CONSTRUCTION OF SYMMETRIC TIGHT WAVELET FRAMES FROM QUASI-INTERPOLATORY SUBDIVISION MASKS AND THEIR APPLICATIONS

2008 ◽  
Vol 06 (01) ◽  
pp. 97-120 ◽  
Author(s):  
BYEONGSEON JEONG ◽  
MYUNGJIN CHOI ◽  
HONG OH KIM
Keyword(s):  
Filter Bank ◽  
Extension Principle ◽  
Discrete Wavelet ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Vanishing Moments ◽  
Frequency Scale ◽  
Unitary Extension ◽  
Signal Image ◽  
Interpolatory Subdivision

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.

On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle

2014 ◽  
Vol 57 (2) ◽  
pp. 254-263 ◽  
Author(s):  
Ole Christensen ◽  
Hong Oh Kim ◽  
Rae Young Kim
Keyword(s):  
Extension Principle ◽  
Sufficient Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Wavelet System ◽  
Number Of Generators ◽  
Unitary Extension

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.

scholarly journals Unitary Extension Principle for Nonuniform Wavelet Frames in L2(ℝ)

2021 ◽  
Vol 17 (1) ◽  
pp. 79-94
Author(s):  
Hari Krishan Malhotra ◽  
◽  
Lalit Kumar Vashisht ◽  
Keyword(s):  
Extension Principle ◽  
Wavelet Frames ◽  
Unitary Extension

The Study of Tight Periodic Wavelet Frames and Wavelet Frame Packets and Applications

2014 ◽  
Vol 977 ◽  
pp. 532-535
Author(s):  
Qing Jiang Chen ◽  
Yu Zhou Chai ◽  
Chuan Li Cai
Keyword(s):  
Information Science ◽  
Approximation Order ◽  
Tight Frame ◽  
Extension Principle ◽  
Wavelet Frames ◽  
Decomposition Scheme ◽  
Periodic Wavelet ◽  
Unitary Extension ◽  
Periodic Wavelet Frames ◽  
Necessary And Sufficient

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.

scholarly journals Polyphase matrix characterization of framelets on local fields of positive characteristic

2017 ◽  
Vol 9 (1) ◽  
pp. 248-259
Author(s):  
F. A. Shah ◽  
M. Y. Bhat
Keyword(s):  
Positive Characteristic ◽  
Local Fields ◽  
Extension Principle ◽  
Wavelet Frames ◽  
Unitary Extension ◽  
Polyphase Matrix

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.

A PAIR OF ORTHOGONAL WAVELET FRAMES IN L2(ℝd)

2014 ◽  
Vol 12 (02) ◽  
pp. 1450011 ◽  
Author(s):  
GHANSHYAM BHATT
Keyword(s):  
Dimensional Case ◽  
Extension Principle ◽  
Higher Dimension ◽  
Wavelet Frames ◽  
Orthogonal Wavelet ◽  
Extra Condition ◽  
Simple Method ◽  
One Dimensional ◽  
Unitary Extension ◽  
Polyphase Matrix

A simple method of construction of a pair of orthogonal wavelet frames in L2(ℝd) is presented. This is a generalization of one-dimensional case to higher dimension. The construction is based on the well-known Unitary Extension Principle (UEP). The presented method produces the polyphase components of the filters of the wavelet functions, and hence the filters. A pair of orthogonal wavelet frames can be constructed with an extra condition. In the construction, the polyphase matrix is used as opposed to the modulation matrix. This is less restrictive and yields a fewer wavelet functions in the system than in the previously known constructions.

scholarly journals Construction of nonuniform periodic wavelet frames on non-Archimedean fields

2020 ◽  
Vol 74 (2) ◽  
pp. 1
Author(s):  
Owais Ahmad
Keyword(s):  
Real Life ◽  
Discrete Subgroup ◽  
Extension Principle ◽  
Mathematical Tool ◽  
Wavelet Frames ◽  
Spectral Pair ◽  
Periodic Wavelet ◽  
Spectral Pairs ◽  
Unitary Extension ◽  
Periodic Wavelet Frames

In real life applications not all signals are obtained by uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. Gabardo and Nashed, and Gabardo and Yu filled this gap by the concept of nonuniform multiresolution analysis and nonuniform wavelets based on the theory of spectral pairs for which the associated translation set \(\Lambda= \{0,r/N\}+2\mathbb{Z}\) is no longer a discrete subgroup of \(\mathbb{R}\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we introduce a notion of nonuniform periodic wavelet frame on non-Archimedean field. Using the Fourier transform technique and the unitary extension principle, we propose an approach for the construction of nonuniform periodic wavelet frames on non-Archimedean fields.

TIGHT FRAME CHARACTERIZATION OF MULTIWAVELET VECTOR FUNCTIONS IN TERMS OF THE POLYPHASE MATRIX

2009 ◽  
Vol 07 (01) ◽  
pp. 9-21 ◽  
Author(s):  
JENS KROMMWEH
Keyword(s):  
Tight Frame ◽  
Extension Principle ◽  
Wavelet Frames ◽  
Unitary Extension ◽  
Directional Wavelets ◽  
Extension Principles ◽  
Polyphase Matrix ◽  
Common Extension ◽  
Polyphase Representation

The extension principles play an important role in characterizing and constructing of wavelet frames. The common extension principles, the unitary extension principle (UEP) or the oblique extension principle (OEP), are based on the unitarity of the modulation matrix. In this paper, we state the UEP and OEP for refinable function vectors in the polyphase representation. Finally, we apply our results to directional wavelets on triangles which we have constructed in a previous work. We will show that the wavelet system generates a tight frame for L2(ℝ2).

scholarly journals Geological Stratigraphy and Spatial Distribution of Microfractures over Costa Rica Convergent Margin, Central America – A Wavelet-Fractal Analysis

2016 ◽  
Author(s):  
Upendra K. Singh ◽  
Thinesh Kumar ◽  
Rahul Prajapati
Keyword(s):  
Fractal Dimension ◽  
Wavelet Transform ◽  
Costa Rica ◽  
Central America ◽  
Gamma Ray ◽  
Total Porosity ◽  
Discrete Wavelet ◽  
Mathematical Tool ◽  
Convergent Margin ◽  
Frequency Scale

Abstract. Identification of spatial variation of lithology, as a function of position and scale, is very critical job for lithology modelling in industry. Wavelet Transform (WT) is an efficacious and powerful mathematical tool for time (position) and frequency (scale) localization. It has numerous advantages over Fourier Transform (FT) to obtain frequency and time information of a signal. Initially Continuous Wavelet Transform (CWT) is applied on gamma ray logs of two different Well sites (Well-1039 & Well-1043) of Costa Rica Convergent Margin, Central America for identifications of lithofacies distribution and fracture zone later Discrete Wavelet Transform (DWT) applied to DPHI log signals to show its efficiency in discriminating small changes along the rock matrix irrespective of the instantaneous magnitude to represent the fracture contribution from the total porosity recorded. Further the data of the appropriate depths partitioned using above mathematical tools are utilized separately for WBFA. As consequences of CWT operation it is found that there are four major sedimentary layers terminated with a concordant igneous intrusion passing through both the wells. In addition of WBFA analysis, it is clearly understanding that the fractal dimension value is persistent in first sedimentary layers and the last gabbroic sill intrusions. Inconsistent value of fractal dimension is attributed to fracture dominant in intermediate sedimentary layers it is also validate through core analysis. Fractal Dimension values suggest that the sedimentary environments persisting in that well locations bears abundant shale content and of low energy environments.

scholarly journals Discrete and Dual Tree Wavelet Features for Real-Time Speech/Music Discrimination

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Timur Düzenli ◽  
Nalan Özkurt
Keyword(s):  
Wavelet Transform ◽  
Real Time ◽  
Wavelet Transforms ◽  
Principal Component ◽  
Discrete Wavelet ◽  
Vanishing Moments ◽  
Zero Crossings ◽  
Time Frequency ◽  
Common Time ◽  
Classification Tool

The performance of wavelet transform-based features for the speech/music discrimination task has been investigated. In order to extract wavelet domain features, discrete and complex orthogonal wavelet transforms have been used. The performance of the proposed feature set has been compared with a feature set constructed from the most common time, frequency and cepstral domain features such as number of zero crossings, spectral centroid, spectral flux, and Mel cepstral coefficients. The artificial neural networks have been used as classification tool. The principal component analysis has been applied to eliminate the correlated features before the classification stage. For discrete wavelet transform, considering the number of vanishing moments and orthogonality, the best performance is obtained with Daubechies8 wavelet among the other members of the Daubechies family. The dual tree wavelet transform has also demonstrated a successful performance both in terms of accuracy and time consumption. Finally, a real-time discrimination system has been implemented using the Daubhecies8 wavelet which has the best accuracy.

scholarly journals Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Fengjuan Zhu ◽  
Qiufu Li ◽  
Yongdong Huang
Keyword(s):  
Scaling Function ◽  
Minimum Energy ◽  
Sufficient Conditions ◽  
Reconstruction Algorithm ◽  
Wavelet Function ◽  
Necessary Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Numerical Examples ◽  
Dilation Matrix

In order to characterize the bivariate signals, minimum-energy bivariate wavelet frames with arbitrary dilation matrix are studied, which are based on superiority of the minimum-energy frame and the significant properties of bivariate wavelet. Firstly, the concept of minimum-energy bivariate wavelet frame is defined, and its equivalent characterizations and a necessary condition are presented. Secondly, based on polyphase form of symbol functions of scaling function and wavelet function, two sufficient conditions and an explicit constructed method are given. Finally, the decomposition algorithm, reconstruction algorithm, and numerical examples are designed.

Sign in / Sign up

Export Citation Format

Share Document