Session TE2 Sixth MDSP Workshop Maximum Likelihood Image Identification and Restoration Based on the EM Algorithm'

In order to restore a degraded image, the first step is to identify the kind of degradation the image has suffered. Modeling the blurred image as the output of a two-dimensional (2D linear space-invariant (LSI) system, the identification problem is the estimation of the unknown point-spread function (PSF) of the LSI system. One approach to blur identification is to obtain the blur parameters (i.e. PSF) from the physics of the distortion process. However, it is hardly the actual case, since usually we do not have enough knowledge of how the images are distorted. INstead, the identification of the parameters needs to be based on the available observed images, which are noisy and blurred. The earliest work on blur identification concentrated on identifying PSFs that have zeros only on the unit bi-circle (motion blur, for example) [l]. The identification algorithm then was searching for these zeros in the frequency domain. Two of the shortcomings of this method are that PSFs which do not satisfy this requirement (truncated Gaussian PSF, for example) can not be identified and that the presence of noise makes the determination of the location of the zeros very difficult. In more recent work [2,3,4], the image and blur model identification problem is specified as a 2D autoregressive moving-average (ARMA) identification problem, where the AR coefficients are related to the image model, and the MA coefficients to the blur model (PSF). In [3] and [5] the blur identification was formulated as a constrained maximum likelihood (ML) problem. The resulting nonlinear minimization problem was solved by employing an iterative gradient based minimization procedure. This paper focuses on simultaneous iterative identification and restoration. It is conceptually advantageous to perform the identification and restoration simultaneously, since the interaction between the two processes may improve both the estimates for the blur and the image. Iterative techniques offer the possibility of incorporating prior knowledge about the original blur, model and image into the identification and restoration procedure. Furthermore, since they act upon one complete image, they are free from the causality restriction imposed by recursive techniques. More specifically, the image and noise are modeled as multivariate Gausian processes. Then we us ML estimation to estimate the parameters which characterize the Gaussian processes, whre the estimationof the conditional mean of the image represnets the restored image. Likelihood functions of observed images are highly nonlinear with respect to those parameters. Therefore, it is in general very difficult to directly maximize them. Here we exploit the expectation-Maximization (EM) algorithm [6] to find those parameters. The EM algorithm is a powerful itertive procedure for computing ML estimates of unknown parameters involved in the likelihood function of the observed data. In applying