  This paper presents a statistical model for stationary ergodic point
processes, estimated from a single realization observed in a square window.
With existing approaches in stochastic geometry, it is very difficult to model
processes with complex geometries formed by a large number of particles.
Inspired by recent works on gradient descent algorithms for sampling
maximum-entropy models, we describe a model that allows for fast sampling of
new configurations reproducing the statistics of the given observation.
Starting from an initial random configuration, its particles are moved
according to the gradient of an energy, in order to match a set of prescribed
moments (functionals). Our moments are defined via a phase harmonic operator on
the wavelet transform of point patterns. They allow one to capture multi-scale
interactions between the particles, while controlling explicitly the number of
moments by the scales of the structures to model. We present numerical
experiments on point processes with various geometric structures, and assess
the quality of the model by spectral and topological data analysis.
