Image identification and restoration based on the expectation-maximization algorithm K. 1. Lay

CONTENTS 1. Introduction 2. Image and blur models 3. Maximum likelihood (ML) parameter identification 3.1. Formulation 3.2. Constraints on the unknown parameters 4. ML parameter identification via the expectation-maximization (EM) algorithm 4.1. The EM algorithm in the linear Gaussian case 4.2. Choices of complete data 4.2.1. {x,y} as the complete data 4.2.2. {x,v} as the complete data Abstract. In this paper, the problem of identifying the image and blur parameters and restoring a noisy blurred image is addressed. Specifying the blurring process by its point spread function (PSF), the blur identification problem is formulated as the maximum likelihood estimation (MLE) of the PSF. Modeling the original image and the additive noise as zero-mean Gaussian processes, the MLE of their covariance matrices is also computed. An iterative approach, called the EM (expectation-maximization) algorithm, is used to find the maximum likelihood estimates ofthe relevant unknown parameters. In applying the EM algorithm, the original image is chosen to be part of the complete data; its estimate is computed in the E-step of the EM iterations and represents the restored image. Two algorithms for identification/restoration, based on two different choices of complete data, are derived and compared. Simultaneous blur identification and restoration is performed by the first algorithm, while the identification of the blur results from a separate minimization in the second algorithm. Experiments with simulated and photographically blurred images are shown.