Marked Point Processes

The network of filaments is modeled by a marked point process, that is to say a random set of objects whose number of data points is also a random variable [6,7] . The objects of this process are segments described by three random variables corresponding to their midpoint, their length and their orientation. The segment distribution is simulated by a density probability. For a uniform distribution, we use a Poisson process. In order to find the segment configuration   that better fits the filamentary network, we define a density probability f(x) which takes into account the interactions between segments. f(x) is given by a Gibbs point process. The configuration of segments composing the filament network is estimated by the minimum of the energy U of the system. U has two components: the prior term Up forces the segment configuration to be a network and the data term Ud helps this network to best fit the data. The estimate of  x = arg min U  is obtained by means of a simulated annealing algorithm. This algorithm iteratively samples the density at some temperature while slowly decreasing the temperature. At high temperature, a lot of configurations are explored. When the temperature goes down to zero, the configuration of minimal energy is reached, assuming that a geometrical cooling scheme is sufficient. The probability density is simulated through a reversible jump Metropolis-Hastings dynamics sampling [8,9,10]. Basically, this dynamics drives the system to the minimal state by means of a set of segment perturbations: birth, death, translation, rotation and dilation. From an initial configuration, the algorithm is, at step t

1. Propose a new configuration y , obtained by a perturbation of the current configuration x,
2. Evaluate the Green acceptance ratio R(T),
3. Move to y with a probability equal to min(1,R(T)),
4. Decrease the temperature T.

The computation of the Green ratio is described in [11,12].