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Harmonic wavelet transform

About: Harmonic wavelet transform is a research topic. Over the lifetime, 9602 publications have been published within this topic receiving 247336 citations.


Papers
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Journal ArticleDOI
TL;DR: The experiments results show that the proposed algorithm based on the fractional Fourier transform (FRFT) is very robust to JPEG compression noise attacks and image manipulation operations, but also can provide protection even under compound attacks.

89 citations

Journal Article
TL;DR: The electroencephalograph (EEG) signal is one of the most widely signal used in the bioinformatics field due to its rich information about human tasks and its classification is achieved using the Discrete Wavelet Transform DWT with Fast Fourier Transform (FFT).
Abstract: The electroencephalograph (EEG) signal is one of the most widely signal used in the bioinformatics field due to its rich information about human tasks. In this work EEG waves classification is achieved using the Discrete Wavelet Transform DWT with Fast Fourier Transform (FFT) by adopting the normalized EEG data. The DWT is used as a classifier of the EEG wave’s frequencies, while FFT is implemented to visualize the EEG waves in multi-resolution of DWT. Several real EEG data sets (real EEG data for both normal and abnormal persons) have been tested and the results improve the validity of the proposed technique. Keywords—Bioinformatics, DWT, EEG waves, FFT.

89 citations

Journal ArticleDOI
TL;DR: This paper introduces a transform that is called the biquaternion Fourier transform (BiQFT), and shows how it can be used to generalize the notion of analytic signal to complex-valued signals.
Abstract: In this paper, we consider the extension of the Fourier transform to biquaternion-valued signals. We introduce a transform that we call the biquaternion Fourier transform (BiQFT). After giving some general properties of this transform, we show how it can be used to generalize the notion of analytic signal to complex-valued signals. We introduce the notion of hyperanalytic signal. We also study the Hermitian symmetries of the BiQFT and their relation to the geometric nature of a biquaternion-valued signal. Finally, we present a fast algorithm for the computation of the BiQFT. This algorithm is based on a (complex) change of basis and four standard complex FFTs.

89 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that a one-dimensional or multidimensional sequence is uniquely specified under mild restrictions by its signed Fourier transform magnitude (magnitude and 1 bit of phase information).
Abstract: In this paper, we show that a one-dimensional or multidimensional sequence is uniquely specified under mild restrictions by its signed Fourier transform magnitude (magnitude and 1 bit of phase information). In addition, we develop a numerical algorithm to reconstruct a one-dimensional or multidimensional sequence from its Fourier transform magnitude. Reconstruction examples obtained using this algorithm are also provided.

88 citations

Book
25 Mar 2004
TL;DR: This chapter discusses Wavelet Transform and Wavelet Packet Transform, which involves Fourier Transformation as Applied to Convolution and Deconvolution, and some Basic Concepts for Two-Dimensional Data from Hyphenated Instrumentation.
Abstract: PREFACE. CHAPTER 1: INTRODUCTION. 1.1. Modern Analytical Chemistry. 1.1.1. Developments in Modern Chemistry. 1.1.2. Modern Analytical Chemistry. 1.1.3. Multidimensional Dataset. 1.2. Chemometrics. 1.2.1. Introduction to Chemometrics. 1.2.2. Instrumental Response and Data Processing. 1.2.3. White, Black, and Gray Systems. 1.3. Chemometrics-Based Signal Processing Techniques. 1.3.1. Common Methods for Processing Chemical Data. 1.3.2. Wavelets in Chemistry. 1.4. Resources Available on Chemometrics and Wavelet Transform. 1.4.1. Books. 1.4.2. Online Resources. 1.4.3. Mathematics Software. CHAPTER 2: ONE-DIMENSIONAL SIGNAL PROCESSING TECHNIQUES IN CHEMISTRY. 2.1. Digital Smoothing and Filtering Methods. 2.1.1. Moving-Window Average Smoothing Method. 2.1.2. Savitsky--Golay Filter. 2.1.3. Kalman Filtering. 2.1.4. Spline Smoothing. 2.2. Transformation Methods of Analytical Signals. 2.2.1. Physical Meaning of the Convolution Algorithm. 2.2.2. Multichannel Advantage in Spectroscopy and Hadamard Transformation. 2.2.3. Fourier Transformation. 2.2.3.1. Discrete Fourier Transformation and Spectral Multiplex Advantage. 2.2.3.2. Fast Fourier Transformation. 2.2.3.3. Fourier Transformation as Applied to Smooth Analytical Signals. 2.2.3.4. Fourier Transformation as Applied to Convolution and Deconvolution. 2.3. Numerical Differentiation. 2.3.1. Simple Difference Method. 2.3.2. Moving-Window Polynomial Least-Squares Fitting Method. 2.4. Data Compression. 2.4.1. Data Compression Based on B-Spline Curve Fitting. 2.4.2. Data Compression Based on Fourier Transformation. 2.4.3. Data Compression Based on Principal-Component Analysis. CHAPTER 3: TWO-DIMENSIONAL SIGNAL PROCESSING TECHNIQUES IN CHEMISTRY. 3.1. General Features of Two-Dimensional Data. 3.2. Some Basic Concepts for Two-Dimensional Data from Hyphenated Instrumentation. 3.2.1. Chemical Rank and Principal-Component Analysis (PCA). 3.2.2. Zero-Component Regions and Estimation of Noise Level and Background. 3.3. Double-Centering Technique for Background Correction. 3.4. Congruence Analysis and Least-Squares Fitting. 3.5. Differentiation Methods for Two-Dimensional Data. 3.6 Resolution Methods for Two-Dimensional Data. 3.6.1. Local Principal-Component Analysis and Rankmap. 3.6.2. Self-Modeling Curve Resolution and Evolving Resolution Methods. 3.6.2.1. Evolving Factor Analysis (EFA). 3.6.2.2. Window Factor Analysis (WFA). 3.6.2.3. Heuristic Evolving Latent Projections (HELP). CHAPTER 4: FUNDAMENTALS OF WAVELET TRANSFORM. 4.1. Introduction to Wavelet Transform and Wavelet Packet Transform. 4.1.1. A Simple Example: Haar Wavelet. 4.1.2. Multiresolution Signal Decomposition. 4.1.3. Basic Properties of Wavelet Function. 4.2. Wavelet Function Examples. 4.2.1. Meyer Wavelet. 4.2.2. B-Spline (Battle-Lemarie) Wavelets. 4.2.3. Daubechies Wavelets. 4.2.4. Coiflet Functions. 4.3. Fast Wavelet Algorithm and Packet Algorithm. 4.3.1. Fast Wavelet Transform. 4.3.2. Inverse Fast Wavelet Transform. 4.3.3. Finite Discrete Signal Handling with Wavelet Transform. 4.3.4. Packet Wavelet Transform. 4.4. Biorthogonal Wavelet Transform. 4.4.1. Multiresolution Signal Decomposition of Biorthogonal Wavelet. 4.4.2. Biorthogonal Spline Wavelets. 4.4.3. A Computing Example. 4.5. Two-Dimensional Wavelet Transform. 4.5.1. Multidimensional Wavelet Analysis. 4.5.2. Implementation of Two-Dimensional Wavelet Transform. CHAPTER 5: APPLICATION OF WAVELET TRANSFORM IN CHEMISTRY. 5.1. Data Compression. 5.1.1. Principle and Algorithm. 5.1.2. Data Compression Using Wavelet Packet Transform. 5.1.3. Best-Basis Selection and Criteria for Coefficient Selection. 5.2. Data Denoising and Smoothing. 5.2.1. Denoising. 5.2.2. Smoothing. 5.2.3. Denoising and Smoothing Using Wavelet Packet Transform. 5.2.4. Comparison between Wavelet Transform and Conventional Methods. 5.3. Baseline/Background Removal. 5.3.1. Principle and Algorithm. 5.3.2. Background Removal. 5.3.3. Baseline Correction. 5.3.4. Background Removal Using Continuous Wavelet Transform. 5.3.5. Background Removal of Two-Dimensional Signals. 5.4. Resolution Enhancement. 5.4.1. Numerical Differentiation Using Discrete Wavelet Transform. 5.4.2. Numerical Differentiation Using Continuous Wavelet Transform. 5.4.3. Comparison between Wavelet Transform and other Numerical Differentiation Methods. 5.4.4. Resolution Enhancement. 5.4.5. Resolution Enhancement by Using Wavelet Packet Transform. 5.4.6. Comparison between Wavelet Transform and Fast Fourier Transform for Resolution Enhancement. 5.5. Combined Techniques. 5.5.1. Combined Method for Regression and Calibration. 5.5.2. Combined Method for Classification and Pattern Recognition. 5.5.3. Combined Method of Wavelet Transform and Chemical Factor Analysis. 5.5.4. Wavelet Neural Network. 5.6. An Overview of the Applications in Chemistry. 5.6.1. Flow Injection Analysis. 5.6.2. Chromatography and Capillary Electrophoresis. 5.6.3. Spectroscopy. 5.6.4. Electrochemistry. 5.6.5. Mass Spectrometry. 5.6.6. Chemical Physics and Quantum Chemistry. 5.6.7. Conclusion. APPENDIX VECTOR AND MATRIX OPERATIONS AND ELEMENTARY MATLAB. A.1. Elementary Knowledge in Linear Algebra. A.1.1. Vectors and Matrices in Analytical Chemistry. A.1.2. Column and Row Vectors. A.1.3. Addition and Subtraction of Vectors. A.1.4. Vector Direction and Length. A.1.5. Scalar Multiplication of Vectors. A.1.6. Inner and Outer Products between Vectors. A.1.7. The Matrix and Its Operations. A.1.8. Matrix Addition and Subtraction. A.1.9. Matrix Multiplication. A.1.10. Zero Matrix and Identity Matrix. A.1.11. Transpose of a Matrix. A.1.12. Determinant of a Matrix. A.1.13. Inverse of a Matrix. A.1.14. Orthogonal Matrix. A.1.15. Trace of a Square Matrix. A.1.16. Rank of a Matrix. A.1.17. Eigenvalues and Eigenvectors of a Matrix. A.1.18. Singular-Value Decomposition. A.1.19. Generalized Inverse. A.1.20. Derivative of a Matrix. A.1.21. Derivative of a Function with Vector as Variable. A.2. Elementary Knowledge of MATLAB. A.2.1. Matrix Construction. A.2.2. Matrix Manipulation. A.2.3. Basic Mathematical Functions. A.2.4. Methods for Generating Vectors and Matrices. A.2.5. Matrix Subscript System. A.2.6. Matrix Decomposition. A.2.6.1. Singular-Value Decomposition (SVD). A.2.6.2. Eigenvalues and Eigenvectors (eig). A.2.7. Graphic Functions 288 INDEX.

88 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202323
202274
20213
20207
20196
201831