scholarly journals Wavelet inversion of the k-plane transform and its application

2006 ◽  
Vol 21 (2) ◽  
pp. 262-267 ◽  
Author(s):  
Gangrong Qu
Keyword(s):  
Wavelet Inversion

scholarly journals Continuous Wavelet Transform of Schwartz Distributions in DL2′(Rn), n ≥ 1

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1106
Author(s):  
Jagdish N. Pandey
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
The Real ◽  
Schwartz Distributions ◽  
Distributional Sense ◽  
Wavelet Inversion

We define a testing function space DL2(Rn) consisting of a class of C∞ functions defined on Rn, n≥1 whose every derivtive is L2(Rn) integrable and equip it with a topology generated by a separating collection of seminorms {γk}|k|=0∞ on DL2(Rn), where |k|=0,1,2,… and γk(ϕ)=∥ϕ(k)∥2,ϕ∈DL2(Rn). We then extend the continuous wavelet transform to distributions in DL2′(Rn), n≥1 and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element ψ(x) of DL2(Rn)∩DL1(Rn), n≥1 which, when integrated along each of the real axes X1,X2,…Xn vanishes, but none of its moments ∫Rnxmψ(x)dx is zero; here xm=x1m1x2m2⋯xnmn, dx=dx1dx2⋯dxn and m=(m1,m2,…mn) and each of m1,m2,…mn is ≥1. The set of such wavelets will be denoted by DM(Rn).

scholarly journals Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n )

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 235 ◽  
Author(s):  
Jagdish Pandey ◽  
Jay Maurya ◽  
Santosh Upadhyay ◽  
Hari Srivastava
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Weak Topology ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
Wavelet Inversion ◽  
Constant Distribution

In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution f ∈ S ′ ( R n ) with wavelet kernel ψ ∈ S ( R n ) and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of S ′ ( R n ) . It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

Simultaneous deconvolution and wavelet inversion as a global optimization

Geophysics ◽  
1999 ◽  
Vol 64 (4) ◽  
pp. 1108-1115 ◽  
Author(s):  
Warren T. Wood
Keyword(s):  
Global Optimization ◽  
Field Data ◽  
Synthetic Data ◽  
Data Sets ◽  
Minimum Entropy ◽  
Desktop Computer ◽  
Highly Nonlinear ◽  
Computer Environment ◽  
Band Limited ◽  
Wavelet Inversion

Estimates of the source wavelet and band‐limited earth reflectivity are obtained simultaneously from an optimization of deconvolution outputs, similar to minimum‐entropy deconvolution (MED). The only inputs required beyond the observed seismogram are wavelet length and an inversion parameter (cooling rate). The objective function to be minimized is a measure of the spikiness of the deconvolved seismogram. I assume that the wavelet whose deconvolution from the data results in the most spike‐like trace is the best wavelet estimate. Because this is a highly nonlinear problem, simulated annealing is used to solve it. The procedure yields excellent results on synthetic data and disparate field data sets, is robust in the presence of noise, and is fast enough to operate in a desktop computer environment.

The continuous wavelet transform in n-dimensions

2016 ◽  
Vol 14 (05) ◽  
pp. 1650037 ◽  
Author(s):  
J. N. Pandey ◽  
N. K. Jha ◽  
O. P. Singh
Keyword(s):  
Image Processing ◽  
Wavelet Transform ◽  
Real Number ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Spherically Symmetric ◽  
Continuous Wavelet ◽  
Wavelet Inversion ◽  
Translation Parameter

Daubechies obtained the [Formula: see text]-dimensional inversion formula for the continuous wavelet transform of spherically symmetric wavelets in [Formula: see text] with convergence interpreted in the [Formula: see text]-norm. From the wavelet [Formula: see text], Daubechies generated a doubly indexed family of wavelets [Formula: see text] by restricting the dilation parameter [Formula: see text] to be a real number greater than zero and the translation parameter [Formula: see text] belonging to [Formula: see text]. We show that [Formula: see text] can be chosen to be in [Formula: see text] with none of the components [Formula: see text] vanishing. Further, we prove that if [Formula: see text] and [Formula: see text] are continuous in [Formula: see text], then the convergence besides being in [Formula: see text] is also pointwise in [Formula: see text]. We advance our theory further to the case when [Formula: see text] and [Formula: see text] both belong to [Formula: see text] then convergence of the wavelet inversion formula is pointwise at all points of continuity of [Formula: see text]. This result significantly enhances the applicability of the wavelet inversion formula to the image processing.

The continuous wavelet inversion formula and regularity of weak solutions to partial differential equations

2020 ◽  
pp. 2050075
Author(s):  
Alfonso Castro ◽  
Dmitry Gokhman ◽  
Jaime Navarro
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Regularity Properties ◽  
Regularity Of Weak Solutions ◽  
Wavelet Inversion

We establish regularity properties of weak solutions to linear partial differential equations in terms of the continuous wavelet transform of the data. Our arguments rely on the existence of radial functions that remain radial under the operator defined by the highest order terms of the linear equation and a variant of the inversion formula introduced by Grossmann, Morlet and Paul.

scholarly journals Continuous Wavelet Transform of Schwartz Tempered Distributions in $S'(\mathbb R^n)$

2019 ◽  
Author(s):  
Jagdish Narayan Pandey ◽  
Jay Singh Maurya ◽  
Santosh Kumar Upadhyay ◽  
Hari Mohan Srivastava
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Weak Topology ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
Tempered Distributions ◽  
Tempered Distribution ◽  
Wavelet Inversion ◽  
Constant Distribution

In this paper we define a continuous wavelet transform of a Schwartz tempered distribution $f \in S^{'}(\mathbb R^n)$ with wavelet kernel $\psi \in S(\mathbb R^n)$ and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of $S^{'}(\mathbb R^n)$. It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

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