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

Particle gradient descent model for point process generation