As this is a conference on mathematical tools for defense, I would like to dedicate this talk to ... more As this is a conference on mathematical tools for defense, I would like to dedicate this talk to the memory of Louis Auslander, who through his insights and visionary leadership, brought powerful new mathematics into DARPA, he has provided the main impetus to the development and insertion of wavelet based processing in defense. My goal here is to describe the evolution of a stream of ideas in Harmonic Analysis, ideas which in the past have been mostly applied for the analysis and extraction of information from physical data, and which now are increasingly applied to organize and extract information and knowledge from any set of digital documents, from text to music to questionnaires. This form of signal processing on digital data, is part of the future of wavelet analysis.
Adapted wave form analysis, refers to a collection of FFT like adapted transform algorithms. Give... more Adapted wave form analysis, refers to a collection of FFT like adapted transform algorithms. Given a signal or an image these methods provide a special orthonormal basis relative to which the image is well represented. The selected basis functions are chosen inside predefined libraries of oscillatory localized functions (waveforms) so as to minimize the number of parameters needed to describe our object. These algorithms are of complexity N log N opening the door for a large range of applications in signal and image processing, such as compression, feature extraction and enhancement. Our goal is to describe and relate traditional Fourier methods to wavelet, wavelet-packet based algorithms by making explicit their relative role in analysis. Starting with a recent refinement of the windowed sine and cosine transforms we will derive an adapted local sine transform, show it''s relation to wavelet and wavelet- packet analysis and describe an analysis tool-kit illustrating the merits of different adaptive and non-adaptive schemes.
In the first part of this thesis, we study the use of anisotropic diffusions on datasets as a too... more In the first part of this thesis, we study the use of anisotropic diffusions on datasets as a tool for signal processing and machine learning. We modify the geometry of the data by adding feature coordinates derived from a function or set of functions which are to be studied. The anisotropic diffusion on the data is just the isotropic diffusion on its modification. We thus transfer some of the complexity of the functions to the geometry of the dataset. This gives a notion of smoothness on the dataset which is adapted to the function(s) under study. We show applications to image denoising and semi-supervised learning problems. In the second part, we study the construction of local and multiscale bases on datasets which are adapted to dyadic partitions of the dataset. Some progress is made towards generalizing the local cosines construction from R . Simpler constructions, like the Haar basis, or block local cosines, generalize easily, and we show applications of the use of these bases in the analysis of functions on datasets and image denoising.
One of the goals of classical Fourier analysis is to obtain L p estimates for linear operators th... more One of the goals of classical Fourier analysis is to obtain L p estimates for linear operators that commute with translations. The case of p = 2 plays a special role, because of Placherel’s theorem, and because of the availability of methods from the Calderon- Zygmund school for deriving L p estimates from the L 2 case. However, the relevance of Plancherel’s theorem fades swiftly to oblivion when we shift our attention to nonlinear objects. (The Calderon-Zygmund technology does not.) In this paper we present an approach to dealing with nonlinear functionals that commute with translations and which satisfy nonlinear versions of the Calderon-Zygmund conditions.
This is an interactive software tool for the Windows operating environment that allows the users ... more This is an interactive software tool for the Windows operating environment that allows the users to explore the properties of the wavelet packet and local trigonometric transforms by performing adapted waveform analysis on digital signals. This package includes a diskette and a users manual. The objective of adapted waveform analysis is to find the "best" representation of a digital signal through its adequate display with a relatively small number of coefficients. The new representation can then be used more effectively in applications such as signal compression and feature extraction. The software features an instructional tutorial to get you started. It is then possible to move on to designing specialized algorithms using the various transforms. The laboratory also includes an applications guide which presents the mathematical background of the program and some practical examples. The package is aimed at mathematicians, engineers and physicists alike.
We enlarge the class of singular integrals of Caldern-Zygmund type by generalizing the usual assu... more We enlarge the class of singular integrals of Caldern-Zygmund type by generalizing the usual assumptions on the kernel. These weaker conditions on the kernel arise naturally in the study of operators depending (linearly or not) on a functional parameter. Examples include the Cauchy integral operator, viewed as a function of the curve, and multilinear operators, viewed as operating on one of the arguments while the others are frozen.
As this is a conference on mathematical tools for defense, I would like to dedicate this talk to ... more As this is a conference on mathematical tools for defense, I would like to dedicate this talk to the memory of Louis Auslander, who through his insights and visionary leadership, brought powerful new mathematics into DARPA, he has provided the main impetus to the development and insertion of wavelet based processing in defense. My goal here is to describe the evolution of a stream of ideas in Harmonic Analysis, ideas which in the past have been mostly applied for the analysis and extraction of information from physical data, and which now are increasingly applied to organize and extract information and knowledge from any set of digital documents, from text to music to questionnaires. This form of signal processing on digital data, is part of the future of wavelet analysis.
Adapted wave form analysis, refers to a collection of FFT like adapted transform algorithms. Give... more Adapted wave form analysis, refers to a collection of FFT like adapted transform algorithms. Given a signal or an image these methods provide a special orthonormal basis relative to which the image is well represented. The selected basis functions are chosen inside predefined libraries of oscillatory localized functions (waveforms) so as to minimize the number of parameters needed to describe our object. These algorithms are of complexity N log N opening the door for a large range of applications in signal and image processing, such as compression, feature extraction and enhancement. Our goal is to describe and relate traditional Fourier methods to wavelet, wavelet-packet based algorithms by making explicit their relative role in analysis. Starting with a recent refinement of the windowed sine and cosine transforms we will derive an adapted local sine transform, show it''s relation to wavelet and wavelet- packet analysis and describe an analysis tool-kit illustrating the merits of different adaptive and non-adaptive schemes.
In the first part of this thesis, we study the use of anisotropic diffusions on datasets as a too... more In the first part of this thesis, we study the use of anisotropic diffusions on datasets as a tool for signal processing and machine learning. We modify the geometry of the data by adding feature coordinates derived from a function or set of functions which are to be studied. The anisotropic diffusion on the data is just the isotropic diffusion on its modification. We thus transfer some of the complexity of the functions to the geometry of the dataset. This gives a notion of smoothness on the dataset which is adapted to the function(s) under study. We show applications to image denoising and semi-supervised learning problems. In the second part, we study the construction of local and multiscale bases on datasets which are adapted to dyadic partitions of the dataset. Some progress is made towards generalizing the local cosines construction from R . Simpler constructions, like the Haar basis, or block local cosines, generalize easily, and we show applications of the use of these bases in the analysis of functions on datasets and image denoising.
One of the goals of classical Fourier analysis is to obtain L p estimates for linear operators th... more One of the goals of classical Fourier analysis is to obtain L p estimates for linear operators that commute with translations. The case of p = 2 plays a special role, because of Placherel’s theorem, and because of the availability of methods from the Calderon- Zygmund school for deriving L p estimates from the L 2 case. However, the relevance of Plancherel’s theorem fades swiftly to oblivion when we shift our attention to nonlinear objects. (The Calderon-Zygmund technology does not.) In this paper we present an approach to dealing with nonlinear functionals that commute with translations and which satisfy nonlinear versions of the Calderon-Zygmund conditions.
This is an interactive software tool for the Windows operating environment that allows the users ... more This is an interactive software tool for the Windows operating environment that allows the users to explore the properties of the wavelet packet and local trigonometric transforms by performing adapted waveform analysis on digital signals. This package includes a diskette and a users manual. The objective of adapted waveform analysis is to find the "best" representation of a digital signal through its adequate display with a relatively small number of coefficients. The new representation can then be used more effectively in applications such as signal compression and feature extraction. The software features an instructional tutorial to get you started. It is then possible to move on to designing specialized algorithms using the various transforms. The laboratory also includes an applications guide which presents the mathematical background of the program and some practical examples. The package is aimed at mathematicians, engineers and physicists alike.
We enlarge the class of singular integrals of Caldern-Zygmund type by generalizing the usual assu... more We enlarge the class of singular integrals of Caldern-Zygmund type by generalizing the usual assumptions on the kernel. These weaker conditions on the kernel arise naturally in the study of operators depending (linearly or not) on a functional parameter. Examples include the Cauchy integral operator, viewed as a function of the curve, and multilinear operators, viewed as operating on one of the arguments while the others are frozen.
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Papers by Ronald Coifman