A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way, by S. Mallat is the improved, revised version of his classic book. It should be noted that much of the work on this third edition was done by Gabriel Peyre. Some of the new developments of the past few years are now discussed in the book, including in Chapter 12, "Sparsity in redundant dictionaries", and Chapter 13, "Inverse problems". I don't know what Caltech does to its graduate students who used the 2nd edition of the book in a certain class, but there is a certain negative review of this book on Amazon that you should take with a grain of salt. Allow me to retort. First, it is said that this is an information dump. Nobody said that you should read the book in linear order --- the author himself lists possible course paths in the preface --- so this argument is very cheap in my view. Second, there is the complaint that the author does not give all the information necessary to do the numerical implementation: I'll rephrase that by saying that most of the information is in the book, but not in the form of pseudo-code. There is a reason why "Numerical recipes in C" is not on my night stand! "A wavelet tour" is a book meant to be read, and in addition, all the code is provided online. Third, it is said that the book has many typos. I agree that this is true for the 2nd edition, but did the reviewer bother to even open the 3rd edition before writing his review? I stand by my view that "A wavelet tour" is still, in 2009, the best book on wavelets for mathematically-inclined people. (Note to Academic Press: restore glossy pages, please.)
Stephane Mallat's book AWavelet Tour of Signal Processing is a must for anybodyinterested in learning about wavelets. It provides a clear and solidtheoretical foundation directed towards applications. Its unusualbreadth makes it interesting to engineers, physicists andmathematicians alike. The subject of wavelets crystallized in theearly 90's so this book (published in 1999) will stay a reference forquite a while. Mallat is one of the main contributors to the theory ofwavelets and multiresolution analysis. This book is used as the mainreference for the class "Wavelets and modern signal processing" atCaltech. My favorite chapters contain material on: Fourier transformsand series, sampling and aliasing, Time-frequency transforms, Frames,Orthonormal bases of wavelets, multiresolution analysis, Waveletpackets, Approximation theory of wavelet thresholding, Statisticalestimation with wavelets, and Coding theory.
In which, x refers to the actual ECG signals, sn refers to noises and y refers to the signals observed. Denoising actually means estimating x in y and drawing x near to y as much as possible. The denoising based on sparse decomposition supposes the ECG signal x is sparse. Consequently, if having constructed an sparse dictionary D, we can find a sparse signal x=Dα^ that approximate the observed signal most and consider it as the original ECG signal's estimation, so as to find the minimum value of L as
where λ=σ2logeN, where σ is covariance of y, and N denotes the length of y. Since wi of different segments of signals vary from others, the sparse-based denosing is non-linear and can minimise the denosing risk .
Course Description: This course is a systematic introduction to mathematical tools from (applied) harmonic analysis, such as wavelets, time-frequency analysis, sparse and redundant representations, etc. These concepts have tremendous impact on many areas in mathematics,engineering, physics, statistics, etc., and have revolutionized applications including signal- and image processing, PDEs, compressive sensing, data analysis, and digital communications.
Syllabus:Fourier series and Fourier transform; Shannon's sampling theorem Orthonormal bases and frames, redundancy and sparsity Short-time Fourier transform and Gabor (Weyl-Heisenberg) systems, Wigner distribution, Uncertainty Principle Multiresolution analysis, the wavelet transform; the continuous wavelettransform, discrete wavelet transforms, fast wavelet transform Sparse representations and compressive sensing Applications to signal and image processing, data analysis, data compression, inverse problems, ... Textbooks: The following textbooks are used as references and good books to keepon one's desk (but are not required):Stephane Mallat: "A Wavelet Tour of Signal Processing, Third edition: The Sparse Way", Academic Press, 2008. Ingrid Daubechies: "Ten Lectures on Wavelets", SIAM, 1992. Karlheinz Groechenig: "Foundations of Time-Frequency Analysis", Birkhaeuser, 2001. Michael Elad: "Sparse and Redundant Representations", Springer 2010. Grading:
Description: Decompositions of functions into frequency components via the Fourier transform, and related sparse representations, are fundamental tools in applied mathematics. These ideas have been important in applications to signal processing, imaging, and the quantitative and qualitative analysis of a broad range of mathematical models of data (including modern approaches to machine learning) and physical systems. Topics to be covered in this course include an overview of classical ideas related to Fourier series and the Fourier transform, wavelet representations of functions and the framework of multiresolution analysis, and applications throughout computational and applied mathematics.
This course is an introduction to Fourier Analysis and Wavelets.It has been specifically designed for engineers, scientists, statisticians and mathematicians interestedin the basic mathematical ideas underlying Fourier analysis, wavelets and their applications. This course integrates the classical Fourier Theory with its latest offspring, the Theory of Wavelets.Wavelets and Fourier analysis are invaluable tools for researchers in many areas of mathematics and the applied sciences, to name a few: signal processing, statistics, physics,differential equations, numerical analysis, geophysics, medicalimaging, fractals, harmonic analysis, etc. It is theirmultidisciplinary nature that makes these theories so appealing.
SParse Optimization Research COde (SPORCO) is an open-source Python packagefor solving optimization problems with sparsity-inducing regularization,consisting primarily of sparse coding and dictionary learning, for bothstandard and convolutional forms of sparse representation. In the currentversion, all optimization problems are solved within the AlternatingDirection Method of Multipliers (ADMM) framework. SPORCO was developed forapplications in signal and image processing, but is also expected to beuseful for problems in computer vision, statistics, and machine learning.
Some people asked about some textbook recommendations:Mallat, S. (2009) "A wavelet tour of signal processing: the sparse way", Academic Press. Chapter 4 is quite good.Boashash, B. (2016) "Time-Frequency Signal Analysis and Processing: A Comprehensive Reference" Acadmic Press. Very comprehensive.
Wavelet, a powerful tool for signal processing, can be used to approximate the target function. For enhancing the sparse property of wavelet approximation, a new algorithm was proposed by using wavelet kernel Support Vector Machines (SVM), which can converge to minimum error with better sparsity. Here, wavelet functions would be firstly used to construct the admitted kernel for SVM according to Mercy theory; then new SVM with this kernel can be used to approximate the target function with better sparsity than wavelet approxiamtion itself. The results obtained by our simulation experiment show the feasibility and validity of wavelet kernel support vector machines.
We propose a time-domain approach to detect frequencies, frequency couplings, and phases using nonlinear correlation functions. For frequency analysis, this approach is a multivariate extension of discrete Fourier transform, and for higher-order spectra, it is a linear and multivariate alternative to multidimensional fast Fourier transform of multidimensional correlations. This method can be applied to short and sparse time series and can be extended to cross-trial and cross-channel spectra (CTS) for electroencephalography data where multiple short data segments from multiple trials of the same experiment are available. There are two versions of CTS. The first one assumes some phase coherency across the trials, while the second one is independent of phase coherency. We demonstrate that the phase-dependent version is more consistent with event-related spectral perturbation analysis and traditional Morlet wavelet analysis. We show that CTS can be applied to short data windows and yields higher temporal resolution than traditional Morlet wavelet analysis. Furthermore, the CTS can be used to reconstruct the event-related potential using all linear components of the CTS.
The vibrations of hands and arms are the main symptoms of Parkinson's ailment. Nevertheless, the affection of the vocal cords leads to troubles and defects in the speech, which is another accurate symptom of the disease. This article presents a diagnostic model of Parkinson's disease (PD) and proposes the time-frequency transform (wavelet WT) and Mel-frequency cepstral coefficients (MFCC) treatment for this disease. The proposed treatment is centered on the vocal signal transformation by a method based on the WT and to extract the coefficients of the MFCC and eventually the categorization of the sick and healthy patients by the use of the classifier K-nearest neighbor (KNN). The analysis used in this article uses a database that contains 18 healthy patients and twenty patients. The Daubechies mother WT is used in treatments to compress the vocal signal and extract the MFCC cepstral coefficients. As far as, the diagnosis of Parkinson's ailment is concerned the KNN classifying performance gives 89% accuracy when applied to 52% of the database as training data, whereas when we increase this percentage from 52% to 73%, we reach 98.68% accuracy which is higher than using the support-vector machine classifier. The KNN is conclusive in the determination of the PD. Moreover, the higher the training data is, the more precise the results are. 2b1af7f3a8