Anisotropic Diffusion in Anisotropic Stepanov Spaces

We consider a problem on the image processing and computer vision. A wide range of methods allows to solve problems of this type. The methods of partial differential equations are the most useful and interesting ones. A non-linear diffusion takes special place in these studies. In this context, fundamental theoretical foundation is a central part of this approach. Therefore, we introduce a new functional class of spaces, formulate and prove the lemma on the equivalent norms in anisotropic Stepanov spaces. Another important result of this study is the lemma that the anisotropic Stepanov spaces are Banach. In addition, we obtain the theorem on the solvability of the equation of anisotropic diffusion in anisotropic Stepanov spaces. The results can be applied to the image processing and computer vision. Also, the obtained results open the new view to this problem.

Keywords

diffusion; Nikol'skii spaces; anisotropic Stepanov spaces; anisotropic diffusion; differential equations.

References

1. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Springer, 2011.
2. Deimling K. Multivalued Differential Equations. Berlin, New York, Walter de Gruyter, 1992.
3. Perona P., Malik J. Scale-Space and Edge Detection Using Anisotropic Diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1990, vol. 12, no. 7, pp. 629-639.
4. Besov O.V., Il'in V.P., Nikol'skii S.M. Integral'nye predstavleniya funkciy i teoremy vlozheniya [Integral Representations of Functions and Embedding Theorems]. Moscow, Nauka, 1975. (in Russian)
5. Nikol'skii S.M. Priblizhenie funkciy mnogih peremennyh i teoremy vlozheniya [Approximation of Functions of Many Variables and Embedding Theorems]. Moscow, Nauka, 1977. (in Russian)
6. Kostin V.A., Kostin A.V. K teorii funkcional'nyh prostranstv Stepanova [To the Theory of Stepanov Function Spaces]. Voronezh, VSU Press, 2007. (in Russian)
7. Levitan B.M. Pochti-periodicheskie funkcii [Almost Periodic Functions]. Moscow, Gosudarstvennoe izdatel'stvo tekhniko-teoreticheskoy literatury, 1953. (in Russian)
8. Hille E., Phillips R.S. Functional Analysis and Semi-Groups. Providence, American Mathematical Society, 1957.