A Hierarchical Markov Random Field Model and Multitemperature Annealing for Parallel Image Classification

cation [6, 7, 14, 15]. It is well known that multigrid methods can improve significantly the convergence rate and the In this paper, we are interested in massively parallel multiscale relaxation algorithms applied to image classification. quality of the final results of iterative relaxation techniques. It is well known that multigrid methods can improve signifiThere are many approaches in multigrid image segmencantly the convergence rate and the quality of the final results tation. A well known approach is the renormalization of iterative relaxation techniques. First, we present a classical group algorithm which is based on renormalization group multiscale model which consists of a label pyramid and a whole ideas from statistical physics. This technique has been observation field. The potential functions of coarser grids are adapted by Gidas [13] to image processing. The main adderived by simple computations. The optimization problem is vantage of the method is that it provides a mechanism for first solved at the higher scale by a parallel relaxation algorithm; relating the processing at different scales with one another. then the next lower scale is initialized by a projection of the This mechanism is a nonlinear transformation—called the result. Second, we propose a hierarchical Markov random field renormalization group (RG) transformation. The coarser model based on this classical model. We introduce new interactions between neighbor levels in the pyramid. It can also be grids and their Hamiltonians are well defined; they are seen as a way to incorporate cliques with far apart sites for a deduced from the original image. The major difficulty is reasonable price. This model results in a relaxation algorithm the computation of the energy functions at coarser grids. with a new annealing scheme: the multitemperature annealing Usually, this computation cannot be done explicitly; one (MTA) scheme, which consists of associating higher temperamust approximate them [10, 25]. Another drawback is the tures to higher levels, in order to be less sensitive to local minima loss of Markovianity at coarser grids [26] since the coarser at coarser grids. The convergence to the global optimum is energy functions obtained by the RG transformation canproved by a generalization of the annealing theorem of S. Genot be decomposed as a sum of clique-potentials. In [13], man and D. Geman (IEEE Trans. Pattern Anal. Mach. Intell. the Hamiltonians are approximated by a sum of clique6, 1984, 721–741).  1996 Academic Press, Inc. potentials, and hence one can use classical relaxation algorithms to minimize the energy at coarser grids. Unfortunately, such approximations are available only for certain