Abstract
Subdivision schemes have been extensively developed since the eighties with very powerful applications for surface generation. To be implemented for compression, subdivision schemes have to be coupled with decimation operators sharing some consistency relation and with detail operators. The flexibility of subdivision schemes (they can be non-stationary, position or zone dependent, non-linear,…) makes that the construction of consistent decimation operators is a difficult task. In this paper, following the first results introduced in Kui et al. (On the coupling of decimation operator with subdivision schemes for multi-scale analysis. In: Lecture notes in computer science, vol. 10521. Springer, Berlin, pp. 162–185, 2016), we present the construction of multiresolution analyses connected to general subdivision schemes with detailed application to a non-interpolatory linear scheme called shifted Lagrange (Dyn et al., A C2 four-point subdivision scheme with fourth order accuracy and its extensions. In: Mathematical methods for curves and surfaces: Tromsø 2004. Citeseer, 2005) and its non-linear version called shifted PPH (Amat et al., Math. Comput. 80:959–959, 2011).
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Notes
- 1.
As it will become clear in the next section, this type of scheme is derived from a perturbation of a linear one. Therefore, a non-linear mask cannot be constructed but it is straightforward to extend the definition of a subdivision scheme provided by Expression (1).
- 2.
For any decimation operator \(\tilde h\) constructed from \((\tilde h_k)_{k\in \mathbb {Z}}\) and any integer t, we define \(T_t(\tilde h)\) the translated decimation operator related to the sequence \((\tilde h_{k-t})_{k\in \mathbb {Z}}\).
References
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Kui, Z., Baccou, J., Liandrat, J. (2019). Subdivision Schemes and Multiresolution Analyses: Focus on the Shifted Lagrange and Shifted PPH Schemes. In: García Guirao, J., Murillo Hernández, J., Periago Esparza, F. (eds) Recent Advances in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-00341-8_9
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