Revisiting Sphere Unfolding Relationships for the Stereological Analysis of Segmented Digital Microstructure Images

ABSTRACT

Sphere unfolding relationships are revisited with a specific focus on the analysis of segmented digital images of microstructures. Since the features of such images are most easily quantified by counting pixels, the required equations are re-derived in terms of the histogram of areas (instead of diameters or radii) as inputs and it is shown that a substitution can be made that simplifies the calculation. A practical method is presented for utilizing negative number fraction bins (which sometimes arise from erroneous assumptions and/or insufficient numbers of observations) for the creation of error bars. The complete algorithm can be implemented in a spreadsheet. The derived unfolding equations are explored using both linear and logarithmic binning schemes, and the pros and cons of both binning schemes are illustrated using simulated data. The effects of the binning schemes on the stereological results are demonstrated and discussed with reference to their consequences for practical materials characterization situations, allowing for the suggestion of guidelines for proper application of this, and other, distribution-free stereological methods.

Sphere unfolding relationships are revisited with a specific focus on the analysis of segmented digital images of microstructures. Since the features of such images are most easily quantified by counting pixels, the required equations are re-derived in terms of the histogram of areas (instead of diameters or radii) as inputs and it is shown that a substitution can be made that simplifies the calculation. A practical method is presented for utilizing negative number fraction bins (which sometimes arise from erroneous assumptions and/or insufficient numbers of observations) for the creation of error bars. The complete algorithm can be implemented in a spreadsheet. The derived unfolding equations are explored using both linear and logarithmic binning schemes, and the pros and cons of both binning schemes are illustrated using simulated data. The effects of the binning schemes on the stereological results are demonstrated and discussed with reference to their consequences for practical materials characterization situations, allowing for the suggestion of guidelines for proper application of this, and other, distribution-free stereological methods.

Cite this paper

E. Payton, "Revisiting Sphere Unfolding Relationships for the Stereological Analysis of Segmented Digital Microstructure Images,"*Journal of Minerals and Materials Characterization and Engineering*, Vol. 11 No. 3, 2012, pp. 221-242. doi: 10.4236/jmmce.2012.113018.

E. Payton, "Revisiting Sphere Unfolding Relationships for the Stereological Analysis of Segmented Digital Microstructure Images,"

References

[1] Abbruzzese G., 1985, “Computer simulated grain growth stagnation.” Acta Metall., Vol. 33, pp. 1329–1337.

[2] Dieter, G.E., 1986, Mechanical Metallurgy, 3rd ed., McGraw-Hill Series in Materials Science and Engineering, McGraw-Hill, Boston, MA.

[3] Porter, D.A. and Easterling, K. E., 1992, Phase Transformations in Metals and Alloys, 2nd ed., CRC press, Boca Raton, FL.

[4] Payton, E.J., 2009, Characterization and Modeling of Grain Coarsening in Powder Metallurgical Nickel-based Superalloys, Ph.D. thesis, The Ohio State University, Columbus, OH.

[5] Wang, G., Xu, D.S., Payton, E.J., Ma, N., Yang, R., Mills, M.J., and Wang, Y., 2011, “Mean-field statistical simulation of grain coarsening in the presence of stable and unstable pinning particles.” Acta Mater., Vol. 59, pp. 4587-4594.

[6] Kahn, H., Mano, E.S., and Tassinari, M.M., 2002, “Image analysis coupled with a SEM-EDS applied to the characterization of a Zn-Pb partially weathered ore.” J. Miner. Mater. Charact. Eng., Vol. 1, pp. 1-9.

[7] Gu, Y., 2003, “Automated scanning electron microscope based mineral liberation analysis: An introduction to JKMRC/FEI mineral liberation analyzer.” J. Miner. Mater. Charact. Eng., Vol. 2, pp. 33-41.

[8] Hagni, A.M., 2008, “Phase identification, phase quantification, and phase association determinations utilizing automated mineralogy technology.” JOM, Vol. 60, pp. 33-37.

[9] Payton, E.J., Phillips, P.J., and Mills, M.J., 2010, “Semi-automated characterization of the gamma prime phase in Ni-based superalloys via high-resolution backscatter imaging.” Mater. Sci. Eng. A, Vol. 527, pp. 2684–2692.

[10] Schouwstra, R.P. and Smit, A.J., 2011, “Developments in mineralogical techniques – What about mineralogists?” Miner. Eng., Vol. 24, pp. 1224-1228.

[11] Payton, E.J., Wang, G., Wang, G., Ma, N., Wang, Y., Mills, M.J., Whitis, D.D., Mourer, D.P., and Wei, D., 2008, “Integration of simulations and experiments for modeling superalloy grain growth,” in: Superalloys 2008, pp. 975-985, The Minerals, Metals & Materials Society, Warrendale, PA.

[12] Wang, G., Xu, D.S., Ma, N., Zhou, N., Payton, E.J., Yang, R., Mills, M.J., and Wang, Y., 2009, “Simulation study of effects of initial particle size distribution on dissolution.” Acta Mater., Vol. 57, pp. 316-325.

[13] Hilliard, J.E. and Cahn, J.W., 1961, “An evaluation of procedures in quantitative metallography for volume-fraction analysis.” Trans. AIME, Vol. 221, pp. 344-352.

[14] Underwood, E.E., 1968, “Particle-size Distribution,” in: Quantitative Microscopy, pp. 149–200, McGraw-Hill Book Company, New York, NY.

[15] Underwood, E.E., 1970, Quantitative Stereology, 2nd ed., Addison-Wesley Publishing Company, Reading, MA.

[16] Cruz-Orive, L.M., 1976, “Particle size-shape distributions: the general spheroid problem.” J. Microsc., Vol. 107, pp. 235–253.

[17] Cruz-Orive, L.M., 1978, “Particle size-shape distributions: the general spheroid problem: II. Stochastic model and practical guide.” J. Microsc., Vol. 112, pp. 153–167.

[18] Cruz-Orive, L.M., 1983, “Distribution-free estimation of sphere size distributions from slabs showing overprojection and truncation, with a review of previous methods.” J. Microsc., Vol. 131, pp. 265–290.

[19] Takahashi, J. and Suito, H., 2003, “Evaluation of the accuracy of the three-dimensional size distribution estimated from the Schwartz-Saltikov method.” Metall. Mater. Trans. A, Vol. 34, pp. 171-181.

[20] Payton, E.J. and Mills, M. J., 2011, “Stereology of backscatter electron images of etched surfaces for characterization of particle size distributions and volume fractions: Estimation of imaging bias via Monte Carlo simulations.” Mater. Charact., Vol. 62, pp. 563-574.

[21] Jeppsson, J., Mannesson, K., Borgenstam, A., and ?gren, J., 2011, “Inverse Saltikov analysis for particle-size distributions and their time evolution.” Acta Mater., Vol. 59, pp. 874-882.

[22] Ohser, J. and Mücklich, F., 2000, Statistical analysis of microstructures in materials science, John Wiley & Sons, New York, NY.

[23] Ohser, J. and Mucklich, F., 1995, “Stereology for some classes of polyhedrons.” Adv. Appl. Prob., Vol. 27, pp. 384-396.

[24] Scott, D.W., 1979, “On optimal and data-based histograms.” Biometrika, Vol. 66, pp. 605-610.

[25] Saltikov, S.A.: “The determination of the size distribution of particles in an opaque material from a measurement of the size distribution of their sections,” in: Stereology: Proceedings of the Second International Congress for Stereology, Chicago, 1967, pp. 163–173.

[26] Heilbronner, R. and Bruhn D., 1998, “The influence of three-dimensional grain size distributions on the rheology of polyphase rocks.” J. Struct. Geol., Vol. 20, pp. 695–705.

[27] Heilbronner, R., 2002, “How to derive size distributions of particles from size distributions of sectional areas,” Conférence Universitaire de Suisse Occidentale, 3ème Cycle Séminaire: “Analyse d’images et morphométrie d’objets géologiques,” Organisé à Neuchatel, Institut de Géologie, Université de Neuchatel.

[28] Ford, W.B., 1922, A Brief Course in College Algebra, The Macmillan Company, New York, NY.

[29] Limpert, E., Stahel, W.A., and Abbt, M., 2001, “Log-normal Distributions across the Sciences: Keys and Clues.” BioScience, Vol. 51, pp. 341-352.

[30] Russ, J.C. and Dehoff, R.T., 2000, Practical Stereology, Kluwer Academic, New York, NY.

[1] Abbruzzese G., 1985, “Computer simulated grain growth stagnation.” Acta Metall., Vol. 33, pp. 1329–1337.

[2] Dieter, G.E., 1986, Mechanical Metallurgy, 3rd ed., McGraw-Hill Series in Materials Science and Engineering, McGraw-Hill, Boston, MA.

[3] Porter, D.A. and Easterling, K. E., 1992, Phase Transformations in Metals and Alloys, 2nd ed., CRC press, Boca Raton, FL.

[4] Payton, E.J., 2009, Characterization and Modeling of Grain Coarsening in Powder Metallurgical Nickel-based Superalloys, Ph.D. thesis, The Ohio State University, Columbus, OH.

[5] Wang, G., Xu, D.S., Payton, E.J., Ma, N., Yang, R., Mills, M.J., and Wang, Y., 2011, “Mean-field statistical simulation of grain coarsening in the presence of stable and unstable pinning particles.” Acta Mater., Vol. 59, pp. 4587-4594.

[6] Kahn, H., Mano, E.S., and Tassinari, M.M., 2002, “Image analysis coupled with a SEM-EDS applied to the characterization of a Zn-Pb partially weathered ore.” J. Miner. Mater. Charact. Eng., Vol. 1, pp. 1-9.

[7] Gu, Y., 2003, “Automated scanning electron microscope based mineral liberation analysis: An introduction to JKMRC/FEI mineral liberation analyzer.” J. Miner. Mater. Charact. Eng., Vol. 2, pp. 33-41.

[8] Hagni, A.M., 2008, “Phase identification, phase quantification, and phase association determinations utilizing automated mineralogy technology.” JOM, Vol. 60, pp. 33-37.

[9] Payton, E.J., Phillips, P.J., and Mills, M.J., 2010, “Semi-automated characterization of the gamma prime phase in Ni-based superalloys via high-resolution backscatter imaging.” Mater. Sci. Eng. A, Vol. 527, pp. 2684–2692.

[10] Schouwstra, R.P. and Smit, A.J., 2011, “Developments in mineralogical techniques – What about mineralogists?” Miner. Eng., Vol. 24, pp. 1224-1228.

[11] Payton, E.J., Wang, G., Wang, G., Ma, N., Wang, Y., Mills, M.J., Whitis, D.D., Mourer, D.P., and Wei, D., 2008, “Integration of simulations and experiments for modeling superalloy grain growth,” in: Superalloys 2008, pp. 975-985, The Minerals, Metals & Materials Society, Warrendale, PA.

[12] Wang, G., Xu, D.S., Ma, N., Zhou, N., Payton, E.J., Yang, R., Mills, M.J., and Wang, Y., 2009, “Simulation study of effects of initial particle size distribution on dissolution.” Acta Mater., Vol. 57, pp. 316-325.

[13] Hilliard, J.E. and Cahn, J.W., 1961, “An evaluation of procedures in quantitative metallography for volume-fraction analysis.” Trans. AIME, Vol. 221, pp. 344-352.

[14] Underwood, E.E., 1968, “Particle-size Distribution,” in: Quantitative Microscopy, pp. 149–200, McGraw-Hill Book Company, New York, NY.

[15] Underwood, E.E., 1970, Quantitative Stereology, 2nd ed., Addison-Wesley Publishing Company, Reading, MA.

[16] Cruz-Orive, L.M., 1976, “Particle size-shape distributions: the general spheroid problem.” J. Microsc., Vol. 107, pp. 235–253.

[17] Cruz-Orive, L.M., 1978, “Particle size-shape distributions: the general spheroid problem: II. Stochastic model and practical guide.” J. Microsc., Vol. 112, pp. 153–167.

[18] Cruz-Orive, L.M., 1983, “Distribution-free estimation of sphere size distributions from slabs showing overprojection and truncation, with a review of previous methods.” J. Microsc., Vol. 131, pp. 265–290.

[19] Takahashi, J. and Suito, H., 2003, “Evaluation of the accuracy of the three-dimensional size distribution estimated from the Schwartz-Saltikov method.” Metall. Mater. Trans. A, Vol. 34, pp. 171-181.

[20] Payton, E.J. and Mills, M. J., 2011, “Stereology of backscatter electron images of etched surfaces for characterization of particle size distributions and volume fractions: Estimation of imaging bias via Monte Carlo simulations.” Mater. Charact., Vol. 62, pp. 563-574.

[21] Jeppsson, J., Mannesson, K., Borgenstam, A., and ?gren, J., 2011, “Inverse Saltikov analysis for particle-size distributions and their time evolution.” Acta Mater., Vol. 59, pp. 874-882.

[22] Ohser, J. and Mücklich, F., 2000, Statistical analysis of microstructures in materials science, John Wiley & Sons, New York, NY.

[23] Ohser, J. and Mucklich, F., 1995, “Stereology for some classes of polyhedrons.” Adv. Appl. Prob., Vol. 27, pp. 384-396.

[24] Scott, D.W., 1979, “On optimal and data-based histograms.” Biometrika, Vol. 66, pp. 605-610.

[25] Saltikov, S.A.: “The determination of the size distribution of particles in an opaque material from a measurement of the size distribution of their sections,” in: Stereology: Proceedings of the Second International Congress for Stereology, Chicago, 1967, pp. 163–173.

[26] Heilbronner, R. and Bruhn D., 1998, “The influence of three-dimensional grain size distributions on the rheology of polyphase rocks.” J. Struct. Geol., Vol. 20, pp. 695–705.

[27] Heilbronner, R., 2002, “How to derive size distributions of particles from size distributions of sectional areas,” Conférence Universitaire de Suisse Occidentale, 3ème Cycle Séminaire: “Analyse d’images et morphométrie d’objets géologiques,” Organisé à Neuchatel, Institut de Géologie, Université de Neuchatel.

[28] Ford, W.B., 1922, A Brief Course in College Algebra, The Macmillan Company, New York, NY.

[29] Limpert, E., Stahel, W.A., and Abbt, M., 2001, “Log-normal Distributions across the Sciences: Keys and Clues.” BioScience, Vol. 51, pp. 341-352.

[30] Russ, J.C. and Dehoff, R.T., 2000, Practical Stereology, Kluwer Academic, New York, NY.