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Topic 6 MAP classification with hidden Markov models: 2 6.1 NP-hard exact energy minimization When energy functions involve mulliple labels per pixel, the exact global minimization is an NP-hard problem, apart from a few special cases with very restrictive conditions on the energy functions. The pixel-wise stochastic global minimization with simulated annealing (SA) or the deterministic pixel-wise local minimization with the “greedy” it- erated conditional modes (ICM) algorithm produce typically very poor results. Even in the simplest case of binary labeling considered in Lecture 4, Section 5.3, these algorithms converge to stable points that arc too far from the global minimum! [14]. The main drawback of these algorithms is that cach iteration changes only one label in a single pixel in accord with the neighboring labels and therefore it results in an extremely small move in the space of possible labelings. Obviously, the convergence rate should become faster under larger moves that simultaneously change labels in a large number of pixels. 1 Although simulated annealing provably converges to the global minimum of energy [11], this could be obtained only in exponential time; this is why no practical implementation closely approaches the goal. 82 
