3 Relaxation labeling process

Our sunspot image is taken at the center of ${\rm H}_{\alpha}$. Our purpose is to divide the image into regions of different activity (e.g., umbra, penumbra, active regions) and therefore the problem now is reduced in a relaxation labeling process.  
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).

3.1 Potts interaction

Let ${\cal I} = \{0, 1, \ldots q-1\}$ be the set of gray levels (labels or classes). A label random variable l_s is associated for each site $s \in {\it S}= \{1, 2 \ldots N\}$,where N is the number of pixels. The true distribution of l_s is unknown, but based on some measurements, a probability distribution P(l_s): ${\cal I} \mapsto [0, 1]$is estimated for such a random variable.  
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 r-pixels is defined as a collection of pixels that are assumed to interact directly with s-pixel. A basic characteristic of most images is that intensity values at neighbor locations r and s are likely to be similar (e.g., Besag 1974; Besag 1986).  
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 U_rs (Wu 1982; Bratsolis & Sigelle 1997).  

$$U_{rs} = - K {\frac {q-1} {q}} ( \hat {u}_s ~.~\hat{u}_r ) - {\frac K q} + {\frac K 2}$$ (2)

where K is a constant and $\hat {u}_s$, $\hat {u}_r$ random (unit) vectors at sites s, r under a (q-1)-dimensional space with values in R.

3.2 Mean-field theory

One of the basic ideas of the mean-field approximation, is to find the best linear approximation, in a statistical sense, of a Hamiltonian expression. So, Eq. (2) is written up to a constant,  
 
$U_{rs} = - \beta\{\hat{u}_s ~.~E(\hat{u}_r) + \hat{u}_r ~.~E(\hat{u}_s) - E(\hat{u}_s) ~.~E(\hat{u}_r)$ 

$$~~~~~~~~~~~+\; [\hat{u}_s - E(\hat{u}_r)] ~.~[\hat{u}_r - E(\hat{u}_s)]\}$$ (3)

where $\beta=K \displaystyle \frac {q-1} {q}$,$E(\hat{u}_s)$ the expected value of vector $\hat {u}_s$.and $E(\hat{u}_r)$ the expected value of vector $\hat {u}_r$.Using mean-field theory, we consider only the first two terms of Eq. (3). Each local energy function is thus approximated by a sum of first-order clique potentials, leading to a quasi-independent vector approximation.  
The total energy of the Potts model with mean-field approximation up to a constant is:  

$$U = - \beta ~\sum_{(s,r)} \; \hat{u}_s ~.~\hat{u}_r \approx ... ... ~.~\left [ \sum_{r\in N_{\mbox{\it s}}}~ E(\hat{u}_r) \right ]$$ (4)

or  

$$U = - \sum_{s \in S} \; {\bf h}_s ~.~\hat{u}_s$$ (5)

where ${\bf h}_s = \beta \sum_{r\in N_{\mbox{\it s}}}~ E(\hat{u}_r)$is the effective magnetic field (in physical sense) produced by neighbors of site s and N_s the set of the r-neighbors of the site s. There is no link between the effective magnetic field ${\bf h}_s$ and sunspot magnetic field.  
Calling now ${\cal N}_s$ the configuration of these neighbor sites, we have

$$E(\hat{u}_s) = \sum_{k=0}^{q-1}P(\hat{u}_s = \hat{v}_k)\hat{... ...sum_{k=0}^{q-1}P(\hat{u}_s = \hat{v}_k ~/~{\cal N}_s)\hat{v}_k.$$ (6)

Here $P(\hat{u}_s=\hat{v}_k ~/~{\cal N}_s)$ is simultaneously the local and global Gibbs-Boltzmann distribution $\hat {u}_s$, on account of our assumption of vector variables independence. The expected value of vector $\hat {u}_s$ at site s is then computed as  

$$E(\hat {u}_s) = \frac {\displaystyle \sum_{k=0}^{q-1} \hat{... ...it s}}}~\hat{v}_k ~.~ E(\hat{u}_r) \right ]} ~~\forall s \in S.$$ (7)

Equation (7) gives rise to a self-consistent nonlinear dynamical system of q equations, where T > 0 is a parameter at our disposal and, in physical terms, represents the "absolute temperature'' of the system. The strict relationship between Markov random-field hypothesis is evident. An extended mathematical description of Potts interaction with mean-field approximation is in Bratsolis & Sigelle (1997). The mean-field equations in a probabilistic interpretation in iterative form is given by:

$[P(l_s= i)]^{\rm new}$

 

$$=\! \! \frac{\displaystyle \exp \! \! \left( {\frac 1 T}\lef... ...= k)]^{\rm old}+B \delta (l_{s}^{0}, k) \right\}\! \! \right)}$$ (8)

$\forall~ i \in \cal I ,~\forall~ \mbox{\it s} \in {\it S}$
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_s⁰ 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:

$[P(l_s= i)]^{\rm new}$

 

$$=\frac{\displaystyle \exp\left( \displaystyle {\frac 1 T}\le... ...(l_r= k)]^{\rm old} + B \delta (l_{s}^{0}, k) \right\}\right)}$$ (9)

$~\forall~ i \in \cal I ,~\forall~ \mbox{\it s} \in {\it S}$.

3.3 Label changes and critical temperature

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 $T_{\rm o}$. 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 $T_{\rm c}$.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 $T_{\rm o}$ a little higher than $T_{\rm c}$ in order to reach convergence. There is no link between the parameters $T_{\rm o}$ and $T_{\rm c}$ 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 $T_{\rm o}$ such that at step k we have $T(k)=\displaystyle \frac{T_{\rm o}}{\log (k)}$. A similar technique developed by Szu & Hartley (1987) employs the following Fast Cooling Schedule: $T(k)=\displaystyle \frac{T_{\rm o}}{k}$. 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.

3.4 The MFFA algorithm

This algorithm propose a relaxation labeling process.

MFFA algorithm (with K as a scalar), (Figs. 5, 6, 7).
1. Define $k_{\rm limit}$: number of sweeps, and $T_{\rm o}$: initial temperature.
2. Initialize q buffers of size N with $P(l_s=i)=10^{-5} ~~\forall i \in \cal I , ~~\forall \mbox{\it s} \in S$.
3. For $k=1...k_{\rm limit}$$T=\displaystyle \frac{T_{\rm o}}{k}$Use [Eq. (9)].
4. If $k=k_{\rm limit}$Round off and display taking $l_s=\displaystyle \arg \max_{i\in \cal I } P(l_s=i)$.

  

Figure 5: Results of MFFA after a misclassification of 35% of pixels by adding uniform channel noise on this image $(128 \times 128)$ with q=4, K=1.0 and B=1.0. The number of sweeps is k=8. $Up\;Left$: Artificial image. $Up\;Right$: Degraded channel noisy image. $Down\;Left$: Relaxation result. $Down\;Right$: Classification error as number of steps

  

Figure 6: Sunspot image $(128 \times 128)$ of Fig. 3 after MFFA application. The labels now are three. Labels of too small isolated areas are disappeared

  

Figure 7: Sunspot image $(128 \times 128)$ 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, $T_{\rm c}=1.92$ and for q=8, $T_{\rm c}=1.36$. In any case we accept $T_{\rm o} \simeq 2T_{\rm c}$.