Fractional Wavelet Transform Research Articles

Fractional wavelet transform for the quantitative spectral resolution of the composite signals of the active compounds in a two-component mixture

Fractional calculus is a powerful tool that has been applied successfully for the analysis of the complex systems. One interesting example of a complex mixture is given by the multicomponent pharmaceutical samples having constant matrix content. The main aim of this study is to develop a new approach based on the combined use of the fractional wavelet transform (FWT) and the continuous wavelet transform (CWT) in order to quantify atorvastatin (ATO) and amlodipine (AML) in their mixtures without requiring a chemical pretreatment. In the first step, the absorption spectra of the compounds and their samples were processed by the FWT method. In the next step, the CWT approach was applied to the fractional wavelet spectra obtained in the above step. The aim of the application of FWT is data reduction corresponding to the spectra of compounds and their commercial samples. In the following step, the CWT was used for the quantitative resolution of the composite signals of the analyzed compounds. After method validation, the proposed signal processing methods based on the combined use of the FWT and the CWT were successfully applied to the resolution of the composite spectra for the quantitation of atorvastatin (ATO) and amlodipine (AML) in tablets.

New approach for simultaneous spectral analysis of a complex mixture using the fractional wavelet transform

Fractional wavelet transform (FWT) is a powerful tool in order to extract largest analytical information from the spectral bands. For the above reason, the combined applications of fractional wavelet transform with principal component regression (PCR) and partial least squares (PLS) calibrations are performed for the quantitative analysis of a pharmaceutical mixture containing trimethoprim (TMP) and sulphamethoxazole (SMX). One interesting example of complex pharmaceutical mixture is given by the multi-component pharmaceutical sample having constant matrix content. In this study, the absorption spectra of the related compounds and their samples were processed by fractional wavelet transform. In the following procedure, FWT-coefficients obtained in the above step are used to construct the PCR and PLS calibrations. These hybrid mathematical approaches are denoted as FWT-PCR and FWT-PLS calibrations. FWT-PCR and FWT-PLS obtained by using the FWT-coefficients for the quantitative analysis of TMP and SMX gives the higher accurate and precise results than classical PCR and PLS based on the use of the original absorbance values.

Multivariate analysis of paracetamol, propiphenazone, caffeine and thiamine in quaternary mixtures by PCR, PLS and ANN calibrations applied on wavelet transform data

The quantitative resolution of a quaternary pharmaceutical mixture consisting of paracetamol, propiphenazone, caffeine and thiamine was performed by the simultaneous use of fractional wavelet transform (FWT) with principal component regression (PCR), partial least squares (PLS) and artificial neural networks (ANN) methods. A calibration set consisting of 22 mixture solutions was prepared by means of an orthogonal experimental design and their absorption spectra were recorded in the spectral range of 210.0–312.3 nm and then transferred into the fractional wavelet domain and processed by FWT. The chemometric calibrations FWT–PCR, FWT–PLS and FWT–ANN were computed by using the relationship between the coefficients provided by FWT method and the concentration data from calibration set. An external validation was carried out by applying the developed methods to the analysis of synthetic mixtures of the related compounds, obtaining successful results. The models were finally used to assay the studied drugs in the commercial pharmaceutical formulations.

Secure Watermarking in Fractional Wavelet Domains

This paper proposes a novel technique for securing and preventing the misuse of images by digital image watermarking. The technique is based on a hybrid approach employing the Wavelet Transform (WT), and the Fractional Fourier transform (FRT) termed as the Fractional Wavelet Transform (FWT). We present experimental evidence to establish the advantages of this scheme over other watermarking schemes when subject to attacks such as noise and compression.

Fractional Wavelet Analysis for the Simultaneous Quantitative Analysis of Lacidipine and Its Photodegradation Product by Continuous Wavelet Transform and Multilinear Regression Calibration

Fractional wavelet transform (FWT) was applied to the original absorption spectra of lacidipine (LAC) and its photodegradation product (LACD), and the resulting FWT spectra were processed by continuous wavelet transform (CWT) and multilinear regression calibration (MLRC) for the simultaneous quantitative analysis of both products in their binary mixtures. These methods do not require any chemical separation step and chemical complex reaction to obtain a detectable signal for the degradation product. By using the Mexican hat function, 2 calibration functions for LAC and LACD were obtained by measuring the CWT transformed signals at 416.1 nm for LAC and 414.6 nm for LACD, after FWT processing of the original absorption spectra. The calibration graphs were linear in the concentration range of 5.08-40.64 microg/mL for LAC and 0.51-8.16 microg/mL for LACD. The limit of detection and the limit of quantitation were found to be 0.289 and 0.956 microg/mL for LAC and 0.036 and 0.118 microg/mL for LACD, respectively. For comparison, the MLRC algorithm was applied to the linear regression functions for the individual drug and its photoproduct. In this approach, a set of linear regression functions was obtained from the relationship between concentrations and FWT signals in the wavelength range 411.0-412.4 nm. Both methods were applied to the quantitative evaluation of LAC and LACD in laboratory and pharmaceutical samples, and produced very satisfactory results.

Application of Fractional Spline Wavelet in Detection of Abrupt Information from Fault Gear System

In this paper the Fractional Spline Wavelet Transform (FrSWT) is introduced based on the definitions and properties of the fractional B-spline and the symmetric fractional B-spline. Then the computation process and approximation properties of the FrSWT are also discussed and especially, the truncation error of computation is obtained analytically. At last, the FrSWT is applied to detect the abrupt information in the simulated and measured signal of fault gear system.

A new fractional wavelet approach for the simultaneous determination of ampicillin sodium and sulbactam sodium in a binary mixture

A new application of the fractional wavelet transform (FWT) was proposed for the simultaneous determination of ampicillin (AP) and sulbactam (SB) in a pharmaceutical combination for injection. FWT approach is a new powerful tool for removing noise and irrelevant information from the absorption spectra. Cardinal information having higher peak amplitude, eliminated noise, sharp peaks with shrinking width of spectral range was obtained by the application of FWT procedure to the original absorption spectra. In this paper, FWT approach was subjected to the data vector of the UV-signals obtained from AP and SB in the wavelength range of 211.5–313.8 nm. Derivative transform was applied to the original absorption signal together with its FWT generalization. The calibration graphs for AP and SB were obtained by measuring the FWT and usual derivative amplitudes at zero-crossing points. The method validation was carried out by using the synthetic mixture analysis. Our proposed FWT approach was compared with the usual derivative spectrophotometry and chemometric methods (CLS, PCR and PLS) and a good agreement was reported.

Optical image encryption based on fractional wavelet transform

Based on the fractional wavelet transform (FWT) a novel method for the image encryption is proposed. We encrypt the image by two fractional orders and a series of scaling factors. There are two series of keys in this method. Only when all of these keys are correct, the image could be decrypted. We can also realize the partial encryption by using this method. Meanwhile, the optical implementation is suggested and some numerical simulations prove its possibility.