Abstract
The effect of the kernel on the smoothing of orientations in a kernel method was studied, and the influence of dependent orientations and the grain sizes on the resulting distribution was analyzed. Discrete central normal distributions on the group SO(3) were smoothed by the kernel method. This problem is motivated by the development of experimental tools for studying the texture of polycrystalline materials, especially electron microscopy, which makes it possible to measure the orientations of individual grains.
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References
H. J. Bunge, Texture Analysis in Material Science: Mathematical Methods (Butterworths, London, 1982).
T. I. Savyolova and T. M. Ivanova, “Overview of Methods for Recovering the Orientation Distribution Function from Pole Figures,” Zavod. Lab. 78(7), 25–33 (2008).
A. A. Borovkov, Mathematical Statistics (Nauka, Moscow, 1984) [in Russian].
L. Devroye and L. Gyorfi, Nonparametric Density Estimation: The L1 View (Wiley, New York, 1985; Mir, Moscow, 1988).
N. N. Chentsov, Statistical Decision Rules and Optimal Inference (Nauka, Moscow, 1972; Am. Math. Soc., Providence, R.I., 1982).
A. V. Kryanev and G. V. Lukin, Mathematical Methods for Stochastic Data Processing (FNL, Moscow, 2003) [in Russian].
E. Guilmeau, C. Henrist, T. S. Suzuki, et al., “Texture of Alumina by Neutron Diffraction and SEM-EBSD” ICOTOM 14, 1395–1400 (2005).
K. G. Boogaart, Statistic for Individual Crystallographic Orientation Measurement (Shaker, Aachen, 2002), pp. 1–161.
K. G. Boogart, “Statistical Errors of Texture Entities Based on EBSD Orientation Measurements,” ICOTOM 14, 179–184 (2005).
N. Bozzolo, F. Gerspach, G. Sawina, and F. Wagner, “Accuracy of Orientation Distribution Function Determination Based on EBSD Data: A Case Study of a Recrystallized Low Alloyed Zr Sheet,” J. Microscopy 227, 245–283 (2007).
S. Prazolo, V. G. Sursaeva, and D. J. Prior, “Optical Grain Size Measurements: What Is Being Measured? Comparative Study of Optical and EBSD Grain Size Determination in 2D Al Foil,” ICOTOM 14, 213–218 (2005).
M. V. Borovkov, T. I. Savyolova, and V. N. Serebryanyi, “Analysis of Statistical Errors in Roentgen Texture Experiment Measuring Pole Figures Using the Monte Carlo Method,” Zavod. Lab. 21(12), 19–24 (2005).
T. I. Savyolova and E. F. Koren’kova, “Estimation of Accuracy of Some Statistical Characteristics in Texture Analysis,” Zavod. Lab. 72(12), 29–34 (2006).
T. I. Savyolova and M. V. Sypchenko, “Calculation of the Orientation Distribution Function from a Set of Individual Orientations on SO(3)|” Zh. Vychisl. Mat. Mat. Fiz. 47, 1015–1028 (2007) [Comput. Math. Math. Phys. 47, 970–982 (2007)].
T. I. Savyolova and M. V. Sypchenko, “Estimation of Accuracy of Kernel and Projective Methods of Probability Density Distribution Restoration from Individual Orientations on Group SO(3),” 3rd French-Russian Seminar on New Achievement in Materials and Environmental Sciences (Metz, 2007), p. 103.
K. P. Aganin and T. I. Savyolova, “Error Estimates for Kernel and Projection Methods of Recovering the Orientation Distribution Function on SO(3),” Zh. Vychisl. Mat. Mat. Fiz. 28, 1087–1102 (2008) [Comput. Math. Math. Phys. 28, 1024–1038 (2008)].
T. I. Savyolova, “Orientation Distribution Function of Grains and Their Gaussian Approximations,” Zavod. Lab. 50(5), 48–52 (1984).
M. V. Borovkov and T. I. Savyolova, “Computation of Normal Distributions on Rotation Groups by the Monte Carlo Method,” Zh. Vychisl. Mat. Mat. Fiz. 42, 112–128 (2002) [Comput. Math. Math. Phys. 42, 108–124 (2002)].
M. Borovkov and T. Savyolova, “The Computational Approaches to Calculate Normal Distributions on the Rotation Group,” J. Appl. Crystalogr. 40, 449–455 (2007).
H. Schaeben, ““Normal” Orientation Distribution,” Textures Microstructures 29, 201–233 (1997).
H. Schaeben and K. G. Boogaart, “Spherical Harmonics in Texture Analysis,” Tectonophysics 370, 253–268 (2003).
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Original Russian Text © T.I. Savyolova, M.V. Sypchenko, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, Vol. 49, No. 5, pp. 879–890.
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Savyolova, T.I., Sypchenko, M.V. Error estimation of grain distribution function recovery for dependent orientations with allowance for grain sizes. Comput. Math. and Math. Phys. 49, 846–856 (2009). https://doi.org/10.1134/S0965542509050108
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DOI: https://doi.org/10.1134/S0965542509050108