Abstract
Methods from the representation theory of finite groups are used to construct efficient processing methods for the special geometries related to the finite subgroups of the rotation group. We motivate the use of these subgroups in computer vision, summarize the necessary facts from the representation theory and develop the basics of Fourier theory for these geometries. We illustrate its usage for data compression in applications where the processes are (on average) symmetrical with respect to these groups. We use the icosahedral group as an example since it is the largest finite subgroup of the 3D rotation group. Other subgroups with fewer group elements can be studied in exactly the same way.
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References
Debevec, P., Hawkins, T., Tchou, C., Duiker, H.P., Sarokin, W., Sagar, M.: Acquiring the reflectance field of a human face. In: Proc. SIGGRAPH 2000, pp. 145–156. ACM Press/Addison-Wesley Publishing Co., New York, NY, USA (2000)
Edström, P.: A fast and stable solution method for the radiative transfer problem. Siam Review 47(3), 447–468 (2005)
Weyrich, T., Matusik, W., Pfister, H., Bickel, B., Donner, C., Tu, C., McAndless, J., Lee, J., Ngan, A., Jensen, H.W., Gross, M.: Analysis of human faces using a measurement-based skin reflectance model. Acm Transactions On Graphics 25(3), 1013–1024 (2006)
Serre, J.P.: Linear representations of finite groups. Springer, Heidelberg (1977)
Stiefel, E., Fässler, A.: Gruppentheoretische Methoden und ihre Anwendungen. Teubner, Stuttgart (1979)
Sternberg, S.: Group Theory and Physics. First paperback (edn.) Cambridge University Press, Cambridge, England (1995)
Kim, S.K.: Group theoretical methods and applications to molecules and crystals. Cambridge University Press, Cambridge (1999)
Henyey, L., Greenstein, J.: Diffuse radiation in the galaxy. Astrophys. Journal 93, 70–83 (1941)
Schechner, Y., Nayar, S., Belhumeur, P.: A theory of multiplexed illumination. In: Proc. Ninth IEEE Int. Conf. on Computer Vision, vol. 2, pp. 808–815. IEEE Computer Society Press, Los Alamitos (2003)
Ratner, N., Schechner, Y.Y.: Illumination multiplexing within fundamental limits. In: Conference on Computer Vision and Pattern Recognition, pp. 1–8. IEEE, Los Alamitos (2007)
Schechner, Y.Y., Nayar, S.K., Belhumeur, P.N.: Multiplexing for optimal lighting. IEEE Trans. Pattern Analysis and Machine Intelligence 29(8), 1339–1354 (2007)
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Lenz, R. (2007). Crystal Vision-Applications of Point Groups in Computer Vision. In: Yagi, Y., Kang, S.B., Kweon, I.S., Zha, H. (eds) Computer Vision – ACCV 2007. ACCV 2007. Lecture Notes in Computer Science, vol 4844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76390-1_73
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DOI: https://doi.org/10.1007/978-3-540-76390-1_73
Publisher Name: Springer, Berlin, Heidelberg
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