Interpolatory Dual Framelets with a General Dilation Matrix
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929580886982656 |
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| author | Lu, Ran |
| author_facet | Lu, Ran |
| contents | Interpolatory filters are of great interest in subdivision schemes and wavelet analysis. Due to the high-order linear-phase moment property, interpolatory refinement filters are often used to construct wavelets and framelets with high-order vanishing moments. In this paper, given a general dilation matrix $\mathsf{M}$, we propose a method that allows us to construct a dual $\mathsf{M}$-framelet from an arbitrary pair of $\mathsf{M}$-interpolatory filters such that all framelet generators/high-pass filters (1) have the interpolatory properties; (2) have high-order vanishing moments. Our method is easy to implement, as the high-pass filters are either given in explicit formulas or can be obtained by solving specific linear systems. Motivated by constructing interpolatory dual framelets, we can further deduce a method to construct an interpolatory quasi-tight framelet from an arbitrary interpolatory filter. If, in addition, the refinement filters have symmetry, we will perform a detailed analysis of the symmetry properties that the high-pass filters can achieve. We will present several examples to demonstrate our theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04485 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interpolatory Dual Framelets with a General Dilation Matrix Lu, Ran Functional Analysis 42C40, 42C15, 41A15, 65D07 Interpolatory filters are of great interest in subdivision schemes and wavelet analysis. Due to the high-order linear-phase moment property, interpolatory refinement filters are often used to construct wavelets and framelets with high-order vanishing moments. In this paper, given a general dilation matrix $\mathsf{M}$, we propose a method that allows us to construct a dual $\mathsf{M}$-framelet from an arbitrary pair of $\mathsf{M}$-interpolatory filters such that all framelet generators/high-pass filters (1) have the interpolatory properties; (2) have high-order vanishing moments. Our method is easy to implement, as the high-pass filters are either given in explicit formulas or can be obtained by solving specific linear systems. Motivated by constructing interpolatory dual framelets, we can further deduce a method to construct an interpolatory quasi-tight framelet from an arbitrary interpolatory filter. If, in addition, the refinement filters have symmetry, we will perform a detailed analysis of the symmetry properties that the high-pass filters can achieve. We will present several examples to demonstrate our theoretical results. |
| title | Interpolatory Dual Framelets with a General Dilation Matrix |
| topic | Functional Analysis 42C40, 42C15, 41A15, 65D07 |
| url | https://arxiv.org/abs/2411.04485 |