) image reconstruction. The 2D image data can be reshaped into a 1D vector X and every projection ray in the sinogram P is computed simultaneously using the system matrix A:

                      AX=P  

In each iteration, for each pixel p_i and each row A_i of A, the following update is performed:

            X^k+1=X^k+(p_i-A_iX^k)A_i^T/(A_iA_i^T)  

Here k is the iteration index and we need to initialize the reconstructed image as X⁰=0. For k_th iteration, A_iX^k stands for the i_th projection of the reconstructed image X^k and (p_i-A_iX^k) stands for the difference between the estimated sinogram ray and the measured sinogram ray, and (p_i-A_iX^k)A_i^T means to back project the sinogram difference. This value is then normalized with the factor A_iA_i^T. The update is performed until convergence or a maximum iteration number is reached.

Another approach for iterative reconstruction is gradient descent. Here, we seek to minimize the following objective function.

         J = ||AX-P||₂²

To find the minimum, we follow the negative gradient direction δX in each iteration: 

        X^k+1 = X^k- λ*δX^k

       δX^k=(p_i-A_iX^k)A_i^T

The objective function for the (k+1)_th iteration is found as follows:

       J^k+1 = ||AX^k+1-P||₂²= ||AX^k-λA*δX^k-P||₂²

We derive the above equation, set it to zero. Next, we solve for λ to find the formula for the optimal step size.

       λ= (AδX^k)^T(AX^k-P)/(AδX^k)^T(AδX^k)

Here AδX^k means the projection of the image update δX^k and AX^k-P denotes the difference between the k_th estimated sinogram and the measured sinogram.

With the adaptive step size, this iterative reconstruction method converges much faster than with a constant step size.