Multiple-point statistics using multi-resolution images

Multiple-point statistics (MPS) is a simulation technique allowing to generate images that reproduce the spatial features present in a training image (TI). MPS algorithms consist in sequentially filling a simulation grid such that patterns around the simulated values come from the TI. Following this principle, joint simulations of multiple variables can be handled and complex heterogeneous fields can be generated. However, inconsistent patterns are often found in the results and some spatial features can be difficult to reproduce. In this paper, a new MPS algorithm based on a multi-resolution representation of the TI is proposed to enhance the quality of the realizations. The method consists in first building a pyramid of images from the TI by successive convolution using Gaussian-like kernels. Secondly, a MPS simulation is done at the lowest resolution level. Then, the result is expanded to the next level of resolution (one rank higher) and used as a conditioning variable for a joint MPS simulation at that level. This last step is repeated up to the initial resolution, where the final simulation is retrieved. The method is implemented in the DeeSse code based on the direct sampling algorithm. Most of the features provided by the direct sampling (conditioning to hard data, uni- or multi-variate simulation of categorical and continuous variables, scaling and rotation of the training structures) are compatible with the proposed method and the usability is maintained. Finally, various examples show that in most of the situations, combining Gaussian pyramids with MPS allows to get results of better quality and in less time compared to direct MPS simulations.