Abstract
In this paper, a Bayesian framework is introduced for pattern modeling and multiple point statistics simulation. The method presented here is a generalized clustering-based method where the patterns can live on a hyper-plane of very low dimensionality in each cluster. The provided generalizationallows a remarkable increase in variability of the model and a significant reduction in the number of necessary clusters for pattern modeling which leads to more computational efficiency compared with clustering-based methods. The Bayesian model employed here is a nonlinear model which is composed of a mixture of linear models. Therefore, the model is stronger than linear models for data modeling and computationally more effective than nonlinear models. Furthermore, the model allows us to extract features from incomplete patterns and to compare patterns in feature space instead of spatial domain. Due to the lower dimensionality of feature space, comparison in feature space results in more computational efficiency as well. Despite most of the previously employed methods, the feature extraction filters employed here are customized for each training image (TI). This causes the features to be more informative and useful. Using a fully Bayesian model, the method does not require extensive parameter setting and tunes its parameters itself in a principled manner. Extensive experiments on different TIs (either continuous or categorical) show that the proposed method is capable of better reproduction of complex geostatistical patterns compared with other clustering-based methods using a very limited number of clusters.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Guardiano, F., Srivastava, M.: Multivariate geostatistics: beyond bivariate moments. In: Geostatistics Troia, pp. 133–144. Kluwer Academic, Dordrecht (1993)
Isaaks, E.: The application of Monte Carlo methods to the analysis of spatially correlated data. Ph.D. thesis, Stanford University (1990)
Journel, AG.: Non-parametric estimation of spatial distributions. Math. Geol. 15(3), 445–468 (1983)
Strebelle, S.: Conditional simulation of complex geological structures using multiple point statistics. Math. Geol. 34(1), 1–22 (2002)
Mariethoz, G., Renard, P., Straubhaar, J.: The Direct Sampling method to perform multiple-points geostatistical simulations. Water Resour. Res. (2010) doi:10.1029/2008WR007621
Mariethoz, G., Renard, P.: Reconstruction of incomplete data sets or images using direct sampling. Math. Geosci. 42(3), 245–268 (2010)
Tjelmeland, H., Eidsvik, J.: Directional Metropolis–Hastings updates for posteriors with nonlinear likelihood. In: Leuangthong, O., Deutsch, C.V. (eds.) Geostatistics, pp. 195–104. Springer, Banff (2004)
Tjelmeland, H.: Stochastic models in reservoir characterization and Markov random fields for compact objects. Doctoral dissertation, Norwegian University of Science and Technology, Trondheim (1996)
Kjønsberg, H., Kolbjørnsen, O.: Markov mesh simulations with data conditioning through indicator kriging. In: Proceedings of the 8th International Geostatistics Congress. Santiago, Chile (2008)
Lyster, S., Deutsch, C.V.: MPS simulation in a Gibbs sampler algorithm. In: Proceedings of the 8th International Geostatistics Congress. Santiago, Chile (2008)
Arpat, BG., Caers, J.,: Stochastic simulation with patterns. Math. Geol. 39(202), 177–203 (2007)
Zhang, T., Switzer, P., Journel, A.G.: Filter-based classification of training image patterns for spatial simulation. Math. Geol. 38, 63–80 (2006)
Wu, J., Zhang, T., Journel, A.G.: A fast FILTERSIM simulation with score-based distance. Math. Geosci. 47, 773–788 (2008)
Honarkhah, M., Caers, J.: Stochastic simulation of patterns using distance-based pattern modeling. Math. Geosci. 42, 487–517 (2010)
Gloaguen, E., Dimitrakopoulos, R.: Two-dimensional conditional simulations based on the wavelet decomposition of training images. Math. Geosci. 41(6), 679–701 (2009)
Tahmasebi, P., Hezarkhani, A., Sahimi, M.: Multiple-point geostatistical modeling based on the cross-correlation functions. Comput. Geosci. 16(3), 779–797 (2012)
Bishop, C.M., Winn, J.: Non-linear Bayesian image modeling. In: Proceedings of the 6th European Conference on Computer Vision. Dublin, Ireland (2000)
Choudrey, R.A., Roberts, S.J.: Variational mixture of Bayesian independent component analyzers. Neural Comput. 15(1), 213–252 (2003)
Chen, M., Silva, J., Paisley, J., Wang, C., Dunson, D., Carin, L.: Compressive sensing on manifolds using a nonparametric mixture of factor analyzers: algorithm and performance bounds. IEEE Trans. Sig. Proc. 58(12), 6140–6155 (2010)
Carin, L., Baraniuk, R.G., Cevher, V., Dunson, D., Jordan, M.I., Sapiro, G., Wakin, M.B.: Learning low-dimensional signal models. IEEE Sig. Proc. Mag. 28(2), 39–51 (2011)
Bouguila, N., Ziou, D.: A Dirichlet process mixture of generalized Dirichlet distributions for proportional data modeling. IEEE Trans. Neural Nets. 21(1) 107–122 (2010)
Caers, J., Journel, A.G.: Stochastic reservoir simulation using neural networks trained on outcrop data. SPE paper 49026 (1998)
Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 3rd edn. Prentice Hall, Upper Saddle River (2008)
Jolliffe, I.T.: Principal Component Analysis, 2nd edn. Springer, Heidelberg (2002)
Bell, A.J., Sejnowski, T.J.: An information maximization approach to blind separation and blind deconvolution. Neural Comput. 7(6), 1129–1159 (1996)
Aharon, M., Elad, M., Bruckstein, A.M.: K-SVD: An algorithm for designing of overcomplete dictionaries for sparse representation. IEEE Trans. Sig. Proc. 54(11), 4311–4322 (2006)
Lewicki, M.S., Olshausen, B.A.: A probabilistic framework for the adaptation and comparison of image codes. J. Opt. Soc. Amer. A: Opt., Image Sci. Vision. 16(7), 1587–1601 (1999)
Tošić, I., Frossard, P.: Dictionary learning what is the right representation for my signal. IEEE Sig. Proc. Mag. 28(2), 27–38 (2011)
Jiang, J.: Image compression with neural networks—a survey. Signal Process. Image Commun. 14, 737–760 (1999)
Tsekouras, G.E, Antonios, M., Anagnostopoulos, C., Gavalas, D., Economou, D.: Improved batch fuzzy learning vector quantization for image compression. Inf. Sci. 178, 3895–3907 (2008)
Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, New York (2006)
Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. (PAMI) 6(6), 721–741 (1984)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abdollahifard, M.J., Faez, K. Stochastic simulation of patterns using Bayesian pattern modeling. Comput Geosci 17, 99–116 (2013). https://doi.org/10.1007/s10596-012-9319-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-012-9319-x