scholarly journals Maximum Vanishing Moment of Compactly Supported B-spline Wavelets

2019 ◽  
pp. 1-8
Author(s):  
Kanchan Lata Gupta ◽  
B. Kunwar ◽  
V. K. Singh
Keyword(s):  
Scaling Function ◽  
Simple Procedure ◽  
Smoothness Property ◽  
Vanishing Moments ◽  
B Spline ◽  
B Splines ◽  
Compactly Supported ◽  
Spline Wavelets ◽  
Vanishing Moment ◽  
Rule Order

Spline function is of very great interest in field of wavelets due to its compactness and smoothness property. As splines have specific formulae in both time and frequency domain, it greatly facilitates their manipulation. We have given a simple procedure to generate compactly supported orthogonal scaling function for higher order B-splines in our previous work. Here we determine the maximum vanishing moments of the formed spline wavelet as established by the new refinable function using sum rule order method.

CONSTRUCTION OF COMPACTLY SUPPORTED WAVELETS FROM TRIGONOMETRIC B-SPLINES

2011 ◽  
Vol 09 (05) ◽  
pp. 843-865 ◽  
Author(s):  
M. K. JENA
Keyword(s):  
Convex Hull ◽  
Autocorrelation Function ◽  
Scaling Function ◽  
Partition Of Unity ◽  
B Spline ◽  
B Splines ◽  
Compactly Supported ◽  
Odd Order ◽  
Subdivision Algorithm ◽  
The Scaling Function

We construct a class of semiorthogonal wavelets by taking a normalized trigonometric B-spline of any order as the scaling function. The construction is based on generalized Euler–Frobenius polynomial and generalized autocorrelation function. We also show that the odd order normalized trigonometric B-spline satisfies convex hull property as well as partition of unity property. Moreover, we also present a subdivision algorithm for the convolution of normalized trigonometric B-splines. Several examples of wavelet are also provided.

scholarly journals Solution of nonlinear Fredholm-Hammerstein integral equations by using semiorthogonal spline wavelets

2005 ◽  
Vol 2005 (1) ◽  
pp. 113-121 ◽  
Author(s):  
M. Lakestani ◽  
M. Razzaghi ◽  
M. Dehghan
Keyword(s):  
Integral Equations ◽  
Algebraic Equations ◽  
B Spline ◽  
Compactly Supported ◽  
Hammerstein Integral Equations ◽  
Spline Wavelets

Compactly supported linear semiorthogonal B-spline wavelets together with their dual wavelets are developed to approximate the solutions of nonlinear Fredholm-Hammerstein integral equations. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through an illustrative example.

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.

scholarly journals ELLIPTIC SCALING FUNCTIONS AS COMPACTLY SUPPORTED MULTIVARIATE ANALOGS OF THE B-SPLINES

2014 ◽  
Vol 12 (02) ◽  
pp. 1450018 ◽  
Author(s):  
VICTOR G. ZAKHAROV
Keyword(s):  
Differential Operators ◽  
Scaling Function ◽  
Null Space ◽  
Partition Of Unity ◽  
Total Positivity ◽  
Scaling Functions ◽  
Algebraic Polynomials ◽  
B Splines ◽  
Compactly Supported ◽  
Elliptic Differential Operators

In the paper, we present a family of multivariate compactly supported scaling functions, which we call as elliptic scaling functions. The elliptic scaling functions are the convolution of elliptic splines, which correspond to homogeneous elliptic differential operators, with distributions. The elliptic scaling functions satisfy refinement relations with real isotropic dilation matrices. The elliptic scaling functions satisfy most of the properties of the univariate cardinal B-splines: compact support, refinement relation, partition of unity, total positivity, order of approximation, convolution relation, Riesz basis formation (under a restriction on the mask), etc. The algebraic polynomials contained in the span of integer shifts of any elliptic scaling function belong to the null-space of a homogeneous elliptic differential operator. Similarly to the properties of the B-splines under differentiation, it is possible to define elliptic (not necessarily differential) operators such that the elliptic scaling functions satisfy relations with these operators. In particular, the elliptic scaling functions can be considered as a composition of segments, where the function inside a segment, like a polynomial in the case of the B-splines, vanishes under the action of the introduced operator.

scholarly journals Fractional nonlinear Volterra–Fredholm integral equations involving Atangana–Baleanu fractional derivative: framelet applications

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mutaz Mohammad ◽  
Alexander Trounev
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Approximation ◽  
Fredholm Integral Equations ◽  
Spline Functions ◽  
Vanishing Moments ◽  
B Spline ◽  
B Splines ◽  
Cas Wavelets

Abstract In this work, we propose a framelet method based on B-spline functions for solving nonlinear Volterra–Fredholm integro-differential equations and by involving Atangana–Baleanu fractional derivative, which can provide a reliable numerical approximation. The framelet systems are generated using the set of B-splines with high vanishing moments. We provide some numerical and graphical evidences to show the efficiency of the proposed method. The obtained numerical results of the proposed method compared with those obtained from CAS wavelets show a great agreement with the exact solution. We confirm that the method achieves accurate, efficient, and robust measurement.

Design of High Dimensional Nonseparable Compactly Supported Wavelets with Special Dilation Matrix

2012 ◽  
Vol 542-543 ◽  
pp. 547-550
Author(s):  
Lan Li
Keyword(s):  
Mild Condition ◽  
Scaling Function ◽  
New Method ◽  
High Dimensional ◽  
Moment Condition ◽  
Compactly Supported ◽  
Vanishing Moment ◽  
Dilation Matrix ◽  
The Scaling Function

In this paper, a new method to construct the compactly supported M- wavelet under a mild condition are given. The constructed wavelet satisfies the vanishing moment condition which is originated from the symbols of the scaling function.

Riesz basis of wavelets constructed from trigonometric B-splines

2015 ◽  
Vol 13 (04) ◽  
pp. 419-436
Author(s):  
Mahendra Kumar Jena
Keyword(s):  
Differential Operator ◽  
Sobolev Space ◽  
Sobolev Spaces ◽  
Riesz Basis ◽  
Scaling Function ◽  
Wavelet Bases ◽  
B Splines ◽  
Compactly Supported ◽  
The Scaling Function

In this paper, we construct a class of compactly supported wavelets by taking trigonometric B-splines as the scaling function. The duals of these wavelets are also constructed. With the help of these duals, we show that the collection of dilations and translations of such a wavelet forms a Riesz basis of 𝕃2(ℝ). Moreover, when a particular differential operator is applied to the wavelet, it also generates a Riesz basis for a particular generalized Sobolev space. Most of the proofs are based on three assumptions which are mild generalizations of three important lemmas of Jia et al. [Compactly supported wavelet bases for Sobolev spaces, Appl. Comput. Harmon. Anal. 15 (2003) 224–241].

scholarly journals Vanishing moments for scaling vectors

2004 ◽  
Vol 2004 (36) ◽  
pp. 1897-1908 ◽  
Author(s):  
David K. Ruch
Keyword(s):  
Necessary Conditions ◽  
Scaling Function ◽  
Vanishing Moments ◽  
Minimal Support ◽  
Vanishing Moment ◽  
The Relationship ◽  
Support Length

One advantage of scaling vectors over a single scaling function is the compatibility of symmetry and orthogonality. This paper investigates the relationship between symmetry, vanishing moments, orthogonality, and support length for a scaling vectorΦ. Some general results on scaling vectors and vanishing moments are developed, as well as some necessary conditions for the symbol entries of a scaling vector with both symmetry and orthogonality. If orthogonal scaling vectorΦhas some kind of symmetry and a given number of vanishing moments, we can characterize the type of symmetry forΦ, give some information about the form of the symbolP(z), and place some bounds on the support of eachϕi. We then construct anL2(ℝ)orthogonal, symmetric scaling vector with one vanishing moment having minimal support.

scholarly journals Atlas-based Reconstruction of 3D Volumes of Lower Extremity from 2D Calibrated X-ray Images

10.29007/qxff ◽  
2018 ◽  
Author(s):  
Weimin Yu ◽  
Guoyan Zheng
Keyword(s):  
Knee Joint ◽  
Lower Extremity ◽  
3D Reconstruction ◽  
Present Approach ◽  
Smoothness Property ◽  
B Spline ◽  
X Ray ◽  
B Splines ◽  
Ct Data

A new atlas-based 2D-3D reconstruction of 3D volumes of lower extremity from a pair of calibrated X-ray images was presented. The approach combines non-rigid 2D- 2D registration based 3D landmark reconstruction with the B-spline parametrization of TPS transformation, incorporating the smoothness property of B-splines for regularization. Efficacy of the present approach was evaluated on the calibrated X-ray images and CT data. Also, we take the knee joint articulation into consideration. Articulated B-spline parameterization leads to the almost same accuracy as individual B-spline parameterization and has the superiority over the latter when it comes to the prevention from the knee joint penetration.

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