Bridges between multiple-point geostatistics and texture synthesis: Review and guidelines for future research

Multiple-Point Simulations (MPS) is a family of geostatistical tools that has received a lot of attention in recent years for the characterization of spatial phenomena in geosciences. It relies on the definition of training images to represent a given type of spatial variability, or texture. We show that the algorithmic tools used are similar in many ways to techniques developed in computer graphics, where there is a need to generate large amounts of realistic textures for applications such as video games and animated movies. Similarly to MPS, these texture synthesis methods use training images, or exemplars, to generate realistic-looking graphical textures.Both domains of multiple-point geostatistics and example-based texture synthesis present similarities in their historic development and share similar concepts. These disciplines have however remained separated, and as a result significant algorithmic innovations in each discipline have not been universally adopted. Texture synthesis algorithms present drastically increased computational efficiency, patterns reproduction and user control. At the same time, MPS developed ways to condition models to spatial data and to produce 3D stochastic realizations, which have not been thoroughly investigated in the field of texture synthesis.In this paper we review the possible links between these disciplines and show the potential and limitations of using concepts and approaches from texture synthesis in MPS. We also provide guidelines on how recent developments could benefit both fields of research, and what challenges remain open. We provide a comparative historical overview of MPS and texture synthesis.Some concepts that were thought as original in geostatistics have actually been used for a long time in computer graphics.We analyze the considerable computational gains that were achieved in texture synthesis in the last decades.Recommendations are given regarding future research directions and potential cross-fertilization between disciplines.

[1]  James F. O'Brien,et al.  Exposing photo manipulation with inconsistent reflections , 2012, TOGS.

[2]  S. De Iaco,et al.  Validation Techniques for Geological Patterns Simulations Based on Variogram and Multiple-Point Statistics , 2011 .

[3]  Yizhou Yu,et al.  Vector solid textures , 2010, ACM Trans. Graph..

[4]  BoucherAlexandre,et al.  A SGeMS code for pattern simulation of continuous and categorical variables , 2008 .

[5]  D. Benson,et al.  Particle tracking and the diffusion‐reaction equation , 2013 .

[6]  K. Faez,et al.  Stochastic simulation of patterns using Bayesian pattern modeling , 2013, Computational Geosciences.

[7]  R. Dimitrakopoulos,et al.  Two-dimensional Conditional Simulations Based on the Wavelet Decomposition of Training Images , 2009 .

[8]  Paul Switzer,et al.  Filter-Based Classification of Training Image Patterns for Spatial Simulation , 2006 .

[9]  Jesús Carrera,et al.  Application of Multiple Point Geostatistics to Non-stationary Images , 2008 .

[10]  Tuanfeng Zhang,et al.  Memory-Efficient Categorical Multi-point Statistics Algorithms Based on Compact Search Trees , 2012, Mathematical Geosciences.

[11]  Ares Lagae,et al.  A Survey of Procedural Noise Functions , 2010, Comput. Graph. Forum.

[12]  Pierre Goovaerts,et al.  Breast and prostate cancer survival in Michigan , 2009, Cancer.

[13]  Alain Dassargues,et al.  Modeling the effect of clay drapes on pumping test response in a cross-bedded aquifer using multiple-point geostatistics , 2012 .

[14]  P. Goovaerts.  Geostatistical approaches for incorporating elevation into the spatial interpolation of rainfall , 2000 .

[15]  Kung-Sik Chan,et al.  Spatial fisheries ecology: Recent progress and future prospects , 2008 .

[16]  Anil K. Jain,et al.  Markov Random Field Texture Models , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[17]  Marita Stien,et al.  Facies Modeling Using a Markov Mesh Model Specification , 2011 .

[18]  G. Matheron.  Éléments pour une théorie des milieux poreux , 1967 .

[19]  Eli Shechtman,et al.  Image melding , 2012, ACM Trans. Graph..

[20]  Pejman Tahmasebi,et al.  Multiple-point geostatistical modeling based on the cross-correlation functions , 2012, Computational Geosciences.

[21]  R. Frodeman.  Geological reasoning: Geology as an interpretive and historical science , 1995 .

[22]  Francesc Maynou,et al.  Assessing the performance of linear geostatistical tools applied to artificial fisheries data , 2006 .

[23]  András Bárdossy,et al.  Effects of non‐Gaussian copula‐based hydraulic conductivity fields on macrodispersion , 2012 .

[24]  Thomas Aigner,et al.  CARBONATE GEOBODIES: HIERARCHICAL CLASSIFICATION AND DATABASE - A NEW WORKFLOW FOR 3D RESERVOIR MODELLING , 2012 .

[25]  Ted Chang,et al.  Introduction to Geostatistics: Applications in Hydrogeology , 2001, Technometrics.

[26]  Heidi Kjønsberg.  MARKOV MESH SIMULATIONS WITH DATA CONDITIONING THROUGH INDICATOR KRIGING , 2008 .

[27]  Steve Marschner,et al.  Structure-aware synthesis for predictive woven fabric appearance , 2012, ACM Trans. Graph..

[28]  Harry Shum,et al.  Image completion with structure propagation , 2005, ACM Trans. Graph..

[29]  Jay M. Ver Hoef,et al.  Spatial methods for plot-based sampling of wildlife populations , 2008, Environmental and Ecological Statistics.

[30]  Peter M. Atkinson,et al.  Multiple-point geostatistical simulation for post-processing a remotely sensed land cover classification , 2013 .

[31]  Gary A. Pope,et al.  Society of Petroleum Engineers of AIME, (Paper) SPE , 1986 .

[32]  Antoine Saucier,et al.  A patchwork approach to stochastic simulation: A route towards the analysis of morphology in multiphase systems , 2008 .

[33]  F. Alabert,et al.  Non-Gaussian data expansion in the Earth Sciences , 1989 .

[34]  Sung Yong Shin,et al.  On pixel-based texture synthesis by non-parametric sampling , 2006, Comput. Graph..

[35]  A. Safekordi,et al.  A multiple-point statistics algorithm for 3D pore space reconstruction from 2D images , 2011 .

[36]  Baining Guo,et al.  Context-aware textures , 2007, TOGS.

[37]  Baining Guo,et al.  Real-time texture synthesis by patch-based sampling , 2001, TOGS.

[38]  P. Renard,et al.  Dealing with spatial heterogeneity , 2005 .

[39]  Giuseppe Patanè,et al.  Topology- and error-driven extension of scalar functions from surfaces to volumes , 2009, TOGS.

[40]  P. Kitanidis.  Quasi‐Linear Geostatistical Theory for Inversing , 1995 .

[41]  Jef Caers,et al.  Direct Pattern-Based Simulation of Non-stationary Geostatistical Models , 2012, Mathematical Geosciences.

[42]  Song-Chun Zhu,et al.  Filters, Random Fields and Maximum Entropy (FRAME): Towards a Unified Theory for Texture Modeling , 1998, International Journal of Computer Vision.

[43]  Baining Guo,et al.  Chaos Mosaic: Fast and Memory Efficient Texture Synthesis , 2000 .

[44]  Jef Caers,et al.  Modeling Uncertainty in the Earth Sciences , 2011 .

[45]  Clayton V. Deutsch,et al.  The Application of Simulated Annealing to Stochastic Reservoir Modeling , 1994 .

[46]  J. Relethford,et al.  Geostatistics and spatial analysis in biological anthropology. , 2008, American journal of physical anthropology.

[47]  Henning Omre,et al.  Petroleum Geostatistics , 1996 .

[48]  Sonia M. Kreidenweis,et al.  Biomass burning as a potential source for atmospheric ice nuclei: Western wildfires and prescribed burns , 2012 .

[49]  Sayali Baheti,et al.  Region Filling and Object Removal by Exemplar based Image Inpainting , 2015 .

[50]  Christian P. Robert,et al.  Statistics for Spatio-Temporal Data , 2014 .

[51]  G. Matheron.  Les variables régionalisées et leur estimation : une application de la théorie de fonctions aléatoires aux sciences de la nature , 1965 .

[52]  Marc Levoy,et al.  Texture synthesis over arbitrary manifold surfaces , 2001, SIGGRAPH.

[53]  Charles F. Harvey,et al.  When good statistical models of aquifer heterogeneity go bad: A comparison of flow, dispersion, and mass transfer in connected and multivariate Gaussian hydraulic conductivity fields , 2003 .

[54]  Michael Garland,et al.  Towards Real-Time Texture Synthesis with the Jump Map , 2002, Rendering Techniques.

[55]  Sylvain Lefebvre,et al.  Instant Texture Synthesis by Numbers , 2010, VMV.

[56]  Dani Lischinski,et al.  Solid texture synthesis from 2D exemplars , 2007, ACM Trans. Graph..

[57]  Alexei A. Efros,et al.  Image quilting for texture synthesis and transfer , 2001, SIGGRAPH.

[58]  A. Bárdossy,et al.  Geostatistical interpolation using copulas , 2008 .

[59]  J. Gómez-Hernández,et al.  To be or not to be multi-Gaussian? A reflection on stochastic hydrogeology , 1998 .

[60]  T. Hansen,et al.  Inverse problems with non-trivial priors: efficient solution through sequential Gibbs sampling , 2012, Computational Geosciences.

[61]  J. Chilès,et al.  Geostatistics: Modeling Spatial Uncertainty , 1999 .

[62]  Hugh G. Lewis,et al.  Super-resolution land cover pattern prediction using a Hopfield neural network , 2002 .

[63]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[64]  Tao Huang,et al.  GPU-based SNESIM implementation for multiple-point statistical simulation , 2013, Comput. Geosci..

[65]  M. Levoy,et al.  Texture synthesis by fixed neighborhood searching , 2001 .

[66]  Snehamoy Chatterjee,et al.  Geologic heterogeneity representation using high‐order spatial cumulants for subsurface flow and transport simulations , 2011 .

[67]  Christian Lantuéjoul,et al.  Geostatistical Simulation: Models and Algorithms , 2001 .

[68]  Xin Tong,et al.  Accelerated Parallel Texture Optimization , 2007, Journal of Computer Science and Technology.

[69]  Sebastien Strebelle,et al.  Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics , 2002 .

[70]  Alexandre Boucher,et al.  Considering complex training images with search tree partitioning , 2009, Comput. Geosci..

[71]  Hans-Jörg Vogel,et al.  Upscaling for unsaturated flow for non‐Gaussian heterogeneous porous media , 2007 .

[72]  Clayton V. Deutsch,et al.  FLUVSIM: a program for object-based stochastic modeling of fluvial depositional systems , 2002 .

[73]  Sylvain Lefebvre,et al.  Appearance-space texture synthesis , 2006, ACM Trans. Graph..

[74]  Klaus Mueller,et al.  Transferring color to greyscale images , 2002, ACM Trans. Graph..

[75]  G. Mariéthoz,et al.  Demonstration of a geostatistical approach to physically consistent downscaling of climate modeling simulations , 2013 .

[76]  Marc Levoy,et al.  Fast texture synthesis using tree-structured vector quantization , 2000, SIGGRAPH.

[77]  A. Journel,et al.  The Necessity of a Multiple-Point Prior Model , 2007 .

[78]  Daniel Cohen-Or,et al.  Fragment-based image completion , 2003, ACM Trans. Graph..

[79]  Eitan Grinspun,et al.  Multiscale texture synthesis , 2008, ACM Trans. Graph..

[80]  Jesús Carrera,et al.  Scale effects in transmissivity , 1996 .

[81]  Alain Dassargues,et al.  Direct Multiple-Point Geostatistical Simulation of Edge Properties for Modeling Thin Irregularly Shaped Surfaces , 2011 .

[82]  Jeremy S. De Bonet,et al.  Multiresolution sampling procedure for analysis and synthesis of texture images , 1997, SIGGRAPH.

[83]  Sylvain Lefebvre,et al.  Lazy Solid Texture Synthesis , 2008, Comput. Graph. Forum.

[84]  H. H. Franssen,et al.  A comparison of seven methods for the inverse modelling of groundwater flow. Application to the characterisation of well catchments , 2009 .

[85]  H. Haldorsen,et al.  NOTES ON STOCHASTIC SHALES; FROM OUTCROP TO SIMULATION MODEL , 1986 .

[86]  Sylvain Lefebvre,et al.  Parallel controllable texture synthesis , 2005, ACM Trans. Graph..

[87]  Irfan A. Essa,et al.  Graphcut textures: image and video synthesis using graph cuts , 2003, ACM Trans. Graph..

[88]  Michael Ashikhmin,et al.  Synthesizing natural textures , 2001, I3D '01.

[89]  G. Mariéthoz,et al.  Modeling complex geological structures with elementary training images and transform‐invariant distances , 2011 .

[90]  Yuhong Liu,et al.  Improving Sequential Simulation with a Structured Path Guided by Information Content , 2004 .

[91]  Pierre Goovaerts,et al.  Visualizing and testing the impact of place on late-stage breast cancer incidence: a non-parametric geostatistical approach. , 2010, Health & place.

[92]  Pejman Tahmasebi,et al.  Accelerating geostatistical simulations using graphics processing units (GPU) , 2012, Comput. Geosci..

[93]  Kris Popat,et al.  Novel cluster-based probability model for texture synthesis, classification, and compression , 1993, Other Conferences.

[94]  L. Hu,et al.  Multiple-Point Simulations Constrained by Continuous Auxiliary Data , 2008 .

[95]  Mario Chica-Olmo,et al.  Downscaling Cokriging for Super-Resolution Mapping of Continua in Remotely Sensed Images , 2008, IEEE Transactions on Geoscience and Remote Sensing.

[96]  R. Grayson,et al.  Toward capturing hydrologically significant connectivity in spatial patterns , 2001 .

[97]  S. Gorelick,et al.  Identifying discrete geologic structures that produce anomalous hydraulic response: An inverse modeling approach , 2008 .

[98]  Yizhou Yu,et al.  Feature matching and deformation for texture synthesis , 2004, ACM Trans. Graph..

[99]  Song-Chun Zhu,et al.  Equivalence of Julesz Ensembles and FRAME Models , 2000, International Journal of Computer Vision.

[100]  L. Y. Hu,et al.  Multiple‐point geostatistics for modeling subsurface heterogeneity: A comprehensive review , 2008 .

[101]  Adam Finkelstein,et al.  PatchMatch: a randomized correspondence algorithm for structural image editing , 2009, SIGGRAPH 2009.

[102]  Neil A. Dodgson,et al.  Self-similarity based texture editing , 2002, SIGGRAPH.

[103]  Alexandre Boucher,et al.  A SGeMS code for pattern simulation of continuous and categorical variables: FILTERSIM , 2008, Comput. Geosci..

[104]  Philippe Renard,et al.  Issues in characterizing heterogeneity and connectivity in non-multiGaussian media , 2008 .

[105]  Yuhong Liu,et al.  Using the Snesim program for multiple-point statistical simulation , 2006, Comput. Geosci..

[106]  Sylvain Lefebvre,et al.  State of the Art in Example-based Texture Synthesis , 2009, Eurographics.

[107]  Mary P. Anderson,et al.  Simulation of Preferential Flow in Three-Dimensional, Heterogeneous Conductivity Fields with Realistic Internal Architecture , 1996 .

[108]  G. Mariéthoz,et al.  Conditioning Facies Simulations with Connectivity Data , 2011 .

[109]  Rupert Paget,et al.  Texture synthesis via a noncausal nonparametric multiscale Markov random field , 1998, IEEE Trans. Image Process..

[110]  R. M. Srivastava,et al.  Multivariate Geostatistics: Beyond Bivariate Moments , 1993 .

[111]  Julián M. Ortiz,et al.  Parallel implementation of simulated annealing to reproduce multiple-point statistics , 2011, Comput. Geosci..

[112]  Philippe Renard,et al.  Integrating collocated auxiliary parameters in geostatistical simulations using joint probability distributions and probability aggregation , 2009 .

[113]  D. A. Zimmerman,et al.  A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow , 1998 .

[114]  Alexei Pozdnoukhov,et al.  Interest rates mapping , 2007, 0709.4361.

[115]  Grégoire Mariethoz,et al.  Bathymetry fusion using multiple-point geostatistics: Novelty and challenges in representing non-stationary bedforms , 2013, Environmental Modelling & Software.

[116]  Tao Huang,et al.  GPU-accelerated Direct Sampling method for multiple-point statistical simulation , 2013, Comput. Geosci..

[117]  Timothy C. Coburn,et al.  Geostatistics for Natural Resources Evaluation , 2000, Technometrics.

[118]  M. Sahimi,et al.  Cross-correlation function for accurate reconstruction of heterogeneous media. , 2013, Physical review letters.

[119]  Roussos Dimitrakopoulos,et al.  High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena , 2009 .

[120]  Gregoire Mariethoz,et al.  Using bivariate multiple-point statistics and proximal soil sensor data to map fossil ice-wedge polygons , 2011 .

[121]  G. Mariéthoz,et al.  An Improved Parallel Multiple-point Algorithm Using a List Approach , 2011 .

[122]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[123]  Rainer Helmig,et al.  Estimation of effective parameters for a two-phase flow problem in non-Gaussian heterogeneous porous media. , 2011, Journal of contaminant hydrology.

[124]  Philippe Renard,et al.  Parallel Multiple-Point Statistics Algorithm Based on List and Tree Structures , 2013, Mathematical Geosciences.

[125]  Burkhard Wünsche,et al.  Fast Spatially Controllable 2D/3D Texture Synthesis and Morphing for Multiple Input Textures , 2009, GRAPP.

[126]  Clayton V. Deutsch,et al.  Multiple Point Metrics to Assess Categorical Variable Models , 2010 .

[127]  J. Caers,et al.  Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling , 2010 .

[128]  Kun Zhou,et al.  Synthesis of progressively-variant textures on arbitrary surfaces , 2003, ACM Trans. Graph..

[129]  C.-C. Jay Kuo,et al.  Texture analysis and classification with tree-structured wavelet transform , 1993, IEEE Trans. Image Process..

[130]  Colin Daly,et al.  Higher Order Models using Entropy, Markov Random Fields and Sequential Simulation , 2005 .

[131]  Sylvain Lefebvre,et al.  Parallel patch-based texture synthesis , 2012, EGGH-HPG'12.

[132]  Christian Lantuéjoul,et al.  Can a Training Image Be a Substitute for a Random Field Model? , 2014, Mathematical Geosciences.

[133]  Snehamoy Chatterjee,et al.  Multi-scale stochastic simulation with a wavelet-based approach , 2012, Comput. Geosci..

[134]  Marc Levoy,et al.  Order-Independent Texture Synthesis , 2014, ArXiv.

[135]  Håkon Tjelmeland,et al.  Construction of Binary Multi-grid Markov Random Field Prior Models from Training Images , 2013, Mathematical Geosciences.

[136]  Knud Skou Cordua,et al.  A Frequency Matching Method: Solving Inverse Problems by Use of Geologically Realistic Prior Information , 2012, Mathematical Geosciences.

[137]  James R. Bergen,et al.  Pyramid-based texture analysis/synthesis , 1995, Proceedings., International Conference on Image Processing.

[138]  P. Renard,et al.  Probability Aggregation Methods in Geoscience , 2012, Mathematical Geosciences.

[139]  Eero P. Simoncelli,et al.  A Parametric Texture Model Based on Joint Statistics of Complex Wavelet Coefficients , 2000, International Journal of Computer Vision.

[140]  Paul Switzer,et al.  Stochastic space‐time regional rainfall modeling adapted to historical rain gauge data , 2006 .

[141]  E. Isaaks,et al.  Indicator Simulation: Application to the Simulation of a High Grade Uranium Mineralization , 1984 .

[142]  Gregoire Mariethoz,et al.  Quantifying the value of laminated stalagmites for paleoclimate reconstructions , 2011 .

[143]  Philippe Renard,et al.  3D multiple-point statistics simulation using 2D training images , 2012, Comput. Geosci..

[144]  Bruno Galerne,et al.  Gabor noise by example , 2012, ACM Trans. Graph..

[145]  Alexei A. Efros,et al.  Photo clip art , 2007, SIGGRAPH 2007.

[146]  Tuan D. Pham,et al.  Supervised restoration of degraded medical images using multiple-point geostatistics , 2012, Comput. Methods Programs Biomed..

[147]  J. Caers,et al.  Conditional Simulation with Patterns , 2007 .

[148]  Irfan Essa,et al.  Texture optimization for example-based synthesis , 2005, SIGGRAPH 2005.

[149]  David Salesin,et al.  Image Analogies , 2001, SIGGRAPH.

[150]  Baining Guo,et al.  Synthesis of bidirectional texture functions on arbitrary surfaces , 2002, SIGGRAPH.

[151]  Long Quan,et al.  Image deblurring with blurred/noisy image pairs , 2007, SIGGRAPH 2007.

[152]  J. Besag,et al.  Bayesian image restoration, with two applications in spatial statistics , 1991 .

[153]  Julián M. Ortiz,et al.  Adapting a texture synthesis algorithm for conditional multiple point geostatistical simulation , 2011 .