Convergence of EM image reconstruction algorithms with Gibbs smoothing

Area of Science:

• Medical Imaging
• Computational Science
• Statistical Modeling

Background:

• Emission tomography image reconstruction often uses the Expectation-Maximization (EM) algorithm.
• The EM algorithm can be computationally intensive, motivating the development of faster alternatives.
• The one-step late (OSL) algorithm provides a modification to the EM algorithm for image reconstruction.

Purpose of the Study:

• To present and analyze the one-step late (OSL) algorithm for emission tomography.
• To demonstrate guaranteed convergence of modified OSL algorithms to the unique maximum of the log posterior.
• To explore potential functions for Gibbs priors and generalize OSL to transmission tomography.

Main Methods:

• The OSL algorithm retains the E-step of the EM algorithm and approximates the M-step.
• Convergence is proven under specific sufficient conditions, including those related to the Gibbs prior's potential function.
• The study identifies candidate potential functions and considers the generalization of OSL.

Main Results:

• The OSL algorithm provides an approximate solution to the M-step in image reconstruction.
• Modified OSL algorithms are shown to converge to the unique maximum of the log posterior function.
• Several potential functions for Gibbs priors are identified, and the algorithm is generalized.

Conclusions:

• The OSL algorithm is a viable and convergent method for emission tomography image reconstruction.
• The choice of potential function in the Gibbs prior is crucial for OSL convergence.
• The OSL algorithm shows promise for application in transmission tomography as well.