Noise properties of the EM algorithm: I. Theory

Area of Science:

• Medical Imaging
• Statistical Modeling
• Image Reconstruction

Background:

• The expectation-maximization (EM) algorithm is crucial for maximum-likelihood estimation and image reconstruction, particularly in medical imaging.
• While its convergence is understood, the impact of data noise on reconstructed image noise remains unclear.
• Existing linear methods like filtered back-projection exhibit global noise patterns.

Purpose of the Study:

• To statistically analyze the influence of data noise on image noise in EM algorithm reconstructions.
• To provide a theoretical framework for understanding noise properties in EM-based medical imaging.
• To compare noise characteristics of EM algorithm with linear reconstruction methods.

Main Methods:

• Detailed statistical treatment of noise properties in EM algorithm reconstructions.
• Derivation of expressions for grey level variance and pixel-to-pixel covariance.
• Approximation of pixel grey level probability density function by a log-normal law.

Main Results:

• The probability density function of pixel grey levels is well approximated by a log-normal distribution.
• Image noise variance increases with iteration number initially, then saturates.
• Noise standard deviation map resembles the object, indicating lower noise in low-intensity regions.

Conclusions:

• The study provides a theoretical basis for calculating objective image quality figures of merit for EM algorithm reconstructions.
• The noise characteristics of the EM algorithm differ significantly from linear methods, offering potential advantages.
• Approximations used are validated by Monte Carlo simulations in emission tomography.