GMRA-BASED CONSTRUCTION OF FRAMELETS IN REDUCING SUBSPACES OF L2(ℝd)

Author(s):  
YUN-ZHANG LI ◽  
FENG-YING ZHOU

This paper develops GMRA-based construction procedures of Parseval framelets in the setting of reducing subspaces of L2(ℝd). A unitary extension principle is established; in particular, for a general expansive matrix A with | det A| = 2, an explicit construction of Parseval framelets is obtained. Some examples are also provided to illustrate the generality of our theory.

2014 ◽  
Vol 57 (2) ◽  
pp. 254-263 ◽  
Author(s):  
Ole Christensen ◽  
Hong Oh Kim ◽  
Rae Young Kim

AbstractThe unitary extension principle (UEP) by A. Ron and Z. Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.


2021 ◽  
Vol 17 (1) ◽  
pp. 79-94
Author(s):  
Hari Krishan Malhotra ◽  
◽  
Lalit Kumar Vashisht ◽  

2014 ◽  
Vol 977 ◽  
pp. 532-535
Author(s):  
Qing Jiang Chen ◽  
Yu Zhou Chai ◽  
Chuan Li Cai

Information science focuses on understanding problems from the perspective of the stake holders involved and then applying information and other technologies as needed. A necessary and sufficient condition is identified in term of refinement masks for applying the unitary extension principle for periodic functions to construct tight wavelet frames. Then a theory on the approxi-mation order of truncated tight frame series is established, which facilitates construction of tight periodic wavelet frames with desirable approximation order. The pyramid decomposition scheme is derived based on the generalized multiresolution structure.


2017 ◽  
Vol 9 (1) ◽  
pp. 248-259
Author(s):  
F. A. Shah ◽  
M. Y. Bhat

AbstractAn important tool for the construction of framelets on local fields of positive characteristic using unitary extension principle was presented by Shah and Debnath [Tight wavelet frames on local fields, Analysis, 33 (2013), 293-307]. In this article, we continue the study of framelets on local fields and present a polyphase matrix characterization of framelets generated by the extension principle.


2015 ◽  
Vol 85 (297) ◽  
pp. 239-270 ◽  
Author(s):  
Zhitao Fan ◽  
Hui Ji ◽  
Zuowei Shen

Author(s):  
Yan Zhang ◽  
Yun-Zhang Li

In wavelet analysis, refinable functions are the bases of extension principles for constructing (weak) dual wavelet frames for [Formula: see text] and its reducing subspaces. This paper addresses refinable function-based dual wavelet frames construction in Walsh reducing subspaces of [Formula: see text]. We obtain a Walsh–Fourier transform domain characterization for weak [Formula: see text]-adic nonhomogeneous dual wavelet frames; and present a mixed oblique extension principle for constructing weak [Formula: see text]-adic nonhomogeneous dual wavelet frames in Walsh reducing subspaces of [Formula: see text].


2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
H. Vosoughi ◽  
S. Abbasbandy

A numerical method along with explicit construction to interpolation of fuzzy data through the extension principle results by widely used fuzzy-valued piecewise Hermite polynomial in general case based on the cardinal basis functions, which satisfy a vanishing property on the successive intervals, has been introduced here. We have provided a numerical method in full detail using the linear space notions for calculating the presented method. In order to illustrate the method in computational examples, we take recourse to three prime cases: linear, cubic, and quintic.


Author(s):  
BYEONGSEON JEONG ◽  
MYUNGJIN CHOI ◽  
HONG OH KIM

This paper presents tight wavelet frames with two compactly supported symmetric generators of more than one vanishing moments in the Unitary Extension Principle. We determine all possible free tension parameters of the quasi-interpolatory subdivision masks whose corresponding refinable functions guarantee our wavelet frame. In order to reduce shift variance of the standard discrete wavelet transform, we use the three times oversampling filter bank and eventually obtain a ternary (low, middle, high) frequency scale. In applications to signal/image denoising and erasure recovery, the results demonstrate reduced shift variance and better performance of our wavelet frame than the usual wavelet systems such as Daubechies wavelets.


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