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.

本文提出了一个静态遍历点过程的统计模型，该模型是根据在方形窗口中观察到的单个实现来估计的。使用随机几何中的现有方法，很难对由大量粒子形成的复杂几何形状的过程进行建模。受最近关于用于采样最大熵模型的梯度下降算法的工作的启发，我们描述了一个模型，该模型允许对新配置进行快速采样，从而再现给定观察的统计数据。从初始随机配置开始，它的粒子根据能量的梯度移动，以匹配一组规定的矩（泛函）。我们的矩是通过点模式的小波变换上的相位谐波算子来定义的。它们允许人们捕捉粒子之间的多尺度相互作用，同时通过要建模的结构的尺度明确控制矩的数量。我们对具有各种几何结构的点过程进行数值实验，并通过光谱和拓扑数据分析评估模型的质量。