Estimating the Parameters of a Stochastic Geometrical Model for Multiphase Flow Images Using Local Measures


  • Leo Theodon Mines Saint-Etienne
  • Tatyana Eremina Mines Saint-Etienne
  • Kassem Dia Mines Saint-Etienne, Univ Montpellier
  • Fabrice Lamadie Univ Montpellier
  • Jean-Charles Pinoli Mines Saint-Etienne
  • Johan Debayle Mines Saint-Etienne



Local Measures, Maximum Likelihood, Minkowski Functionals, Statistical Inference, Stochastic Geometry


This paper presents a new method for estimating the parameters of a stochastic geometric model for multiphase flow image processing using local measures. Local measures differ from global measures in that they are only based on a small part of a binary image and consequently provide different information of certain properties such as area and perimeter. Since local measures have been shown to be helpful in estimating the typical grain elongation ratio of a homogeneous Boolean model, the objective of this study was to use these local measures to statistically infer the parameters of a more complex non-Boolean model from a sample of observations. An optimization algorithm is used to minimize a cost function based on the likelihood of a probability density of local measurements. The performance of the model is analysed using numerical experiments and real observations. The errors relative to real images of most of the properties of the model-generated images are less than 2%. The covariance and particle size distribution are also calculated and compared.


Bai W., Deen N.G., Kuipers J.A.M.(2012) Numerical investigation of gas holdup and phase mixing in bubble column reactors. IND ENG CHEM RES 51(14):1949-61.

Bello R. Ade, Robinson Campbell W., Moo-Young Murray(1985). Gas holdup and overall volumetric oxygen transfer coefficient in airlift contactors. BIOTECHNOL BIOENG 27(3):369-81

Benassi C., Bianchi G., D'Ercole G. (2010). Covariogram of non-convex sets. MATHEMATIKA 56(2):267-84.

Bettaieb A., Filali N., Filali T., Ben Aissia H. (2020).

An efficient algorithm for overlapping bubbles segmentation. Computer Optics 44(3):363-74.

Buffo A., Vanni V., Marchisio, D. (2017) Simulation of reacting gas-liquid bubbly flow with CFD and PBM: validation with experiments. APPL MATH MODEL 44:43-60.

Cerqueira R.F.L., Paladino E.E., Ynumaru B.K., Maliska C.R. (2018). Image processing techniques for the measurement of two-phase bubbly pipe flows using particle image and tracking velocimetry ({PIV}/{PTV}). CHEM ENG SCI 189:1-23.

Cerqueira R.F.L., Paladino E.E. (2021). Development of a deep learning-based image processing technique for bubble pattern recognition and shape reconstruction in dense bubbly flows. CHEM ENG SCI 230:116-63.

Chiu S.N., Stoyan D., Kendall W.S., Mecke J. (2013).

Stochastic geometry and its applications, 3rd Ed.

Chichester: Wiley.

De Langlard M., Al-Saddik H., Charton S., Debayle J., Lamadie F. (2018). An efficiency improved recognition algorithm for highly overlapping ellipses: Application to dense bubbly flows. PATTERN RECOGN LETT 101:88-95.

De Langlard M., Lamadie F., Charton S., Debayle J. (2018). A 3D stochastic model for geometrical characterization of particles in two-phase flow applications. IMAGE ANAL STEREOL 37(3):233-47.

De Langlard M., Lamadie F., Charton S., Debayle J. (2021). Bayesian Inference of a Parametric Random Spheroid from its Orthogonal Projections. METHODOL COMPUT APPL 23.

De Langlard M. (2019). La géométrie aléatoire pour la caractérisation de populations denses de particules: application aux écoulements diphasiques. Autre. Université de Lyon. Français. NNT: 2019LYSEM001.

Dereudre D., Lavancier F., Helisova K.S. (2014). Estimation of the intensity parameter of the germ-grain Quermass-interaction model when the number of germs is not observed. SCAND J STAT 41:809-29.

Diggle P.J. (1981). Binary mosaics and the spatial pattern of heather. BIOMETRICS 37:531-9.

Epanechnikov V.A. (1969). Non-Parametric Estimation of a Multivariate Probability Density. THEOR PROBAB APPL+, 1969, Vol. 14, No. 1 : pp. 153-58.

Eremina T., Debayle J., Gruy F., Pinoli J.C. (2021). Numerical estimation of local Minkowski measures on Boolean models, 2020 10th International Symposium on Signal, Image, Video and Communications (ISIVC), 2021, pp. 1-5, doi: 10.1109/ISIVC49222.2021.9487537.

Eremina T., Debayle J., Gruy F., Pinoli J.C. (2021). Local Measures Distribution for the Estimation of the Elongation Ratio of the Typical Grain in Homogeneous Boolean Models. IMAGE ANAL STEREOL 40(2):95-103.

Haas T., Schubert C., Eickhoff M., Pfeifer H. (2020). {BubCNN}: Bubble detection using Faster {RCNN} and shape regression network. CHEM ENG SCI 216:1154-67.

Kracht W., Emery X., Paredes C. (2013). A stochastic approach for measuring bubble size distribution via image analysis. INT J MINER PROCESS 121:6-11.

Kurz G., Gilitschenski I., Siegwart R., Hanebeck U.D. (2016). Methods for Deterministic Approximation of Circular Densities. Journal of Advances in Information Fusion 11:138-56.

Lagarias J.C., Reeds J.A., Wright H., Wright P.E. (1998). Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions. SIAM J OPTIMIZ 9(1):112-47.

Lefèvre S. (2009). Beyond Morphological Size-distribution 18. J ELECTRON IMAGING.

Mardia K.V., Jupp P.E. (1999). Directional Statistics. Wiley.

Matheron G. (1972). Ensembles alétoires et géométrie intégrale. Les cahiers du centre de morphologie mathématiques de Fontainebleau.

Molchanov I.S. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley.

Paolinelli L.D., Rashedi A., Yao J. (2018). Characterization of droplet sizes in large scale oil–water flow downstream from a globe valve. INT J MULTIPHAS FLOW 99:132-50.

Rataj J. (2004). On set covariance and three-point test sets. CZECH MATH J 54.1 : 205-14.

Schmidt T. (2007). Coping with copulas. Copulas - From Theory to Application in Finance.

Serra J.A. (1988). Image Analysis and Mathematical Morphology: Theoretical Advances. Academic Press, Cornell University.

Torisaki S., Miwa S. (2020). Robust bubble feature extraction in gas-liquid two-phase flow using object detection technique. J NUCL SCI TECHNOL 57(11):1231-44.

Vinnett L., Yianatos J., Arismendi L., Waters K.E. (2020). Assessment of two automated image processing methods to estimate bubble size in industrial flotation machines. MINER ENG 159:106636.

Zhou H., Niu X. (2020). An image processing algorithm for the measurement of multiphase bubbly flow using predictor-corrector method. INT J MULTIPHAS FLOW 128:103277.




How to Cite

Theodon, L., Eremina, T., Dia, K., Lamadie, F., Pinoli, J.-C., & Debayle, J. (2021). Estimating the Parameters of a Stochastic Geometrical Model for Multiphase Flow Images Using Local Measures. Image Analysis and Stereology, 40(3), 115–125.



Original Research Paper

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