This paper introduces a generative model for planar point processes in a square window, built upon a single realization of a stationary, ergodic point process observed in this window. Inspired by recent advances in gradient descent methods for maximum entropy models, we propose a method to generate similar point patterns by jointly moving particles of an initial Poisson configuration towards a target counting measure. The target measure is generated via a deterministic gradient descent algorithm, so as to match a set of statistics of the given, observed realization. Our statistics are estimators of the multi-scale wavelet phase harmonic covariance, recently proposed in image modeling. They allow one to capture geometric structures through multi-scale interactions between wavelet coefficients. Both our statistics and the gradient descent algorithm scale better with the number of observed points than the classical k-nearest neighbour distances previously used in generative models for point processes, based on the rejection sampling or simulated-annealing. The overall quality of our model is evaluated on point processes with various geometric structures through spectral and topological data analysis.
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