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Recent Bayesian approaches for spatial analysis of 2-D images with application to environmental modelling

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Abstract

From remote sensing of the environment, to brain scans in medicine, the growth in the use of image data has motivated a parallel increase in statistical techniques for analysing these images. A particular area of growth has been in Bayesian models and corresponding computational methods. Bayesian approaches have been proposed to address the gamut of supervised and unsupervised inferential aims in image analysis. In this article we provide a general review of these approaches, with a focus on unsupervised analysis of 2-D images. Four exemplar methods that canvas the broad aims of image modelling and analysis are described. An exposition of these approaches is provided by applying them to an environmental case study involving the use of satellite data to assess water quality in the Great Barrier Reef, Australia. The techniques considered in detail are hidden Markov random fields (MRF), Gaussian MRF, Poisson/gamma random fields, and Voronoi tessellations. We also consider a variety of enabling computational algorithms, including MCMC, variational Bayes and integrated nested Laplace approximations. We compare the different aims and inferential capabilities of the models and discuss the advantages and drawbacks of the corresponding computational algorithms.

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References

  • Alston C, Mengersen K (2010) Allowing for the effect of data binning in a Bayesian normal mixture model. Comput Stat Data Anal 54(4):916–923

    Article  Google Scholar 

  • Alston CL, Mengersen KL, Thompson JM, Littlefield PJ, Perry D, Ball AJ (2005) Extending the Bayesian mixture model to incorporate spatial information in analysing sheep CAT scan images. Aust J Agric Res 56(4):373–388

    Article  Google Scholar 

  • Antoniak C (1974) Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann Stat 2(6):1152–1174

    Article  Google Scholar 

  • Banerjee S, Carlin B, Gelfand A (2004) Hierarchical modeling and analysis for spatial data. Chapman & Hall/CRC Press, Boca Raton

    Google Scholar 

  • Beal MJ, Ghahramani Z (2003) The variational Bayesian EM algorithm for incomplete data: with application to scoring graphical model structures. In: Proceedings of the seventh Valencia international meeting, volume 7 of Bayesian statistics. Oxford University Press, London, pp 453–464

  • Besag J (1974) Spatial interaction and statistical analysis of lattice systems (with discussion). J R Stat Soc Ser B Stat Methodol 36(2):192–236

    Google Scholar 

  • Besag J (1975) Statistical analysis of non-lattice data. J R Stat Soc D 24(3):179–195

    Google Scholar 

  • Besag J (1986) On the statistical analysis of dirty pictures. JRSS B 48(3):259–302

    Google Scholar 

  • Best NG, Ickstadt K, Wolpert RL (2000) Spatial Poisson regression for health and exposure data measured at disparate resolutions. J Am Stat Assoc 95(452):1076–1088

    Article  Google Scholar 

  • Blondeau-Patissier D, Brando VE, Oubelkheir K, Dekker AG, Clementson LA, Daniel P (2009) Bio-optical variability of the absorption and scattering properties of the Queensland inshore and reef waters, Australia. J Geophys Res 114:C05003. doi:10.1029/2008JC005039

    Google Scholar 

  • Brando VE, Dekker AG, Schroeder T, Park YJ, Clementson LA, Steven A, Blondeau-Patissier D (2008) Satellite retrieval of chlorophyll CDOM and NAP in optically complex waters using a semi-analytical inversion based on specific inherent optical properties. A case study for Great Barrier Reef coastal waters. In: Proceedings of the XIX ocean optics conference

  • Chan KS (1993) Asymptotic behavior of the Gibbs sampler. J Am Stat Assoc 88(421):320–326

  • Cox DR (1955) Some statistical methods connected with series of events. JRSS B 17(2):129–164

    Google Scholar 

  • Cressie N, Wikle C (2011) Statistics for spatio-temporal data. Wiley, New York

    Google Scholar 

  • Cucala J, Marin JM, Robert CP, Titterington DM (2009) A Bayesian reassessment of nearest-neighbour classification. J Am Stat Assoc 104(485):263–273

    Article  CAS  Google Scholar 

  • de Berg M, van Kreveld M, Overmars M, Schwarzkopf O (2000) Computational geometry: algorithms and applications, 2nd edn. Springer, Berlin

    Book  Google Scholar 

  • Dempster A, Laird N, Rubin D (1977) Maximum likelihood from incomplete data via the EM algorithm. JRSS B 39(1):1–38

    Google Scholar 

  • Denham R, Falk M, Mengersen K (2011) The Bayesian conditional independence model for measurement error: applications in ecology. Environ Ecol Stat 18:239–255

    Article  Google Scholar 

  • Denison DGT, Adams NM, Holmes CC, Hand DJ (2002) Bayesian partition modelling. Comput Stat Data Anal 38(4):475–485

    Article  Google Scholar 

  • Denison DGT, Holmes CC (2001) Bayesian partitioning for estimating disease risk. Biometrics 57(1):143–149

    Article  CAS  PubMed  Google Scholar 

  • Engen S (1978) Stochastic abundance models, with emphasis on biological communities and species diversity., Monographs on applied probability and statisticsChapman and Hall, London

    Book  Google Scholar 

  • Ferguson T (1973) A Bayesian analysis of some nonparametric problems. Ann Stat 1(2):209–230

    Article  Google Scholar 

  • Friel N, Pettitt AN (2004) Likelihood estimation and inference for the autologistic model. J Comput Graph Stat 13(1):232–246

    Article  Google Scholar 

  • Gelfand A (1996) Model determination using sampling based methods. In: Gilks W, Richardson S, Spiegehalter D (eds) Markov Chain Monte Carlo in Practice, chap 9. Chapman and Hall, Boca Raton, FL

  • Gelfand AE, Smith AFM (1990) Sampling-based approaches to calculating marginal densities. J Am Stat Assoc 85(410):398–409

  • Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans PAMI 6:721–741

    Article  CAS  Google Scholar 

  • Gilks W, Richardson S, Spiegelhalter D (eds) (1996) Markov chain Monte Carlo in practice., Interdisciplinary statistics seriesChapman & Hall/CRC Press, Boca Raton

    Google Scholar 

  • Goldsmith J, Wand MP, Crainceanu C (2011) Functional regression via variational Bayes. Electron J Stat 5:572–602

    Article  PubMed Central  PubMed  Google Scholar 

  • Green P, Richardson S (2002) Hidden Markov models and disease mapping. J Am Stat Assoc 97(460):1055–1070

    Article  Google Scholar 

  • Green PJ (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4):711–732

    Article  Google Scholar 

  • Grenander U (1983) Tutorial in Pattern Theory. Technical report, Division of Applied Mathematics. Brown University

  • Grenander U, Miller M (2007) Pattern theory: from representation to inference. Oxford University Press, Oxford

    Google Scholar 

  • Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1):97–109

    Article  Google Scholar 

  • Heron EA, Walsh CD (2008) A continuous latent spatial model for crack initiation in bone cement. J R Stat Soc Ser C Appl Stat 57(1):25–42

    Article  Google Scholar 

  • Ickstadt K, Wolpert R (1999) Spatial regression for marked point processes (with discussion). In: Bernado J, Berger J, Dawid A, Smith A (eds) Bayesian statistics 6. Oxford University Press, Oxford, pp 323–341

    Google Scholar 

  • Illian JB, Sørbye SH, Rue H (2012) A toolbox for fitting complex spatial point process models using integrated nested Laplace approximation (INLA). Ann Appl Stat 6(4):1499–1530

  • Jordan M, Ghahramani Z, Jaakkola T, Saul L (1999) An introduction to variational methods for graphical models. Mach Learn 37:183–233

    Article  Google Scholar 

  • Lang S, Brezger A (2004) Bayesian P-Splines. J Comput Graph Stat 13(1):183–212

    Article  Google Scholar 

  • Leonte D, Nott DJ (2006) Bayesian spatial modelling of gamma ray count data. Math Geol 38(2):135–154

    Article  Google Scholar 

  • Liang S, Banerjee S, Carlin BP (2009) Bayesian wombling for spatial point processes. Biometrics 65(4):1243–1253

    Article  PubMed Central  PubMed  Google Scholar 

  • Lindgren F, Rue H, Lindström J (2011) An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. J R Stat Soc B 73:423–498

  • Lu H, Carlin BP (2005) Bayesian areal wombling for geographical boundary analysis. Geogr Anal 37(3):265–285

    Article  Google Scholar 

  • Ma H, Carlin BP (2007) Bayesian multivariate areal wombling for multiple disease boundary analysis. Bayesian Anal 2(2):281–302

    Article  Google Scholar 

  • Marin J-M, Robert C (2007) Bayesian core: a practical approach to computational Bayesian statistics., Springer Texts in StatisticsSpringer, Berlin

    Google Scholar 

  • Martin A, Quinn K, Park J (2011) MCMCpack: Markov chain Monte Carlo in R. J Stat Softw 42(9):1–21

    Google Scholar 

  • Matérn B (1960) Spatial variation. stochastic models and their application to some problems in forest surveys and other sampling investigations. Meddelanden fran statens Skogsforskningsinstitut 49:1–144

    Google Scholar 

  • McGrory CA, Titterington DM, Reeves R, Pettitt AN (2009) Variational Bayes for estimating the parameters of a hidden Potts model. Stat Comput 19(3):329–340

    Article  Google Scholar 

  • Mengersen KL, Tweedie RL (1996) Rates of convergence of the Hastings and Metropolis algorithms. Ann Stat 24(1):101–121

    Article  Google Scholar 

  • Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092

    Article  CAS  Google Scholar 

  • Møller J, Pettitt AN, Reeves R, Berthelsen KK (2006) An efficient Markov chain monte carlo method for distributions with intractable normalising constants. Biometrika 93(2):451–458

    Article  Google Scholar 

  • Møller J, Syversveen A, Waagepetersen R (1998) Log Gaussian Cox processes. Scand J Stat 25(3):451–482

    Article  Google Scholar 

  • Møller J, Waagepetersen R (2004) Statistical inference and simulation for spatial point processes. Chapman & Hall/CRC Press, Boca Raton

    Google Scholar 

  • Neal RM (2003) Slice sampling. Ann Stat 31(3):705–741

    Article  Google Scholar 

  • Nicholls GK (1998) Bayesian image analysis with Markov chain Monte Carlo and coloured continuum triangulation models. JRSS B 60(3):643–659

    Article  Google Scholar 

  • Okabe A, Boots B, Sugihara K, Chiu SN (2000) Spatial tessellations: concepts and applications of Voronoi diagrams, 2nd edn. Wiley, Chichester

    Book  Google Scholar 

  • Orbanz P, Buhmann, J (2006) Smooth image segmentation by nonparametric Bayesian inference. In: Proceedings of the ECCV 2006, volume 3951 of Lecture Notes in Computer Science. Springer, Berlin, Heidelberg, pp 444–457

  • Ormerod JT, Wand MP (2010) Explaining variational approximations. Am Stat 64(2):140–153

    Article  Google Scholar 

  • O’Rourke J (1998) Computational geometry in C, 2nd edn. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Pitman J, Yor M (1997) The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann Probab 25(2):855–900

    Article  Google Scholar 

  • Plummer M (2003) JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling. In Hornik K, Leisch F, Zeileis A (eds) Proceedings of the 3rd international workshop on distributed statistical computing

  • Potts RB (1952) Some generalized order-disorder transitions. Proc Camb Philos Soc 48:106–109

    Article  Google Scholar 

  • Robert C, Casella G (2004) Monte Carlo statistical methods, 2nd edn., Springer Texts in StatisticsSpringer, New York

    Book  Google Scholar 

  • Rue H (2001) Fast sampling of Gaussian Markov random fields. JRSS B 63(2):325–338

    Article  Google Scholar 

  • Rue H, Held L (2005) Gaussian Markov random fields: theory and applications, volume 104 of Monographs on statistics and applied probability. Chapman & Hall/CRC Press, Boca Raton

    Book  Google Scholar 

  • Rue H, Martino S (December, 2010) INLA: functions which allow to perform a full Bayesian analysis of structured additive models using Integrated Nested Laplace Approximaxion (0.0 ed.)

  • Rue H, Martino S, Chopin N (2009) Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J R Stat Soc B 71:319–392

    Article  Google Scholar 

  • Rue H, Tjelmeland H (2002) Fitting Gaussian Markov random fields to Gaussian fields. Scand J Stat 29(1):31–49

    Article  Google Scholar 

  • Rydén T, Titterington D (1998) Computational Bayesian analysis of hidden Markov models. J Comput Graph Stat 7(2):194–211

    Google Scholar 

  • Shyr A, Darrell T, Jordan M, Urtasun R (2011) Supervised hierarchical Pitman-Yor process for natural scene segmentation. In: Proceedings of the IEEE CVPR, pp 2281–2288

  • Smith M, Kohn R (1996) Nonparametric regression using Bayesian variable selection. J Econom 75(2):317–343

    Article  Google Scholar 

  • Strickland C, Denham R, Alston C, Mengersen K (2012) A Python package for Bayesian estimation using Markov chain Monte Carlo. In: Alston C, Mengersen K, Pettitt A (eds) Case atudies in Bayesian statistical modelling and analysis. Wiley, New York, pp 421–460

    Chapter  Google Scholar 

  • Sudderth E, Jordan M (2009) Shared segmentation of natural scenes using dependent Pitman-Yor processes. Proc NIPS 21:1585–1592

    Google Scholar 

  • Tanner MA, Wong WH (1987) The calculation of posterior distributions by data augmentation. J Am Stat Assoc 82(398):528–540

  • Teh Y, Jordan M, Beal M, Blei D (2006) Hierarchical Dirichlet processes. J Am Stat Assoc 101(476):1566–1581

  • Thomas A, Best N, Lunn D, Spiegelhalter D (2012) The BUGS book: a practical introduction to Bayesian analysis, volume 98 of Chapman & Hall/CRC Texts in Statistical Science. Taylor & Francis, London

    Google Scholar 

  • Tierney L, Kadane JB (1986) Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc 81(393):82–86

  • Winkler G (2003) Image analysis, random fields and Markov Chain Monte Carlo methods: a mathematical introduction 2nd edn. Springer, Berlin

    Book  Google Scholar 

  • Wolpert RL, Ickstadt K (1998a) Poisson/gamma random field models for spatial statistics. Biometrika 85(2):251–267

    Article  Google Scholar 

  • Wolpert RL, Ickstadt K (1998b) Simulation of Lévy Random Fields. In: Müller P, Dey D, Sinha D (eds) Practical nonparametric and semiparametric Bayesian statistics. Springer, Berlin, pp 227–242

    Chapter  Google Scholar 

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Acknowledgments

The authors would like to thank Dr. Andy Steven, CSIRO, for the data used in this paper. The authors would also like to thank two anonymous reviewers for their comments which greatly improved the manuscript. The research was funded from an Australian Research Council (ARC) Linkage Grant (LP0668185) and by the ARC Centre of Excellence for Complex Dynamic Systems and Control (CDSC).

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Authors and Affiliations

  1. Mathematical Sciences, Science and Engineering, Queensland University of Technology, GPO Box 2434, Brisbane, QLD, 4001, Australia

    M. G. Falk, C. L. Alston, C. A. McGrory, S. Clifford, E. A. Heron, D. Leonte, M. Moores, C. D. Walsh, A. N. Pettitt & K. L. Mengersen

  2. Department of Psychiatry, Trinity College Dublin, Dublin 2, Ireland

    E. A. Heron

  3. National Industry Chemicals Notification and Assessment Scheme, GPO Box 58, Sydney, NSW, 2001, Australia

    D. Leonte

  4. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK

    M. Moores

  5. Department of Statistics, Trinity College Dublin, Dublin 2, Ireland

    C. D. Walsh

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  1. M. G. Falk
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  3. C. A. McGrory
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Correspondence to M. G. Falk.

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Falk, M.G., Alston, C.L., McGrory, C.A. et al. Recent Bayesian approaches for spatial analysis of 2-D images with application to environmental modelling. Environ Ecol Stat 22, 571–600 (2015). https://doi.org/10.1007/s10651-015-0311-1

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