Unlike wavelet, shearlets have the capability to detect directional discontinuities together with their directions. To achieve this, the considered scaling matrices have to be not only expanding, but also anisotropic. Shearlets allow for the definition of a directional multiple multiresolution analysis, where perfect reconstruction of the filterbank can be easily ensured by choosing an appropriate interpolatory multiple subdivision scheme. A drawback of shearlets is their relative large determinant that leads to a substantial complexity. The aim of the paper is to find scaling matrices in $\Z^{d \times d}$ which share the properties of shearlet matrices, i.e. anisotropic expanding matrices with the so-called slope resolution property, but with a smaller determinant. The proposed matrices provide a directional multiple multiresolution analysis whose behaviour is illustrated by some numerical tests on images.

Bozzini, M., Rossini, M., Sauer, T., Volontè, E. (2017). Anisotropic scaling matrices and subdivision schemes. IMA JOURNAL OF NUMERICAL ANALYSIS, 1-28.

### Anisotropic scaling matrices and subdivision schemes

#### Abstract

Unlike wavelet, shearlets have the capability to detect directional discontinuities together with their directions. To achieve this, the considered scaling matrices have to be not only expanding, but also anisotropic. Shearlets allow for the definition of a directional multiple multiresolution analysis, where perfect reconstruction of the filterbank can be easily ensured by choosing an appropriate interpolatory multiple subdivision scheme. A drawback of shearlets is their relative large determinant that leads to a substantial complexity. The aim of the paper is to find scaling matrices in $\Z^{d \times d}$ which share the properties of shearlet matrices, i.e. anisotropic expanding matrices with the so-called slope resolution property, but with a smaller determinant. The proposed matrices provide a directional multiple multiresolution analysis whose behaviour is illustrated by some numerical tests on images.
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Articolo in rivista - Articolo scientifico
Shearlets; subdivision schemes; multiple multiresolution analysis; filterbanks
English
2017
1
28
28
Submitted to IMA Journal of Numerica Analysis, under revision
Bozzini, M., Rossini, M., Sauer, T., Volontè, E. (2017). Anisotropic scaling matrices and subdivision schemes. IMA JOURNAL OF NUMERICAL ANALYSIS, 1-28.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/190382