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Parameter estimation for two-dimensional Ising fields corrupted by noise. (English) Zbl 0702.62096

The studied random field on a finite subset of \({\mathbb{Z}}^ 2\) is a possible model for describing images. The suggested estimation of its parameters can serve for recognition of images. The process arises as the sitewise product of a free-boundary Ising model (with values \(\pm 1)\) with inverse temperature \(\beta\) and of the Bernoulli field with the probability \(\epsilon\) for -1 being ordered to a site from a finite \(\Lambda \subset {\mathbb{Z}}^ 2\). Estimators for \(\beta\) and \(\epsilon\) are suggested which are consistent as \(\Lambda\) tends to \({\mathbb{Z}}^ 2.\)
The proof of the consistency contains some inequalities for elliptic integrals occurring in the expressions for correlations of the limit Ising random field in \({\mathbb{Z}}^ 2\). Numerical results are introduced where the “true” Gibbs field was simulated by a numerical algorithm. The conditional probability and maximum likelihood method are used to estimate the value at each site of the “true” image, and the true and estimated values of the parameters \(\beta\) and \(\epsilon\) are compared.
Reviewer: P.Holicky

MSC:

62M40 Random fields; image analysis
60K35 Interacting random processes; statistical mechanics type models; percolation theory
65C99 Probabilistic methods, stochastic differential equations
68U10 Computing methodologies for image processing
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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References:

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