6.2
The methods described in the previous section are applicable to
situations where the data contains either a single MRF or several but
segmented MRFs so that the estimation can be done for each MRF
separately. In practice, it is normal that the data is an observation,
such as an image, of the underlying, unobservable MRFs. There can be
several complications compared to the clean, single MRF case. Firstly,
the data is a transformed version of the true MRF, for example, due to
the additive noise observation model 
. Secondly, the noise
parameters can be unknown and hence also a part of the parameters to be
estimated. In this case, the set of unknown parameters is composed of the
two parts, those for the MRF and those due to the noise, collectively
denoted as 
. Thirdly, the data d can
contain realizations of more than one MRF and the origin of each data
point is unknown. Since the parameters for an MRF should be estimated by
using only realizations of that MRF, the data has to be segmented before
used for the estimation. However, to perform good segmentation, the
parameter values of the MRFs have to be available. This is a dilemma
between segmentation and estimation. The problem of
restoration-estimation is similar: The estimation should be performed
using smoothed or restored data; however, to obtain good restoration,
the parameter values of the MRF have to be known. We consider these as
the dilemma between labeling and estimation.
One strategy is to perform parameter estimation after labeling (segmentation or restoration). For example, to perform segmentation using clustering techniques [Silverman and Cooper 1988,Hu and Fahmy 1987]. This partitions the input data into blocks; the blocks are then merged into clusters according to some optimization criterion, yielding a segmentation; each cluster contains only blocks due to a single MRF (or due to ``boundary blocks''); now parameters for each MRF can be estimated separately. However, such strategy may not yield the best result because optimal labeling relies on correct parameter values.
In a more sophisticated paradigm, iterative labeling-estimation [Besag 1986,Kelly et al. 1988,Lakshmanan and Derin 1988,Qian and Titterington 1992, Pappas 1992,Won and Derin 1992,Zhang 1992,Bouman and Shapiro 1994] is adopted; labeling and estimation are performed alternately. Initial labeling is chosen by using some scheme possibly with initial parameters; a parameter estimate is computed based on this labeling; it is then used to get a hopefully better labeling; a hopefully better estimate is computed after that; and so on. The expectation-maximization algorithm [Dempster et al. 1977] provides a justifiable formalism for such labeling-estimation schemes.