CONVOLUTION THEOREMS FOR WAVELET TRANSFORM ON TEMPERED DISTRIBUTIONS AND THEIR EXTENSION TO TEMPERED BOEHMIANS

2009 ◽  
Vol 02 (01) ◽  
pp. 117-127 ◽  
Author(s):  
R. Roopkumar
Keyword(s):  
Wavelet Transform ◽  
Tempered Distributions ◽  
Continuous Maps ◽  
Dual Wavelet

We define a new convolution ⊗ : 𝒮'(ℝ × ℝ+) × 𝒟(ℝ) → 𝒮'(ℝ × ℝ+) and derive the convolution theorems for wavelet transform and dual wavelet transform in the context of tempered distributions. By using the new convolution, we construct a Boehmian space containing the tempered distributions on ℝ × ℝ+. Applying the convolution theorems in the context of tempered distributions, we also extend the wavelet transform and dual wavelet transform between the tempered Boehmian space and the new Boehmian space as linear continuous maps with respect to δ-convergence and Δ-convergence, satisfying the convolution theorems.

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.

The Continuous Fractional Wavelet Transform on W-Type Spaces

2018 ◽  
Vol 85 (3-4) ◽  
pp. 377
Author(s):  
Anuj Kumar ◽  
S. K. Upadhyay
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Fractional Fourier Transform ◽  
Euclidean Spaces ◽  
Continuous Maps ◽  
Type Spaces ◽  
Fractional Wavelet Transform

An n-dimensional continuous fractional wavelet transform involving <em>n</em>-dimensional fractional Fourier transform is studied and its properties are obtained on Gel'fand and Shilov spaces of type <em>W<sub>M</sub></em>(R<sup>n</sup>), <em>W</em><sup>Ω</sup> (C<sup>n</sup>) and W<sup>Ω</sup><sub>M</sub> (C<sup>n</sup>). It is shown that continuous fractional wavelet transform, W<sup>α</sup><sub>ψ</sub>Φ : W<sub>M</sub>(R<sup>n</sup>) → W<sub>M</sub>(R<sup>n</sup> × R<sub>+</sub>), W<sup>α</sup><sub>ψ</sub>Φ : W<sup>Ω</sup> (C<sup>n</sup>) → W<sup>Ω</sup> (C<sup>n</sup> × R<sub>+</sub>) and W<sup>α</sup><sub>ψ</sub>Φ : W<sup>Ω</sup><sub>M</sub> (C<sup>n</sup>) → W<sup>Ω</sup><sub>M</sub> (C<sup>n</sup> × R<sub>+</sub>) are linear and continuous maps, where R<sup>n</sup> and C<sup>n</sup> are the usual Euclidean spaces.

Abelian theorems for fractional wavelet transform

2017 ◽  
Vol 10 (01) ◽  
pp. 1750019
Author(s):  
Akhilesh Prasad ◽  
Praveen Kumar
Keyword(s):  
Wavelet Transform ◽  
Wavelet Function ◽  
Tempered Distributions ◽  
Final Value ◽  
Mexican Hat Wavelet ◽  
Mexican Hat ◽  
Fractional Wavelet Transform ◽  
Abelian Theorems

In this paper, initial and final value Abelian theorems for fractional wavelet transform of function and tempered distributions are obtained. Using Mexican hat wavelet function, an application for Abelian theorems is investigated.

scholarly journals Combining dual-tree complex wavelets and multiresolution in iterative CT reconstruction with application to metal artifact reduction

2019 ◽  
Vol 18 (1) ◽  
Author(s):  
Defne Us ◽  
Ulla Ruotsalainen ◽  
Sampsa Pursiainen
Keyword(s):  
Wavelet Transform ◽  
Noise Suppression ◽  
Artifact Reduction ◽  
Wavelet Basis ◽  
Metal Artifact ◽  
Metal Artifact Reduction ◽  
Ct Reconstruction ◽  
Inversion Algorithm ◽  
Complex Wavelet Transform ◽  
Dual Wavelet

Abstract Background This paper investigates the benefits of data filtering via complex dual wavelet transform for metal artifact reduction (MAR). The advantage of using complex dual wavelet basis for MAR was studied on simulated dental computed tomography (CT) data for its efficiency in terms of noise suppression and removal of secondary artifacts. Dual-tree complex wavelet transform (DT-CWT) was selected due to its enhanced directional analysis of image details compared to the ordinary wavelet transform. DT-CWT was used for multiresolution decomposition within a modified total variation (TV) regularized inversion algorithm. Methods In this study, we have tested the multiresolution TV (MRTV) approach with DT-CWT on a 2D polychromatic jaw phantom model with Gaussian and Poisson noise. High noise and sparse measurement settings were used to assess the performance of DT-CWT. The results were compared to the outcome of the single-resolution reconstruction and filtered back-projection (FBP) techniques as well as reconstructions with Haar wavelet basis. Results The results indicate that filtering of wavelet coefficients with DT-CWT effectively removes the noise without introducing new artifacts after inpainting. Furthermore, adoption of multiple resolution levels yield to a more robust algorithm compared to varying the regularization strength. Conclusions The multiresolution reconstruction with DT-CWT is also more robust when reconstructing the data with sparse projections compared to the single-resolution approach and Haar wavelets.

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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