Geodesic
Calculation of the orientation distribution function from a set of individual orientations on $\mathrm{SO}(3)$
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 6, pp. 1015-1028 
T. I. Savyolova; M. V. Sypchenko. Calculation of the orientation distribution function from a set of individual orientations on $\mathrm{SO}(3)$. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 6, pp. 1015-1028. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_6_a7/
@article{ZVMMF_2007_47_6_a7,
     author = {T. I. Savyolova and M. V. Sypchenko},
     title = {Calculation of the orientation distribution function from a set of individual orientations on $\mathrm{SO}(3)$},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1015--1028},
     year = {2007},
     volume = {47},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_6_a7/}
}
TY  - JOUR
AU  - T. I. Savyolova
AU  - M. V. Sypchenko
TI  - Calculation of the orientation distribution function from a set of individual orientations on $\mathrm{SO}(3)$
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2007
SP  - 1015
EP  - 1028
VL  - 47
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_6_a7/
LA  - ru
ID  - ZVMMF_2007_47_6_a7
ER  - 
%0 Journal Article
%A T. I. Savyolova
%A M. V. Sypchenko
%T Calculation of the orientation distribution function from a set of individual orientations on $\mathrm{SO}(3)$
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2007
%P 1015-1028
%V 47
%N 6
%U http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_6_a7/
%G ru
%F ZVMMF_2007_47_6_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

Kernel and projection methods for recovering the density function on the rotation group $\mathrm{SO}(3)$ are considered. Numerical examples are presented in which the density function is estimated depending on the sample size, a smoothing parameter (in the case of kernel methods), the approximation kernel, and the error in the input data. A set of orientations is specified by normally distributed rotations on $\mathrm{SO}(3)$ derived by the Monte Carlo method.

[1] Guilmeau E., Henrist C., Suzuku T. S. et al., “Texture of alumina by neutron diffraction and SEM-EBSD”, Textures of Materials – ICOTOM 14, Material Sci. Forum, 495–497, 2005, 1395–1400

[2] Bunge H. J., Texture analysis in material science. Mathematical Methods, Butterworths, London, 1982

[3] Matthies S., Vinel G. W., Helming K., Standard distributions in texture anatysis, v. I–III, Akad. Verl., Berlin, 1987–1990 | MR

[4] Schaeben H., “A unifield view of methods to resolve the inverse problem of texture goniometry”, Textures and Microstructures, 25:2–4 (1996), 171–181

[5] Savelova T. I., “Funktsiya raspredeleniya zeren po orientatsiyam i ikh gaussovskie priblizheniya”, Zavodskaya laboratoriya, 50:5 (1984), 48–52

[6] Nikolaev D. I., Savelova T. I., “Approksimatsiya resheniya obratnoi zadachi difraktsii $\delta$-funktsiyami i gaussovskimi raspredeleniyami”, Zh. vychisl. matem. i matem. fiz., 27:5 (1987), 791–793

[7] Shaeben H., “Texture approximation or texture modeling with components represented by the von Mises–Fisher matrix distribution on $SO(3)$ and the Bingham distribution of $S_4$”, J. Appl. Crystallogr., 29 (1996), 516–525 | DOI

[8] Nikolaxev D. I., Savyolova T. I., “Normal distribution on the rotation group $SO(3)$”, Textures and Microstructures, 29 (1997), 201–233 | DOI

[9] Borovkov M. V., Savelova T. I., “Vychislenie normalnykh raspredelenii na gruppe vraschenii metodom Monte-Karlo”, Zh. vychisl. matem. i matem. fiz., 42:1 (2002), 112–128 | MR | Zbl

[10] Borovkov M. V., Savelova T. I., Serebryanyi V. N., “Issledovanie statisticheskikh oshibok rentgenovskogo teksturnogo eksperimenta po izmereniyu polyusnykh figur s ispolzovaniem metoda Monte-Karlo”, Zavodskaya laboratoriya, 21:12 (2005), 19–24

[11] Boogart K. G., “Statistical errors of texture entities based on EBSD orientation measurements”, Textures of Materials – ICOTOM 14, Material Sci. Forum, 495–497, 2005, 179–184

[12] Borovkov A. A., Matematicheskaya statistika, Nauka, M., 1984 | MR

[13] Devroi L., Derfi L., Neparametricheskoe otsenivanie plotnosti, Mir, M., 1988 | MR

[14] Chentsov H. H., Statisticheskie reshayuschie pravila i optimalnye vyvody, Nauka, M., 1972 | Zbl

[15] Kryanev A. V., Lukin G. V., Matematicheskie metody obrabotki neopredelennykh dannykh, Fizmatlit, M., 2003

[16] Vilenkin N. Ya., Spetsialnye funktsii i teoriya predstavleniya grupp, Nauka, M., 1965 | MR

[17] Khyuber P., Robastnost v statistike, Mir, M., 1984 | MR

[18] Savelova T. I., Korenkova E. F., “Otsenka tochnosti nekotorykh statisticheskikh kharakteristik v teksturnom analize”, Zavodskaya laboratoriya, 72:12 (2006), 29–34