Constructing totally interpolating wavelet frame systems

2015 ◽  
Vol 13 (03) ◽  
pp. 1550017
Author(s):  
Baobin Li
Keyword(s):  
Filter Banks ◽  
Scaling Function ◽  
Tight Frame ◽  
Special Structure ◽  
Dual Frame ◽  
Wavelet Frame ◽  
Continuous Linear ◽  
Wavelet Frames ◽  
Frame Systems ◽  
The Scaling Function

The system of totally interpolating wavelet frames is discussed in this paper, in which both the scaling function and one of wavelet functions are interpolating. It will be shown that corresponding filter banks possess the special structure, and the parametrization of filter banks is present. Moreover, we show that when considering tight frame systems with two generators, the Ron–Shen's continuous-linear-spline-based tight frame is the only one with totally interpolating property and symmetry. But in the dual frame context, more good examples of bi-frames with symmetric/antisymmetric property can be obtained and constructed, which in particular, include frames with the uniform symmetry.

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.

scholarly journals Tight wavelet frames in Lebesgue and Sobolev spaces

2004 ◽  
Vol 2 (3) ◽  
pp. 227-252 ◽  
Author(s):  
L. Borup ◽  
R. Gribonval ◽  
M. Nielsen
Keyword(s):  
Sobolev Space ◽  
Jackson Inequality ◽  
Wavelet Frame ◽  
Approximation Rate ◽  
Wavelet Frames ◽  
Atomic Decompositions ◽  
Approximation Spaces ◽  
Tight Wavelet Frame ◽  
Frame Systems ◽  
Term Approximation

We study tight wavelet frame systems inLp(ℝd)and prove that such systems (under mild hypotheses) give atomic decompositions ofLp(ℝd)for1≺p≺∞. We also characterizeLp(ℝd)and Sobolev space norms by the analysis coefficients for the frame. We consider Jackson inequalities for bestm-term approximation with the systems inLp(ℝd)and prove that such inequalities exist. Moreover, it is proved that the approximation rate given by the Jackson inequality can be realized by thresholding the frame coefficients. Finally, we show that in certain restricted cases, the approximation spaces, for bestm-term approximation, associated with tight wavelet frames can be characterized in terms of (essentially) Besov spaces.

scholarly journals Frame Multiresolution Analysis on Local Fields of Positive Characteristic

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Firdous A. Shah
Keyword(s):  
Multiresolution Analysis ◽  
Positive Characteristic ◽  
Scaling Function ◽  
Local Fields ◽  
Wavelet Frames ◽  
Shift Invariant Spaces ◽  
Frame Multiresolution Analysis ◽  
Invariant Spaces ◽  
The Scaling Function

We present a notion of frame multiresolution analysis on local fields of positive characteristic based on the theory of shift-invariant spaces. In contrast to the standard setting, the associated subspace V0 of L2(K) has a frame, a collection of translates of the scaling function φ of the form φ(·-u(k)):k∈N0, where N0 is the set of nonnegative integers. We investigate certain properties of multiresolution subspaces which provides the quantitative criteria for the construction of frame multiresolution analysis (FMRA) on local fields of positive characteristic. Finally, we provide a characterization of wavelet frames associated with FMRA on local field K of positive characteristic using the shift-invariant space theory.

Study of Periodic Frames and Trivariate Tight Wavelet Frames and Applications in Materials Engineering

2014 ◽  
Vol 1079-1080 ◽  
pp. 878-881
Author(s):  
Song Zhen Sun ◽  
Yi Guo
Keyword(s):  
Time Domain ◽  
Dual Frame ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Wavelet System ◽  
Materials Engineering ◽  
Number Of Generators ◽  
Dual Wavelet Frames ◽  
The Time Domain ◽  
Dual Wavelet

It is shown that there exists a frame wavelet with fast decay in the time domain and compact support in the frequency domain generating a wavelet system whose canonical dual frame cannot be generated by an arbitrary number of generators. We show that there exist wavelet frame generated by two functions which have good dual wavelet frames, but for which the canonical dual wavelet frame does not consist of wavelets, according to scaling functions.

Assessing Model Uncertainty for the Scaling Function Inversion of Potential Fields

Geophysics ◽  
2021 ◽  
pp. 1-39
Author(s):  
Mahak Singh Chauhan ◽  
Ivano Pierri ◽  
Mrinal K. Sen ◽  
Maurizio FEDI
Keyword(s):  
Simulated Annealing Algorithm ◽  
Scaling Function ◽  
Vertical Section ◽  
Gravity Data ◽  
Geometric Model ◽  
Salt Dome ◽  
The Body ◽  
Gravity Inversion ◽  
Geometrical Parameters ◽  
The Scaling Function

We use the very fast simulated annealing algorithm to invert the scaling function along selected ridges, lying in a vertical section formed by upward continuing gravity data to a set of altitudes. The scaling function is formed by the ratio of the field derivative by the field itself and it is evaluated along the lines formed by the zeroes of the horizontal field derivative at a set of altitudes. We also use the same algorithm to invert gravity anomalies only at the measurement altitude. Our goal is analyzing the different models obtained through the two different inversions and evaluating the relative uncertainties. One main difference is that the scaling function inversion is independent on density and the unknowns are the geometrical parameters of the source. The gravity data are instead inverted for the source geometry and the density simultaneously. A priori information used for both the inversions is that the source has a known depth to the top. We examine the results over the synthetic examples of a salt dome structure generated by Talwani’s approach and real gravity datasets over the Mors salt dome and the Decorah (USA) basin. For all these cases, the scaling function inversion yielded models with a better sensitivity to specific features of the sources, such as the tilt of the body, and reduced uncertainty. We finally analyzed the density, which is one of the unknowns for the gravity inversion and it is estimated from the geometric model for the scaling function inversion. The histograms over the density estimated at many iterations show a very concentrated distribution for the scaling function, while the density contrast retrieved by the gravity inversion, according to the fundamental ambiguity density/volume, is widely dispersed, this making difficult to assess its best estimate.

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.

THE CONSTRUCTION OF A CLASS OF TRIVARIATE NONSEPARABLE COMPACTLY SUPPORTED WAVELETS

2009 ◽  
Vol 07 (03) ◽  
pp. 255-267 ◽  
Author(s):  
YONGDONG HUANG ◽  
SHOUZHI YANG ◽  
ZHENGXING CHENG
Keyword(s):  
General Theory ◽  
Mild Condition ◽  
Scaling Function ◽  
Moment Condition ◽  
Compactly Supported ◽  
Vanishing Moment ◽  
The Scaling Function

In this paper, under a mild condition, the construction of compactly supported [Formula: see text]-wavelets is obtained. Wavelets inherit the symmetry of the corresponding scaling function and satisfy the vanishing moment condition originating in the symbols of the scaling function. An example is also given to demonstrate the general theory.

A note on the standard dual frame of a wavelet frame with three-scale☆

2009 ◽  
Vol 42 (2) ◽  
pp. 931-937 ◽  
Author(s):  
Qingjiang Chen ◽  
Zongtian Wei ◽  
Jinshun Feng
Keyword(s):  
Dual Frame ◽  
Wavelet Frame

NEW AMY AND DELPHI MULTIPLICITY DATA AND THE LOGNORMAL DISTRIBUTION

1991 ◽  
Vol 06 (03) ◽  
pp. 245-257 ◽  
Author(s):  
R. SZWED ◽  
G. WROCHNA ◽  
A.K. WRÓBLEWSKI
Keyword(s):  
Energy Dependence ◽  
Lognormal Distribution ◽  
Scaling Function ◽  
Average Multiplicity ◽  
Linear Functions ◽  
The Mean ◽  
Multiplicity Distributions ◽  
The Scaling Function ◽  
Skewness And Kurtosis

Multiplicity distributions for e+e−→ hadrons recently reported by the AMY and DELPHI collaborations are compared with the data obtained at lower energies. It is proven that the new data obey the KNO-G scaling and the scaling function can be described by the lognormal distribution. The dispersions are linear functions of the mean as for the data measured at lower energies and the standardized moments (such as skewness and kurtosis) are independent of the energy. The energy dependence of the average multiplicity is described by <nch>=β sα−1.

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