Medical Image Noise Reduction Using the Sylvester–Lyapunov Equation

Multiplicative noise is often present in medical and biological imaging, such as magnetic resonance imaging (MRI), ultrasound, positron emission tomography (PET), single photon emission computed tomography (SPECT), and fluorescence microscopy. Noise reduction in medical images is a difficult task in which linear filtering algorithms usually fail. Bayesian algorithms have been used with success but they are time consuming and computationally demanding. In addition, the increasing importance of the 3-D and 4-D medical image analysis in medical diagnosis procedures increases the amount of data that must be efficiently processed. This paper presents a Bayesian denoising algorithm which copes with additive white Gaussian and multiplicative noise described by Poisson and Rayleigh distributions. The algorithm is based on the maximum a posteriori (MAP) criterion, and edge preserving priors which avoid the distortion of relevant anatomical details. The main contribution of the paper is the unification of a set of Bayesian denoising algorithms for additive and multiplicative noise using a well-known mathematical framework, the Sylvester-Lyapunov equation, developed in the context of the control theory.