A new flexible prior for Bayesian image analysis and reservoir modelling is defined
in terms of interacting coloured Voronoi cells described by a certain
nearest-neighbour Markov point process. This prior can be defined in both two and
three (as well as higher) dimensions, and simple MCMC algorithms can be used for
drawing inference from the posterior distribution. Various 2D and 3D applications
are considered.
Ahuja N
,
Schachter BJ
(1983) Pattern models.
New York: John Wiley & Sons
.
2.
Arak T
,
Clifford P
,
Surgailis D
(1993)
Point based polygonal models for random graphs
. Advances in Applied Probability, 25,
348-372
.
3.
Baddeley A
,
Møller J
(1989)
Nearest-neighbour Markov point processes and random sets
. International Statistical Review, 57,
89-121
.
4.
Baddeley A
,
Turner R
(2000)
Practical maximum pseudolikelihood for spatial point patterns
. Australian and New Zealand Journal of Statistics,
42,
283-322
.
5.
Baddeley AJ
,
Van Lieshout MNM
(1993) Stochastic geometry models in high-level
vision. In
Marida, KV
,
Kanjii GK
, eds. Statistics and image, special issue Journal of Applied
Statistics, 231-256.
6.
Besag J
(1975)
Statistical analysis of non-lattice data
. The Statistician, 24,
179-195
.
7.
Besag J
(1977)
Some methods of statistical analysis for spatial data
. Bulletin of the International Statistical Institute,
47,
77-92
.
8.
Besag J
(1986)
On the statistical analysis of dirty pictures
. Journal of the Royal Statistical Society, Series B,
48,
259-302
.
9.
Bjørlykke K
(1989) Sedimentology and petroleum geology.
Berlin: Springer-Verlag
.
10.
Damsleth E
,
Tjølsen CB
,
Omre H
,
Haldorsen HH
(1992)
A two-stage stochastic model applied to a North Sea reservoir
. Journal of Petroleum Technology, 44(4),
402-402
.
11.
Geman S
,
Geman D
(1984)
Stochastic relaxation, Gibbs distribution, and Bayesian restoration of images
. IEEE Transactions on Pattern Analysis and Machine
Intelligence, 6,
721-741
.
12.
Geyer CJ
(1992)
Practical Markov chain Monte Carlo
. Statistical Science, 7,
473-483
.
13.
Geyer CJ
(1999) Likelihood inference for spatial point
processes. In
Barndorff-Nielsen O
,
Kendall W
,
van Lieshout M
, eds. Stochastic geometry: likelihood and computation.
Boca Raton: Chapman and Hall/CRC
, 79-140.
14.
Geyer G
,
Møller J
(1994)
Simulation procedures and likelihood inferences for spatial point processes
. Scandinavian Journal of Statistics, 21
(4),
369-373
.
15.
Geyer CJ
,
Thompson EA
(1992)
Constrained Monte Carlo maximum likelihood for dependent data
. Journal of the Royal Statistical Society, Series B,
54,
657-699
.
16.
Geyer CJ
,
Thompson ET
(1995)
Annealing Markov chain Monte Carlo with applications to ancestral inference
. Journal of the Amercian Statistical Association,
90,
909-920
.
17.
Green PJ
(1995)
Reversible jump Markov chain Monte Carlo computation and Bayesian model determination
. Biometrika, 82,
711-732
.
18.
Grenander U
,
Miller M
(1994)
Representation of knowledge in complex systems
. Journal of The Royal Statistical Society, Series B,
56,
549-603
.
19.
Heikkinen J
,
Arjas E
(1998)
Non-parametric Bayesian estimation of a spatial Poisson intensity
. Scandinavian Journal of Statistics, 25,
435-450
.
20.
Heikkinen J
,
Arjas E
(1999)
Modeling a poisson forest in variable elevations: a nonparametric Bayesian approach
. Biometrics, 55,
738-745
.
21.
Heikkinen J
,
Högmander H
(1994)
Fully Bayesian approach to image restoration with an application in biogeography
. Applied Statistics, 43(4),
569-582
.
22.
Higdon D (1994) Spatial applications of Markov chain Monte Carlo for
Bayesian inferences. PhD thesis, Department of Statistics, University of Washington.
23.
Hjort NL
,
Omre H
(1994)
Topics in spatial statistics
. Scandinavian Journal of Statistics,
21(4),
289-357
.
24.
Holden L
,
Hauge R
,
Skare ø
,
Skorstad A
(1998)
Modelling of fluvial reservoirs with object models
. Mathamatical Geology, 30(5),
473-496
.
25.
Jensen J
,
Møller J
(1991)
Pseudolikelihood for exponential family models of spatial point processes
. Annals of Applied Probability, 3,
445-461
.
26.
Kendall W
,
Møller J
(2000)
Perfect simulation using dominating processes on ordered spaces, with
application to locally stable point processes
. Advances in Applied Probability, 32,
844-865
.
27.
Lia O
,
Tjelmeland H
,
Kjellesvik L
(1997)
Modeling of facies architecture by marked point models
. In
Baafi EY
,
Schofield NA
, eds. Geostatistics Wollongon ’96 Proc. 15th Int. Geostat.
Congr. Wollongong Australia, 1996.
Dordrecht: Kluwer,
386-397
.
28.
Marinari E
,
Parisi G
(1992)
Simulated tempering: a new Monte Carlo scheme
. Europhysics Letters, 19,
451-458
.
29.
Mase S
,
Møller J
,
Stoyan D
,
Waagepetersen R
,
Döge G
(2000)
Packing densities and simulated tempering for hard core Gibbs point processes
. Annals of the Institute of Statistical Mathematics, in press.
30.
Møller J
(1994) Lectures on Random Voronoi Tessellations, Vol. 87. In
Lecture notes in statistics.
New York: Springer-Verlag
.
31.
Møller J
(1999) Markov chain Monte Carlo and spatial point
processes. In
Barndorff-Nielsen O
,
Kendall W
,
van Lieshout M
, eds. Stochastic geometry: likelihood and computation.
Boca Raton: Chapman and Hall/CRC
, 79-140.
32.
Møller J
,
Waagepetersen RP
(1998)
Markov connected component fields
. Advances in Applied Probability, 30,
1-35
.
33.
Mücke E (1993) Shapes and implementations in three-dimensional
geometry. PhD thesis, University of Illinois.
34.
Munthe K
,
Holden L
,
Mostad P
,
Townsend C
(1994)
Modelling sub-seismic fault patterns using a marked point process
. In ECMOR IV, 4th European Conference on the Mathematics of Oil
Recovery. Norway: Røros.
35.
Proceedings of the Leeds Annual Statistics Research Workshop1997.
36.
Nicholls GK
(1998)
Bayesian image analysis with Markov chain Monte Carlo and coloured
continuum triangulation models
. Journal of the Royal Statistical Society, Series B,
60,
643-659
.
37.
Okabe A
,
Boots B
,
Sugihara K
(1992) Spatial tessellations - concepts and applications of
Voronoi diagrams.
Chichester: John Wiley & Sons
.
38.
Ripley BD
(1988) Statistical inference for spatial processes.
Cambridge: Cambridge University Press
.
39.
Rue H
,
Hurn M
(1999)
Bayesian object identifications
. Biometrika, 86(3),
649-660
.
40.
Scheike TH
(2000)
Anisotropic growth of Voronoi cells
. Advances in Applied Probability, 26,
43-53
.
41.
Syversveen A
,
Omre H
(1997)
Conditioning of marked point processes within a Bayesian framework
. Scandinavian Journal of Statistics,
24(3),
341-352
.
42.
Tjelmeland H (1996) Stochastic models in reservoir characterization and
Markov random fields for compact objects. PhD thesis, Department of
Mathematical Sciences, Norwegian University of Science and Technology.
43.
Tjelmeland H
,
Besag JE
(1998)
Markov random fields with higher-order interactions
. Scandinavian Journal of Statistics,
25(3),
415-433
.
44.
Tjelmeland H
,
Omre H
(1997)
A complex sand-shale facies model conditioned on observations from wells,
seismics and production
. In
Baafi EY
,
Schofield NA
, eds. Geostatistics Wollogong ‘96. Proceedings of the Fifth
International Geostatistical Congress. Amsterdam: Kluwer,
634-643
.
