- 3.1 Potts interaction
- 3.2 Mean-field theory
- 3.3 Label changes and critical temperature
- 3.4 The MFFA algorithm

Relaxation labeling processes were originally developed to reduce ambiguity and noise and select the best label among several possible choices in vision systems (Hummel & Zucker 1983). We consider it as a particular case of combinatorial optimization problems (e.g., Titterington 1985; Jeffrey & Rosner 1986) for which the cost function corresponds to the global energy of a complex physical system (e.g., Carnevali et al. 1985; Hiriyannaiah et al. 1989; Bilbro et al. 1991; Hérault & Horaud 1993; Bratsolis & Sigelle 1997).

Real images are often inhomogeneous, with nearly uniform regions separated by relatively sharp edges. To get round this problem, local energy functions have been introduced. To express the local properties of an image, a neighborhood system of

The Hamiltonian of our system is the cost function of the lattice. Using some ideas from Statistical Physics, we can use a Potts potential interaction model

(2) |

(3) |

The total energy of the Potts model with mean-field approximation up to a constant is:

(4) |

(5) |

Calling now the configuration of these neighbor sites, we have

(6) |

(7) |

(8) |

where *P*(*l*_{s}= *i*) is the probability for the label of
the site *s* to be at the state *i*,
*K*_{ii}^{rs} are the potential coefficients
as elements of a diagonal matrix (in image processing plays the role of
a smoothness factor), *B* is the external field (here plays the role of
a roughness factor) and *l*_{s}^{0}
is the initial label of the site *s*, *new* refers to instant *t*+1
and *old* to instant *t*.

In the simplest case we take *K* as a scalar and we have:

(9) |

.

Combinatorial optimization problems having multiple conflicting
constraints are called ill-posed or nonconvex. Nonconvex problems
typically have numerous suboptimal solutions manifested as local
minima in the energy function.
The powerful tool of simulated annealing, which, in theory, eventually
converges to an optimal solution, has been applied to a variety of
nonconvex combinatorial optimization problems. The control parameter
*T* takes an initial value .
The deterministic nature of mean-field method acts on the system
so that the major part of label changes occur
in a small temperature region below critical temperature .The temperature *T* here is a parameter that reduced following a cooling schedule.
Thus it's enough to start mean field annealing at some temperature
a little higher than in order to reach convergence. There is no link
between the parameters and and the measured solar temperatures.

Simulated annealing algorithm reaches an optimal configuration
the same way the annealing process of a solid does into its (globally) lowest
energy state. In Conventional Simulated Annealing
(Geman & Geman 1984) there
is an initial temperature such that at step *k* we have
.
A similar technique developed by
Szu & Hartley (1987)
employs the following Fast Cooling Schedule:
.
Both temperature schedules have been tested within our deterministic
frame, leading to
very similar
results (Bratsolis & Sigelle 1997).
The composition of probabilistic mean-field approximation with the fast
cooling schedule gives the following Mean Field Fast Annealing (MFFA)
algorithm.

*MFFA algorithm
(with K as a scalar), (Figs. 5,
6, 7)*.

1. Define : number of sweeps, and : initial temperature.

2. Initialize

3. For Use [Eq. (9)].

4. If Round off and display taking .

Figure 6:
Sunspot image of Fig. 3
after MFFA application. The labels now
are three. Labels of too small isolated areas are disappeared |

Figure 7:
Sunspot image of Fig. 4
after MFFA application. The labels now
are six. Labels of too small isolated areas are disappeared |

Taking *K* = 1.0 and *B* = 1.0 in four-connectivity, i.e.,
the neighborhood of the site *s* contains the four nearest
pixels, experimental tests give
(Bratsolis & Sigelle 1997)
for *q*=4, and for *q*=8, . In any case we
accept .

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