Line Integral Alternating Minimization Algorithm for Dual-Energy X-Ray CT Image Reconstruction

. Author manuscript; available in PMC: 2019 Feb 28.

Published in final edited form as: IEEE Trans Med Imaging. 2015 Oct 14;35(2):685–698. doi: 10.1109/TMI.2015.2490658

Set k = 0. Initialize $ĉ_{i}^{(0)} (x), {\hat{L}}_{i}^{(0)} (y)$ .
Update ${\hat{p}}_{j}^{(k)} (y, E)$ according to ${\hat{f}}_{j}^{(k)} (y, E) = I_{0 j} (y, E) exp [- \sum_{i = 1}^{2} {\hat{L}}_{i}^{(k)} (y) μ_{i} (E)], {\hat{p}}_{j}^{(k)} (y, E) = d_{j} (y) \frac{{\hat{f}}_{j}^{(k)} (y, E)}{\sum_{E'} {\hat{f}}_{j}^{(k)} (y, E')} .$
Update ${\hat{L}}_{i}^{(k + 1)} (y)$ using Newton’s method.
1. Set m = 0. Let ${\hat{L}}_{i_NEWTON}^{(m = 0)} (y) = {\hat{L}}_{i}^{(k)} (y)$ .
2. ${\hat{L}}_{i_NEWTON}^{(m + 1)} (y) = {\hat{L}}_{i_NEWTON}^{(m)} (y) - {[\nabla^{2} (y)]}^{- 1} \nabla (y)$ , where ∇²(y), ∇(y) are evaluated at ${\hat{L}}_{i_NEWTON}^{(m)} (y)$ .
3. Iterate until convergence to obtain nonnegative ${\hat{L}}_{i}^{(k + 1)} (y) = max (0, {\hat{L}}_{i_NEWTON}^{(m + 1)} (y))$ .
Update $ĉ_{i}^{(k + 1)} (x)$ using the ID algorithm.
1. Set n = 0. Let $ĉ_{i_I D}^{(n = 0)} (x) = ĉ_{i}^{(k)} (x)$ .
2. $ĉ_{i_I D}^{(n + 1)} (x) = \frac{ĉ_{i_I D}^{(n)} (x)}{\sum_{y} h (y, x)} \sum_{y} h (y, x) \frac{{\hat{L}}_{i}^{(k + 1)} (y)}{\sum_{x} h (y, x) ĉ_{i_I D}^{(n)} (x)}$ .
3. Iterate until convergence, $ĉ_{i}^{(k + 1)} (x) = ĉ_{i_I D}^{(n + 1)} (x)$ .
Iterate B through D until convergence.