So far, we have seen 2 image reconstruction methods:
These may not perform optimally when the sinogram is undersampled or noisy. We therefore consider an alternative, which attempts to solve the system of equations explicitly.
We can attempt to solve a linear system of equations
$$Ku = f,$$using the Richardon iteration
$$u^{(k+1)} = u^{(k)} - \alpha K^T(Ku^{(k)} - f).$$When $u^{(0)} = 0$ and $0 < \alpha < 2/\|K\|^2$ it converges to the minimum-norm solution of the normal equations
$$K^T\!Ku = K^Tf.$$Convergence to a solution of the normal equations:
$$e^{(k+1)} = \left(I - \alpha K^T\!K \right)e^{(k)},$$so error components corresponding to eigenvalues $\approx \alpha^{-1}$ decay fastest.
Convergence to the true solution:
$$e^{(k+1)} = \left(I - \alpha K^T\!K \right)e^{(k)} - \alpha K^T\epsilon.$$The computational cost of Richardson's method is dominated by the cost of the foward and adjoint Radon transform. Can we work with a single ray at a time?
The Kaczmarz method proceeds as follows:
$$u^{(k+1)} = u^{(k)} - \frac{f_i - k_i^Tu^{(k)}}{\|k_i\|_2^2} k_i^T,$$with $k_i$ the $i^{\text{th}}$ row of $K$.
Compare the iterative methods introduced in this lecture with the previously defined reconstruction methods in terms if computational efficiency and reconstruction quality.