Multiple Point Statistics algorithm based on direct sampling and multi-resolution images

Multiple Point Statistics (MPS) has become popular for more than one decade in Earth Sciences, because these methods allow to generate random fields reproducing highly complex spatial features given in a conceptual model, the training image, while classical geostatistics techniques based on bi-point statistics (covariance or variogram) fail to generate realistic models. Among MPS methods, the direct sampling consists in borrowing patterns from the training image to populate a simulation grid. This latter is sequentially filled by visiting each of these nodes in a random order, and then the patterns, whose the number of nodes is fixed, become narrower during the simulation process, as the simulation grid is more densely informed. Hence, large scale structures are caught in the beginning of the simulation and small scale ones in the end. However, MPS may mix spatial characteristics distinguishable at different scales in the training image, and then loose the spatial arrangement of different structures. To overcome this limitation, we propose to perform MPS simulation using a decomposition of the training image in a set of images at multiple resolutions. Applying a Gaussian kernel onto the training image (convolution) results in a lower resolution image, and iterating this process, a pyramid of images depicting fewer details at each level is built, as it can be done in image processing for example to lighten the space storage of a photography. The direct sampling is then employed to simulate the lowest resolution level, and then to simulate each level, up to the finest resolution, conditioned to the level one rank coarser. This scheme helps reproduce the spatial structures at any scale of the training image and then generate more realistic models. We illustrate the method with aerial photographies (satellite images) and natural textures. Indeed, these kinds of images often display typical structures at different scales and are well-suited for MPS simulation techniques.