scholarly journals Wavelet transformation on a finite interval

Informatics ◽  
2021 ◽  
Vol 17 (4) ◽  
pp. 22-35
Author(s):  
V. M. Romanchak
Keyword(s):  
Finite Interval ◽  
Discrete Wavelet ◽  
Convergence Conditions ◽  
Average Value ◽  
Basic Wavelet ◽  
Basis Wavelet ◽  
Validity Condition ◽  
Periodic Continuation ◽  
Approximation Of A Function ◽  
Wavelet Transformations

Integral transformations on a finite interval with a singular basis wavelet are considered. Using a sequence of such transformations, the problem of nonparametric approximation of a function is solved. Traditionally, it is assumed that the validity condition must be met for a basic wavelet (the average value of the wavelet must be zero). The paper develops the previously proposed method of singular wavelets when the tolerance condition is not met. In this case Delta-shaped functions that participate in Parzen – Rosenblatt and Nadaray – Watson estimations can be used as a basic wavelet. The set of wavelet transformations for a function defined on a numeric axis, defined locally, and on a finite interval were previously investigated. However, the study of the convergence of the decomposition on a finite interval was carried out only in one particular case. It was due to technical difficulties when trying to solve this problem directly. In the paper the idea of evaluating the periodic continuation of a function defined initially on a finite interval is implemented. It allowed to formulate sufficient convergence conditions for the expansion of the function in a series. An example of approximation of a function defined on a finite interval using the sum of discrete wavelet transformations is given.

scholarly journals Local transformations with a singular wavelet

Informatics ◽  
2020 ◽  
Vol 17 (1) ◽  
pp. 39-46
Author(s):  
V. M. Romanchak
Keyword(s):  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Finite Interval ◽  
Average Value ◽  
Basis Wavelet ◽  
Approximation Of A Function

The paper considers a local wavelet transform with a singular basis wavelet. The problem of nonparametric approximation of a function is solved by the use of the  sequence of local wavelet transforms. Traditionally believed that the wavelet should have an average equal to zero. Earlier, the author considered  singular wavelets when the average value is not equal to zero. As an example, the delta-shaped functions, participated in the estimates of Parzen – Rosenblatt and Nadara – Watson, were used as a wavelet. Previously,  a sequence of wavelet transforms for the entire numerical axis and finite interval was constructed for singular wavelets. The paper proposes a sequence of local wavelet transforms, a local wavelet transform is defined, the theorems that formulate the properties of a local wavelet transform are proved. To confirm the effectiveness of the algorithm an example of approximating the function by use of  the sum of discrete local wavelet transforms is given. 

scholarly journals Isolation of a periodic component by singular wavelet decomposition

2020 ◽  
pp. 4-8
Author(s):  
V. M. Romanchak ◽  
M. A. Hundzina
Keyword(s):  
Wavelet Transform ◽  
Discrete Wavelet Transform ◽  
Wavelet Decomposition ◽  
Periodic Component ◽  
Discrete Wavelet ◽  
Real Problem ◽  
Wavelet Method ◽  
Parametric Regression ◽  
Validity Condition ◽  
Wavelet Transformations

In this paper, we propose to use a discrete wavelet transform with a singular wavelet to isolate the periodic component from the signal. Traditionally, it is assumed that the validity condition must be met for a basic wavelet (the average value of the wavelet is zero). For singular wavelets, the validity condition is not met. As a singular wavelet, you can use the Delta-shaped functions, which are involved in the estimates of Parzen-Rosenblatt, Nadaraya-Watson. Using singular value of a wavelet is determined by the discrete wavelet transform. This transformation was studied earlier for the continuous case. Theoretical estimates of the convergence rate of the sum of wavelet transformations were obtained; various variants were proposed and a theoretical justification was given for the use of the singular wavelet method; sufficient conditions for uniform convergence of the sum of wavelet transformations were formulated. It is shown that the wavelet transform can be used to solve the problem of nonparametric approximation of the function. Singular wavelet decomposition is a new method and there are currently no examples of its application to solving applied problems. This paper analyzes the possibilities of the singular wavelet method. It is assumed that in some cases a slow and fast component can be distinguished from the signal, and this hypothesis is confirmed by the numerical solution of the real problem. A similar analysis is performed using a parametric regression equation, which allows you to select the periodic component of the signal. Comparison of the calculation results confirms that nonparametric approximation based on singular wavelets and the application of parametric regression can lead to similar results.

scholarly journals Discrete Wavelet Transform on Color Picture Interpolation of Digital Still Camera

VLSI Design ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Yu-Cheng Fan ◽  
Yi-Feng Chiang
Keyword(s):  
Wavelet Transform ◽  
Discrete Wavelet Transform ◽  
Color Filter ◽  
Edge Weight ◽  
Discrete Wavelet ◽  
Digital Information ◽  
Color Filter Array ◽  
Image Information ◽  
Basic Wavelet ◽  
Filter Array

Many people use digital still cameras to take photographs in contemporary society. Significant amounts of digital information have led to the emergence of a digital era. Because of the small size and low cost of the product hardware, most image sensors use a color filter array to obtain image information. However, employing a color filter array results in the loss of image information; thus, a color interpolation technique must be employed to retrieve the original picture. Numerous researchers have developed interpolation algorithms in response to various image problems. The method proposed in this study involves integrating discrete wavelet transform (DWT) into the interpolation algorithm. The method was developed based on edge weight and partial gain characteristics and uses the basic wavelet function to enhance the edge performance and processes of the nearest or larger and smaller direction gradients. The experiment results were compared to those of other methods to verify that the proposed method can improve image quality.

scholarly journals Solutions to Some Real-Life Problems Based on Mathematical Modeling and Functional Minimization

2021 ◽  
Vol 10 (4) ◽  
Author(s):  
Oleksii Babaskin ◽  
Danilo Tadeo
Keyword(s):  
Finite Interval ◽  
Real Life ◽  
Functional Dependency ◽  
Functional Dependencies ◽  
Significant Information ◽  
Average Value ◽  
Life Problems ◽  
Dependency Model ◽  
Life Phenomena ◽  
Real World Problems

Building mathematical models that can describe, predict, and explain real-life phenomena is useful. This paper features the functional dependency model and the square of this functional dependency which hold significant information. A mathematical model that relates these functional dependencies with the average value of the function was developed to show that for an arbitrary well-behaved function, the definite integral of the square of the function over a finite interval is minimal when the function is constant over the interval. Finally, the model’s validity and accuracy in representing real-world problems for different situations in physics like mechanics, quantum mechanics, and electricity in economics were evaluated.

A WAVELET ANALYSIS TOOLBOX FOR EXCEL

2005 ◽  
Vol 03 (01) ◽  
pp. 103-123
Author(s):  
HASSAN A. ARTAIL
Keyword(s):  
Wavelet Transform ◽  
Wavelet Analysis ◽  
Discrete Wavelet Transform ◽  
User Interfaces ◽  
Multiresolution Analysis ◽  
Simple Technique ◽  
Visual Basic ◽  
Discrete Wavelet ◽  
Visual Basic For Applications ◽  
Basic Wavelet

This paper presents an implementation of a wavelet analysis toolbox and its integration within Excel. The toolbox includes the discrete wavelet transform, inverse wavelet transform, wavelet-based de-noising, and an associated plotting utility. The transforms and the de-noising algorithms were implemented in a DLL using C++ while the user interfaces were developed using Visual Basic for Applications (VBA) forms. A simple technique was used to automate the transfer of data, associated properties, and user selections from the worksheet to the DLL, and the computed values from the DLL back to the worksheet. We show how the grouping and presentation of computed wavelet coefficients allow for multiresolution analysis and further processing within Excel. We highlight the benefits behind implementing the basic wavelet analysis functions in Excel with reference to MATLAB.

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