Scholars@Duke
Ingrid Daubechies

Ingrid Daubechies

James B. Duke Distinguished Professor Emeritus of Mathematics
Mathematics

Selected Publications

[Update on manual bronchial clearance techniques (adults and adolescents)].

Journal Article Revue des maladies respiratoires · January 2024 In adults and teenagers, airway clearance physiotherapy techniques (ACPT) are various and numerous. However, they for still awaiting scientific validation. Among ACPTs, Slow Expiration with the Glottis Opened in the Lateral Posture (ELTGOL), Autogenic Drai ... Full text Cite

PiPs: A kernel-based optimization scheme for analyzing non-stationary 1D signals

Journal Article Applied and Computational Harmonic Analysis · September 1, 2023 This paper proposes a novel kernel-based optimization scheme to handle tasks in the analysis, e.g., signal spectral estimation and single-channel source separation of 1D non-stationary oscillatory data. The key insight of our optimization scheme for recons ... Full text Cite

Neural Network Approximation of Refinable Functions

Journal Article IEEE Transactions on Information Theory · January 1, 2023 In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of r ... Full text Cite

Image Separation With Side Information: A Connected Auto-Encoders Based Approach.

Journal Article IEEE transactions on image processing : a publication of the IEEE Signal Processing Society · January 2023 X-radiography (X-ray imaging) is a widely used imaging technique in art investigation. It can provide information about the condition of a painting as well as insights into an artist's techniques and working methods, often revealing hidden information invi ... Full text Cite

Diffusion Maps: Using the Semigroup Property for Parameter Tuning

Chapter · January 1, 2023 Diffusion maps (DM) constitute a classic dimension reduction technique, for data lying on or close to a (relatively) low-dimensional manifold embedded in a much larger dimensional space. It consists in constructing a spectral parametrization for the manifo ... Full text Cite

Nonlinear Approximation and (Deep) ReLU Networks

Journal Article Constructive Approximation · February 1, 2022 This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural networks approx ... Full text Cite

Mixed X-Ray Image Separation for Artworks With Concealed Designs.

Journal Article IEEE transactions on image processing : a publication of the IEEE Signal Processing Society · January 2022 In this paper, we focus on X-ray images (X-radiographs) of paintings with concealed sub-surface designs (e.g., deriving from reuse of the painting support or revision of a composition by the artist), which therefore include contributions from both the surf ... Full text Cite

Reconstructing dietary ecology of extinct strepsirrhines (Primates, Mammalia) with new approaches for characterizing and analyzing tooth shape

Journal Article Paleobiology · November 27, 2021 The morphological and ecological diversity of lemurs and lorisiformes once rivaled that of the rest of the primate order. Here, we assemble a dataset of 3D models representing the second mandibular molars of a wide range of extant and fossil strepsirrhines ... Full text Cite

Stable phase retrieval from locally stable and conditionally connected measurements

Journal Article Applied and Computational Harmonic Analysis · November 1, 2021 In this paper, we study the stability of phase retrieval problems via a family of locally stable phase retrieval frame measurements in Banach spaces, which we call “locally stable and conditionally connected” (LSCC) measurement schemes. For any signal f in ... Full text Open Access Cite

Wavelets at your service

Chapter · June 23, 2021 Cite

Robust and resource efficient identification of shallow neural networks by fewest samples

Journal Article Information and Inference · June 1, 2021 We address the structure identification and the uniform approximation of sums of ridge functions f(x)=∑ i=1m gi(ai,x) on Rd, representing a general form of a shallow feed-forward neural network, from a small number of query samples. Higher order differenti ... Full text Cite

Insights from macroevolutionary modelling and ancestral state reconstruction into the radiation and historical dietary ecology of Lemuriformes (Primates, Mammalia).

Journal Article BMC ecology and evolution · April 2021 BackgroundLemurs once rivalled the diversity of rest of the primate order despite thier confinement to the island of Madagascar. We test the adaptive radiation model of Malagasy lemur diversity using a novel combination of phylogenetic comparative ... Full text Cite

A Learning Based Approach to Separate Mixed X-Ray Images Associated with Artwork with Concealed Designs

Conference European Signal Processing Conference · January 1, 2021 X-ray images are widely used in the study of paintings. When a painting has hidden sub-surface features (e.g., reuse of the canvas or revision of a composition by the artist), the resulting X-ray images can be hard to interpret as they include contribution ... Full text Cite

Expression of fractals through neural network functions

Journal Article IEEE Journal on Selected Areas in Information Theory · May 1, 2020 To help understand the underlying mechanisms of neural networks (NNs), several groups have studied the number of linear regions l of piecewise linear (PwL) functions, generated by deep neural networks (DNN). In particular, they showed that l can grow expon ... Full text Open Access Cite

A connected auto-encoders based approach for image separation with side information: With applications to art investigation

Conference ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings · May 1, 2020 X-radiography is a widely used imaging technique in art investigation, whether to investigate the condition of a painting or provide insights into artists' techniques and working methods. In this paper, we propose a new architecture based on the use of 'co ... Full text Cite

Stable Phase Retrieval in Infinite Dimensions

Journal Article Foundations of Computational Mathematics · August 15, 2019 The problem of phase retrieval is to determine a signal f∈ H, with H a Hilbert space, from intensity measurements | F(ω) | , where F(ω) : = ⟨ f, φω⟩ are measurements of f with respect to a measurement system (φω)ω∈Ω⊂H. Although phase retrieval is always st ... Full text Cite

Artificial intelligence for art investigation: Meeting the challenge of separating x-ray images of the Ghent Altarpiece.

Journal Article Science advances · August 2019 X-ray images of polyptych wings, or other artworks painted on both sides of their support, contain in one image content from both paintings, making them difficult for experts to "read." To improve the utility of these x-ray images in studying these artwork ... Full text Open Access Cite

ariaDNE: A robustly implemented algorithm for Dirichlet energy of the normal

Journal Article Methods in Ecology and Evolution · April 1, 2019 Shape characterizers are metrics that quantify aspects of the overall geometry of a three-dimensional (3D) digital surface. When computed for biological objects, the values of a shape characterizer are largely independent of homology interpretations and of ... Full text Cite

LDMNet: Low Dimensional Manifold Regularized Neural Networks

Journal Article Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition · December 14, 2018 Deep neural networks have proved very successful on archetypal tasks for which large training sets are available, but when the training data are scarce, their performance suffers from overfitting. Many existing methods of reducing overfitting are data-inde ... Full text Cite

Directional Wavelet Bases Constructions with Dyadic Quincunx Subsampling

Journal Article Journal of Fourier Analysis and Applications · June 1, 2018 We construct directional wavelet systems that will enable building efficient signal representation schemes with good direction selectivity. In particular, we focus on wavelet bases with dyadic quincunx subsampling. In our previous work (Yin, in: Proceeding ... Full text Cite

Development and Assessment of Fully Automated and Globally Transitive Geometric Morphometric Methods, With Application to a Biological Comparative Dataset With High Interspecific Variation.

Journal Article Anat Rec (Hoboken) · April 2018 Automated geometric morphometric methods are promising tools for shape analysis in comparative biology, improving researchers' abilities to quantify variation extensively (by permitting more specimens to be analyzed) and intensively (by characterizing shap ... Full text Open Access Link to item Cite

Reconstructing Real-Valued Functions from Unsigned Coefficients with Respect to Wavelet and Other Frames

Journal Article Journal of Fourier Analysis and Applications · December 1, 2017 In this paper we consider the following problem of phase retrieval: given a collection of real-valued band-limited functions {ψλ}L2(Rd) that constitutes a semi-discrete frame, we ask whether any real-valued function f∈ L2(Rd) can be uniquely recovered from ... Full text Cite

Multi-Modal Dictionary Learning for Image Separation With Application in Art Investigation.

Conference IEEE transactions on image processing : a publication of the IEEE Signal Processing Society · February 2017 In support of art investigation, we propose a new source separation method that unmixes a single X-ray scan acquired from double-sided paintings. In this problem, the X-ray signals to be separated have similar morphological characteristics, which brings pr ... Full text Cite

A tale of two bases: Local-nonlocal regularization on image patches with convolution framelets

Journal Article SIAM Journal on Imaging Sciences · January 1, 2017 We propose an image representation scheme combining the local and nonlocal characterization of patches in an image. Our representation scheme can be shown to be equivalent to a tight frame constructed from convolving local bases (e.g., wavelet frames, disc ... Full text Cite

Cradle removal in X-ray images of panel paintings

Journal Article Image Processing On Line · January 1, 2017 We address the problem of mitigating the visually displeasing effects of cradling in X-ray images of panel paintings. The proposed algorithm consists of three stages. In the first stage the location of the cradling is detected semi-automatically and the gr ... Full text Cite

An iteratively reweighted least squares algorithm for sparse regularization

Chapter · January 1, 2017 We present a new algorithm and the corresponding convergence analysis for the regularization of linear inverse problems with sparsity constraints, applied to a new generalized sparsity promoting functional. The algorithm is based on the idea of iteratively ... Full text Cite

Optimizing Estimates of Instantaneous Heart Rate from Pulse Wave Signals with the Synchrosqueezing Transform.

Journal Article Methods of information in medicine · October 2016 BackgroundWith recent advances in sensor and computer technologies, the ability to monitor peripheral pulse activity is no longer limited to the laboratory and clinic. Now inexpensive sensors, which interface with smartphones or other computer-bas ... Full text Cite

Electrocardiographic J Wave and Cardiovascular Outcomes in the General Population (from the Atherosclerosis Risk In Communities Study).

Journal Article The American journal of cardiology · September 2016 The association between the J wave, a key component of the early repolarization pattern, and adverse cardiovascular outcomes remains unclear. Inconsistencies have stemmed from the different methods used to measure the J wave. We examined the association be ... Full text Open Access Cite

Removing cradle artifacts in X-ray images of paintings

Journal Article SIAM Journal on Imaging Sciences · August 30, 2016 We propose an algorithm that removes the visually unpleasant effects of cradling in X-ray images of panel paintings, with the goal of improving the X-ray image readability by art experts. The algorithm consists of three stages. In the first stage the locat ... Full text Cite

Sparsity-enforcing regularisation and ISTA revisited

Journal Article Inverse Problems · August 5, 2016 About two decades ago, the concept of sparsity emerged in different disciplines such as statistics, imaging, signal processing and inverse problems, and proved to be useful for several applications. Sparsity-enforcing constraints or penalties were then sho ... Full text Cite

X-ray image separation via coupled dictionary learning

Conference Proceedings - International Conference on Image Processing, ICIP · August 3, 2016 In support of art investigation, we propose a new source separation method that unmixes a single X-ray scan acquired from double-sided paintings. Unlike prior source separation methods, which are based on statistical or structural incoherence of the source ... Full text Cite

Object recognition in art drawings: Transfer of a neural network

Conference ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings · May 18, 2016 We consider the problem of recognizing objects in collections of art works, in view of automatically labeling, searching and organizing databases of art works. To avoid manually labelling objects, we introduce a framework for transferring a convolutional n ... Full text Cite

Adaptive data analysis: theory and applications.

Journal Article Philosophical transactions. Series A, Mathematical, physical, and engineering sciences · April 2016 Full text Cite

ConceFT: concentration of frequency and time via a multitapered synchrosqueezed transform.

Journal Article Philosophical transactions. Series A, Mathematical, physical, and engineering sciences · April 2016 A new method is proposed to determine the time-frequency content of time-dependent signals consisting of multiple oscillatory components, with time-varying amplitudes and instantaneous frequencies. Numerical experiments as well as a theoretical analysis ar ... Full text Cite

A Deterministic Analysis of Decimation for Sigma-Delta Quantization of Bandlimited Functions

Journal Article IEEE Signal Processing Letters · November 1, 2015 We study Sigma-Delta (Σ Δ) quantization of oversampled bandlimited functions. We prove that digitally integrating blocks of bits and then down-sampling, a process known as decimation, can efficiently encode the associated Σ Δ bit-stream. It allows a large ... Full text Cite

Digital image processing of the ghent altarpiece: Supporting the painting's study and conservation treatment

Journal Article IEEE Signal Processing Magazine · July 1, 2015 Hanging in the Saint Bavo Cathedral in Ghent, Belgium, is The Ghent Altarpiece, also known as The Adoration of the Mystic Lamb (see Figure 1). According to an inscription on the outer frames, it was painted by brothers Hubert and Jan van Eyck for Joos Vijd ... Full text Cite

Quantitative Canvas Weave Analysis Using 2-D Synchrosqueezed Transforms: Application of time-frequency analysis to art investigation

Journal Article Signal Processing Magazine, IEEE · July 2015 Quantitative canvas weave analysis has many applications in art investigations of paintings, including dating, forensics, and canvas rollmate identification. Traditionally, canvas analysis is based on X-radiographs. Prior to serving as a painting canvas, a ... Full text Open Access Cite

A Bayesian Nonparametric Approach to Image Super-Resolution.

Journal Article IEEE transactions on pattern analysis and machine intelligence · February 2015 Super-resolution methods form high-resolution images from low-resolution images. In this paper, we develop a new Bayesian nonparametric model for super-resolution. Our method uses a beta-Bernoulli process to learn a set of recurring visual patterns, called ... Full text Cite

Automated J wave detection from digital 12-lead electrocardiogram.

Journal Article Journal of electrocardiology · January 2015 In this report we provide a method for automated detection of J wave, defined as a notch or slur in the descending slope of the terminal positive wave of the QRS complex, using signal processing and functional data analysis techniques. Two different sets o ... Full text Cite

A new fully automated approach for aligning and comparing shapes.

Journal Article Anat Rec (Hoboken) · January 2015 Three-dimensional geometric morphometric (3DGM) methods for placing landmarks on digitized bones have become increasingly sophisticated in the last 20 years, including greater degrees of automation. One aspect shared by all 3DGM methods is that the researc ... Full text Open Access Link to item Cite

Evaluating physiological dynamics via synchrosqueezing: prediction of ventilator weaning.

Journal Article IEEE transactions on bio-medical engineering · March 2014 Oscillatory phenomena abound in many types of signals. Identifying the individual oscillatory components that constitute an observed biological signal leads to profound understanding about the biological system. The instantaneous frequency (IF), the amplit ... Full text Cite

Digital cradle removal in X-ray images of art paintings

Conference 2014 IEEE International Conference on Image Processing, ICIP 2014 · January 28, 2014 We introduce an algorithm that removes the deleterious effect of cradling on X-ray images of paintings on wooden panels. The algorithm consists of a three stage procedure. Firstly, the cradled regions are located automatically. The second step consists of ... Full text Cite

Why STEM (science, technology, engineering and mathematics)?

Conference Proceeding of the International Congress of Mathematicans, ICM 2014 · January 1, 2014 Cite

Bayesian crack detection in ultra high resolution multimodal images of paintings

Journal Article 2013 18th International Conference on Digital Signal Processing, DSP 2013 · December 6, 2013 The preservation of our cultural heritage is of paramount importance. Thanks to recent developments in digital acquisition techniques, powerful image analysis algorithms are developed which can be useful non-invasive tools to assist in the restoration and ... Full text Open Access Cite

Global seismic tomography with sparsity constraints: Comparison with smoothing and damping regularization

Journal Article Journal of Geophysical Research: Planets · September 1, 2013 We present a realistic application of an inversion scheme for global seismic tomography that uses as prior information the sparsity of a solution, defined as having few nonzero coefficients under the action of a linear transformation. In this paper, the sp ... Full text Cite

Restoration of X-ray fluorescence images of hidden paintings

Journal Article Signal Processing · March 1, 2013 This paper describes our methods for repairing and restoring images of hidden paintings (paintings that have been painted over and are now covered by a new surface painting) that have been obtained via noninvasive X-ray fluorescence imaging of their canvas ... Full text Cite

Crack detection and inpainting for virtual restoration of paintings: The case of the Ghent Altarpiece

Journal Article Signal Processing · March 1, 2013 Digital image processing is proving to be of great help in the analysis and documentation of our vast cultural heritage. In this paper, we present a new method for the virtual restoration of digitized paintings with special attention for the Ghent Altarpie ... Full text Cite

Conformal Wasserstein distance: II. Computational aspects and extensions

Journal Article Mathematics of Computation · January 17, 2013 This paper is a companion paper to [Yaron Lipman and Ingrid Daubechies, Conformal Wasserstein distances: Comparing surfaces in polynomial time, Adv. in Math. (ELS), 227 (2011), no. 3, 1047-1077, (2011)]. We provide numerical procedures and algorithms for c ... Full text Cite

Painting analysis using wavelets and probabilistic topic models

Journal Article 2013 IEEE International Conference on Image Processing, ICIP 2013 - Proceedings · January 1, 2013 In this paper, computer-based techniques for stylistic analysis of paintings are applied to the five panels of the 14th century Peruzzi Altarpiece by Giotto di Bondone. Features are extracted by combining a dual-tree complex wavelet transform with a hidden ... Full text Cite

Automated approaches to geometric morphometrics.

Conference AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY · January 1, 2013 Link to item Cite

Variational Bayesian learning of sparse representations and its application in functional neuroimaging

Journal Article Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) · November 30, 2012 Recent theoretical and experimental work in imaging neuroscience reveals that activations inferred from functional MRI data have sparse structure. We view sparse representation as a problem in Bayesian inference, following a machine learning approach, and ... Full text Cite

Capturing Ridge Functions in High Dimensions from Point Queries

Journal Article Constructive Approximation · April 1, 2012 Constructing a good approximation to a function of many variables suffers from the "curse of dimensionality". Namely, functions on ℝ N with smoothness of order s can in general be captured with accuracy at most O(n -s/N) using linear spaces or nonlinear ma ... Full text Cite

Virtual underpainting reconstruction from X-ray fluorescence imaging data

Journal Article European Signal Processing Conference · December 1, 2011 This paper describes our work on the problem of reconstructing the original visual appearance of underpaintings (paintings that have been painted over and are now covered by a new surface painting) from noninvasive X-ray fluorescence imaging data of their ... Cite

Spatiogram features to characterize pearls in paintings

Journal Article Proceedings - International Conference on Image Processing, ICIP · December 1, 2011 Objective characterization of jewels in paintings, especially pearls, has been a long lasting challenge for art historians. The way an artist painted pearls reflects his ability to observing nature and his knowledge of contemporary optical theory. Moreover ... Full text Cite

Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion

Journal Article Proceedings of SPIE - The International Society for Optical Engineering · November 2, 2011 Many flexible parameterizations exist to represent data on the sphere. In addition to the venerable spherical harmonics, we have the Slepian basis, harmonic splines, wavelets and wavelet-like Slepian frames. In this paper we focus on the latter two: spheri ... Full text Cite

Algorithms to automatically quantify the geometric similarity of anatomical surfaces

Journal Article Proceedings of the National Academy of Sciences · November 2011 http://arxiv.org/abs/1110.3649 ... Cite

Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity

Journal Article Geophysical Journal International · November 1, 2011 We propose a class of spherical wavelet bases for the analysis of geophysical models and for the tomographic inversion of global seismic data. Its multiresolution character allows for modelling with an effective spatial resolution that varies with position ... Full text Cite

Virtual restoration of the Ghent altarpiece using crack detection and inpainting

Journal Article Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) · September 2, 2011 In this paper, we present a new method for virtual restoration of digitized paintings, with the special focus on the Ghent Altarpiece (1432), one of Belgium's greatest masterpieces. The goal of the work is to remove cracks from the digitized painting there ... Full text Cite

Uncovering elements of style

Journal Article ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings · August 18, 2011 This paper relates the style of 16th century Flemish paintings by Goossen van der Weyden (GvdW) to the style of preliminary sketches or underpaintings made prior to executing the painting. Van der Weyden made underpaintings in markedly different styles for ... Full text Cite

Conformal Wasserstein distances: Comparing surfaces in polynomial time

Journal Article Advances in Mathematics · June 20, 2011 We present a constructive approach to surface comparison realizable by a polynomial-time algorithm. We determine the "similarity" of two given surfaces by solving a mass-transportation problem between their conformal densities. This mass transportation pro ... Full text Cite

Comparing Dirichlet normal surface energy of tooth crowns, a new technique of molar shape quantification for dietary inference, with previous methods in isolation and in combination.

Journal Article American journal of physical anthropology · June 2011 Inferred dietary preference is a major component of paleoecologies of extinct primates. Molar occlusal shape correlates with diet in living mammals, so teeth are a potentially useful structure from which to reconstruct diet in extinct taxa. We assess the e ... Full text Cite

Wavelets

Journal Article · April 29, 2011 Full text Cite

One or two frequencies? the synchrosqueezing answers

Journal Article Advances in Adaptive Data Analysis · April 1, 2011 The synchrosqueezed transform was proposed recently in [Daubechies et al. (2009)] as an alternative to the empirical mode decomposition (EMD) [Huang et al. (1998)], to decompose composite signals into a sum of "modes" that each have well-defined instantane ... Full text Cite

Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool

Journal Article Applied and Computational Harmonic Analysis · March 2011 Full text Cite

The continuous Procrustes distance between two surfaces

Journal Article Communications in Pure and Applied Mathematics · 2011 http://arxiv.org/abs/1106.4588 ... Cite

The work of Yves Meyer

Journal Article Proceedings of the International Congress of Mathematicians 2010, ICM 2010 · December 1, 2010 Yves Meyer has made numerous contributions to mathematics, several of which will be reviewed here, in particular in number theory, harmonic analysis and partial differential equations. His work in harmonic analysis led him naturally to take an interest in ... Cite

The golden ratio encoder

Journal Article IEEE Transactions on Information Theory · October 1, 2010 This paper proposes a novel Nyquist-rate analog-to-digital (A/D) conversion algorithm which achieves exponential accuracy in the bit-rate despite using imperfect components. The proposed algorithm is based on a robust implementation of a beta-encoder with ... Full text Cite

Symmetry factored embedding and distance

Journal Article ACM SIGGRAPH 2010 Papers, SIGGRAPH 2010 · July 26, 2010 We introduce the Symmetry Factored Embedding (SFE) and the Symmetry Factored Distance (SFD) as new tools to analyze and represent symmetries in a point set. The SFE provides new coordinates in which symmetry is "factored out," and the SFD is the Euclidean ... Full text Cite

Wavelets and applications

Journal Article · July 18, 2010 Cite

Nonlinear regularization techniques for seismic tomography

Journal Article Journal of Computational Physics · February 1, 2010 The effects of several nonlinear regularization techniques are discussed in the framework of 3D seismic tomography. Traditional, linear, ℓ2 penalties are compared to so-called sparsity promoting ℓ1 and ℓ0 penalties, and a total variation penalty. Which of ... Full text Cite

Herpesviruses carrying a Brainbow cassette reveal replication and expression of limited numbers of incoming genomes.

Journal Article Nature communications · January 2010 Whether all the infectious herpesvirus particles entering a cell are able to replicate and/or express their genomes is not known. Here, we developed a general method to determine the number of viral genomes expressed in an infected cell. We constructed and ... Full text Cite

Iteratively reweighted least squares minimization for sparse recovery

Journal Article Communications on Pure and Applied Mathematics · January 1, 2010 Under certain conditions (known as the restricted isometry property, or RIP) on the m × N matrix Φ (wherem < N), vectors x ∈ R{double-struck}N that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y:= Φx even though Φ- ... Full text Cite

Stylistic analysis of paintings using wavelets and machine learning

Journal Article European Signal Processing Conference · December 1, 2009 Wavelet transforms and machine learning tools can be used to assist art experts in the stylistic analysis of paintings. A dual-tree complex wavelet transform, Hidden Markov Tree modeling and Random Forest classifiers are used here for a stylistic analysis ... Cite

Sparse and stable Markowitz portfolios.

Journal Article Proceedings of the National Academy of Sciences of the United States of America · July 2009 We consider the problem of portfolio selection within the classical Markowitz mean-variance framework, reformulated as a constrained least-squares regression problem. We propose to add to the objective function a penalty proportional to the sum of the abso ... Full text Cite

Independent component analysis for brain fMRI does not select for independence.

Journal Article Proceedings of the National Academy of Sciences of the United States of America · June 2009 InfoMax and FastICA are the independent component analysis algorithms most used and apparently most effective for brain fMRI. We show that this is linked to their ability to handle effectively sparse components rather than independent components as such. T ... Full text Cite

Painless nonorthogonal expansions

Journal Article · January 10, 2009 In a Hilbert space {Hilbert space}, discrete families of vectors {hj} with the property that f = ΣJ hJ for every f in {Hilbert space} are considered. This expansion formula is obviously true if the family is an orthonorma1 basis of {Hilbert space}, ... Cite

The wavelet transform, time-frequency localization and signal analysis

Journal Article · January 10, 2009 Two different procedures are studied by which a rrequency analysis of a time-dependenl signal can be effected, locally in lime. The lirst procedure is the short-time or windowed Fourier transform, the second is the "wavelet transform," in which high freque ... Cite

Executive Summary of Dagstuhl Seminar on Structured Decompositions and Efficient Algorithms (08492)

Conference Dagstuhl Seminar Proceedings · January 1, 2009 New emerging technologies such as high-precision sensors or new MRI machines drive us towards a challenging quest for new, more effective, and more daring mathematical models and algorithms. Therefore, in the last few years researchers have started to inve ... Cite

Detection of forgery in paintings using supervised learning

Journal Article Proceedings - International Conference on Image Processing, ICIP · January 1, 2009 This paper examines whether machine learning and image analysis tools can be used to assist art experts in the authentication of unknown or disputed paintings. Recent work on this topic [1] has presented some promising initial results. Our reexamination of ... Full text Cite

Accelerated projected gradient method for linear inverse problems with sparsity constraints

Journal Article Journal of Fourier Analysis and Applications · December 1, 2008 Regularization of ill-posed linear inverse problems via ℓ1 penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an ℓ1 penalized functional is via an iterative soft-thresholding ... Full text Cite

Iteratively Re-weighted Least Squares minimization: Proof of faster than linear rate for sparse recovery

Journal Article CISS 2008, The 42nd Annual Conference on Information Sciences and Systems · September 24, 2008 Given an m × N matrix Φ, with m < N, the system of equations Φx = y is typically underdetermined and has infinitely many solutions. Various forms of optimization can extract a "best" solution. One of the oldest is to select the one with minimal l2 norm. It ... Full text Cite

Image processing for artist identification: Computerized analysis of Vincent van Gogh's painting brushstrokes

Journal Article IEEE Signal Processing Magazine · January 1, 2008 A description on the approaches to brushwork analysis and artist identification within the framework of data set is given. Image processing is now a reality in painting analysis as high resolution and richer data are also available. A summary on the result ... Full text Cite

On some iterative concepts for image restoration

Journal Article Advances in Imaging and Electron Physics · January 1, 2008 Several iterative strategies for solving inverse problems in the context of signal and image processing are discussed. Problems for which it is reasonable to assume that the solution has a sparse expansion with respect to a wavelet basis or frame are focus ... Full text Cite

Analysis of boosting algorithms using the smooth margin function

Journal Article Annals of Statistics · December 1, 2007 We introduce a useful tool for analyzing boosting algorithms called the "smooth margin function," a differentiable approximation of the usual margin for boosting algorithms. We present two boosting algorithms based on this smooth margin, "coordinate ascent ... Full text Cite

Single-bit oversampled A/D conversion with exponential accuracy in the bit rate

Journal Article IEEE Transactions on Information Theory · November 1, 2007 A scheme for simple oversampled analog-to-digital (A/D) conversion using single-bit quantization is presented. The scheme is based on recording positions of zero-crossings of the input signal added to a deterministic dither function. This information can b ... Full text Cite

A sparse spectral method for homogenization multiscale problems

Journal Article Multiscale Modeling and Simulation · August 1, 2007 We develop a new sparse spectral method, in which the fast Fourier transform (FFT) is replaced by RAℓSFA (randomized algorithm of sparse Fourier analysis); this is a sublinear randomized algorithm that takes time O(B log N) to recover a B-term Fourier repr ... Full text Cite

Tomographic inversion using ℓ1-norm regularization of wavelet coefficients

Journal Article Geophysical Journal International · July 1, 2007 We propose the use of ℓ1 regularization in a wavelet basis for the solution of linearized seismic tomography problems Am = d, allowing for the possibility of sharp discontinuities superimposed on a smoothly varying background. An iterative method is used t ... Full text Cite

Reply to Davey, Henriksen, Marković and Pratt [4]

Journal Article Notices of the American Mathematical Society · June 1, 2007 Cite

Iteratively solving linear inverse problems under general convex constraints

Journal Article Inverse Problems and Imaging · January 1, 2007 We consider linear inverse problems where the solution is assumed to fulfill some general homogeneous convex constraint. We develop an algorithm that amounts to a projected Landweber iteration and that provides and iterative approach to the solution of thi ... Full text Cite

Simpler alternatives to information theoretic similarity metrics for multimodal image alignment

Journal Article Proceedings - International Conference on Image Processing, ICIP · December 1, 2006 Mutual information (MI) based methods for image registration enjoy great experimental success and are becoming widely used. However, they impose a large computational burden that limits their use; many applications would benefit from a reduction of the com ... Full text Cite

Robust and practical analog-to-digital conversion with exponential precision

Journal Article IEEE Transactions on Information Theory · August 1, 2006 Beta-encoders with error correction were introduced by Daubechies, DeVore, Güntürk and Vaishampayan as an alternative to pulse-code modulation (PCM) for analog-to-digital conversion. An N -bit beta-encoder quantizes a real number by computing one of its N- ... Full text Cite

A/D conversion with imperfect quantizers

Journal Article IEEE Transactions on Information Theory · March 1, 2006 This paper analyzes mathematically the effect of quantizer threshold imperfection commonly encountered in the circuit implementation of analog-to-digltal (A/D) converters such as pulse code modulation (PCM) and sigma-delta (∑Δ) modulation. ∑Δ modulation, w ... Full text Cite

Theoretical and experimental analysis of a randomized algorithm for Sparse Fourier transform analysis

Journal Article Journal of Computational Physics · January 20, 2006 We analyze a sublinear RAℓSFA (randomized algorithm for Sparse Fourier analysis) that finds a near-optimal B-term Sparse representation R for a given discrete signal S of length N, in time and space poly (B, log(N)), following the approach given in [A.C. G ... Full text Cite

Variational Bayesian learning for wavelet independent component analysis

Journal Article AIP Conference Proceedings · November 23, 2005 In an exploratory approach to data analysis, it is often useful to consider the observations as generated from a set of latent generators or "sources" via a generally unknown mapping. For the noisy overcomplete case, where we have more sources than observa ... Full text Cite

Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising

Journal Article Applied and Computational Harmonic Analysis · July 1, 2005 Inspired by papers of Vese-Osher [Modeling textures with total variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and Osher-Solé-Vese [Image decomposition and restoration using total variation minimization an ... Full text Cite

Reconstructing Sunyaev-Zel'dovich clusters in future cosmic microwave background experiments

Journal Article Monthly Notices of the Royal Astronomical Society · May 1, 2005 We present a new method for component separation aimed at extracting Sunyaev-Zel'dovich (SZ) galaxy clusters from multifrequency maps of cosmic microwave background (CMB) experiments. This method is designed to recover non-Gaussian, spatially localized and ... Full text Cite

One-electron relativistic molecules with coulomb interaction

Journal Article · January 1, 2005 As an approximation to a relativistic one-electron molecule, we study the operator H=(-Δ+m2)1/2-e2 Z j|x-Rj|-1 with Zj0, e -2=137.04. H is bounded below if and only if e2 Z j>2/π, all j. Assuming this condition, the system is unstable when e2ΣZj>2/π in the ... Full text Cite

A detailed study of the attachment strategies of new autonomous systems in the as connectivity graph

Journal Article Internet Mathematics · January 1, 2005 The connectivity of the autonomous systems (ASs) in the Internet can be modeled as a time-evolving random graph, whose nodes represent ASs and whose edges represent direct connections between them. Even though this graph has some random aspects, its proper ... Full text Cite

The dynamics of AdaBoost: Cyclic behavior and convergence of margins

Journal Article Journal of Machine Learning Research · December 1, 2004 In order to study the convergence properties of the AdaBoost algorithm, we reduce AdaBoost to a nonlinear iterated map and study the evolution of its weight vectors. This dynamical systems approach allows us to understand AdaBoost's convergence properties ... Cite

An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

Journal Article Communications on Pure and Applied Mathematics · November 1, 2004 We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted of ℓP - penalties on the coefficients ... Full text Cite

Wavelet based image decomposition by variational functionals

Journal Article Proceedings of SPIE - The International Society for Optical Engineering · May 10, 2004 We discuss a wavelet based treatment of variational problems arising in the context of image processing, inspired by papers of Vese-Osher and Osher-Solé-Vese, in particular, we introduce a special class of variational functionals, that induce a decompositi ... Full text Cite

Normal multiresolution approximation of curves

Journal Article Constructive Approximation · January 1, 2004 A multiresolution analysis of a curve is normal if each wavelet detail vector with respect to a certain subdivision scheme lies in the local normal direction. In this paper we study properties such as regularity, convergence, and stability of a normal mult ... Full text Cite

Boosting based on a smooth margin

Journal Article Lecture Notes in Artificial Intelligence (Subseries of Lecture Notes in Computer Science) · January 1, 2004 We study two boosting algorithms, Coordinate Ascent Boosting and Approximate Coordinate Ascent Boosting, which are explicitly designed to produce maximum margins. To derive these algorithms, we introduce a smooth approximation of the margin that one can ma ... Full text Cite

Pairs of dual wavelet frames from any two refinable functions

Journal Article Constructive Approximation · January 1, 2004 Starting from any two compactly supported refutable functions in L 2 (R) with dilation factor d, we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L2 (R). Mo ... Full text Cite

On the dynamics of boosting

Conference Advances in Neural Information Processing Systems · January 1, 2004 In order to understand AdaBoost's dynamics, especially its ability to maximize margins, we derive an associated simplified nonlinear iterated map and analyze its behavior in low-dimensional cases. We find stable cycles for these cases, which can explicitly ... Cite

Harmonic analysis of the space BV

Journal Article Revista Matematica Iberoamericana · January 1, 2003 We establish new results on the space BV of functions with bounded variation. While it is well known that this space admits no unconditional basis, we show that it is "almost" characterized by wavelet expansions in the following sense: if a function f is i ... Full text Cite

Framelets: MRA-based constructions of wavelet frames

Journal Article Applied and Computational Harmonic Analysis · January 1, 2003 We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can ... Full text Cite

On the importance of combining wavelet-based nonlinear approximation with coding strategies

Journal Article IEEE Transactions on Information Theory · July 1, 2002 This paper provides a mathematical analysis of transform compression in its relationship to linear and nonlinear approximation theory. Contrasting linear and nonlinear approximation spaces, we show that there are interesting classes of functions/random pro ... Full text Cite

Adaptive Gabor transforms

Journal Article Applied and Computational Harmonic Analysis · July 1, 2002 We aim to provide time-frequency representations of a one-dimensional signal where the window is locally adapted to the signal, thus providing a better readability of the representation. © 2002 Elsevier Science (USA). All rights reserved. ... Full text Cite

The pros and cons of democracy

Journal Article IEEE Transactions on Information Theory · June 1, 2002 The concept of democracy was introduced in which the individual bits in a coarsely quantized representation of a signal were given equal weight in the approximation to the original signal. It was proved that such democratic representations could not achiev ... Full text Cite

The canonical dual frame of a wavelet frame

Journal Article Applied and Computational Harmonic Analysis · May 1, 2002 In this paper we show that there exist wavelet frames that have nice dual wavelet frames, but for which the canonical dual frame does not consist of wavelets, i.e., cannot be generated by the translates and dilates of a single function. © 2002 Elsevier Sci ... Full text Cite

Interpolation of bandlimited functions from quantized irregular samples

Conference Data Compression Conference Proceedings · January 1, 2002 The problem of reconstructing a π-bandlimited signal f from its quantized samples taken at an irregular sequence of points (tk)k∈ZZ arises in oversampled analog-to-digital conversion. The input signal can be reconstructed from the quantized samples (f(tk)) ... Full text Cite

Beta expansions: A new approach to digitally corrected A/D conversion

Journal Article Proceedings - IEEE International Symposium on Circuits and Systems · January 1, 2002 We introduce a new architecture for pipelined (and also algorithmic) A/D converters that give exponentially accurate conversion using inaccurate comparators. An error analysis of a sigma-delta converter with an imperfect comparator and a constant input rev ... Full text Cite

Commutation for irregular subdivision

Journal Article Constructive Approximation · December 1, 2001 We present a generalization of the commutation formula to irregular subdivision schemes and wavelets. We show how, in the noninterpolating case, the divided differences need to be adapted to the subdivision scheme. As an example we include the construction ... Full text Cite

Tree Approximation and Optimal Encoding

Journal Article Applied and Computational Harmonic Analysis · September 1, 2001 Tree approximation is a new form of nonlinear approximation which appears naturally in some applications such as image processing and adaptive numerical methods. It is somewhat more restrictive than the usual n-term approximation. We show that the restrict ... Full text Cite

The analysis and design of windowed fourier frame based multiple description source coding schemes

Journal Article IEEE Transactions on Information Theory · December 1, 2000 In this paper the windowed Fourier encoding-decoding scheme applied to the multiple description compression problem is analyzed. In the general case, four window functions are needed to define the encoder and decoder, although this number can be reduced to ... Full text Cite

Single-bit oversampled A/D conversion with exponential accuracy in the bit-rate

Journal Article Data Compression Conference Proceedings · January 1, 2000 We present a scheme for simple oversampled analog-to-digital conversion, with single bit quantization and exponential error decay in the bit-rate. The scheme is based on recording positions of zero-crossings of the input signal added to a deterministic dit ... Cite

Regularity of irregular subdivision

Journal Article Constructive Approximation · January 1, 1999 We study the smoothness of the limit function for one-dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced at every level can lie in irregularly spaced locations between old, adjacent grid points and not only midwa ... Full text Cite

Wavelets on irregular point sets

Journal Article Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences · January 1, 1999 In this article we review techniques for building and analysing wavelets on irregular point sets in one and two dimensions. We discuss current results on both the practical and theoretical sides. In particular, we focus on subdivision schemes and commutati ... Full text Cite

Preface

Journal Article Wavelet Analysis and Its Applications · December 1, 1998 Full text Cite

Data compression and harmonic analysis

Journal Article IEEE Transactions on Information Theory · December 1, 1998 In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon's R(D) theory in the case of Gaussian stationary processes, which says that transforming into a Fourier basis followed ... Full text Cite

Recent results in wavelet applications

Conference Proceedings of SPIE - The International Society for Optical Engineering · March 26, 1998 We present three recent developments in wavelets and subdivision: wavelet-type transforms that map integers to integers, with an application to lossless coding for images; rate-distortion bounds that realize the compression given by nonlinear approximation ... Full text Cite

Wavelet Transforms That Map Integers to Integers

Journal Article Applied and Computational Harmonic Analysis · January 1, 1998 Invertible wavelet transforms that map integers to integers have important applications in lossless coding. In this paper we present two approaches to build integer to integer wavelet transforms. The first approach is to adapt the precoder of Laroia et al. ... Full text Cite

Recent results in wavelet applications

Journal Article Journal of Electronic Imaging · January 1, 1998 We present three recent developments in wavelets and subdivision: wavelet-type transforms that map integers to integers, with an application to lossless coding for images; rate-distortion bounds that realize the compression given by nonlinear approximation ... Full text Cite

Factoring Wavelet Transforms into Lifting Steps

Journal Article Journal of Fourier Analysis and Applications · January 1, 1998 This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also k ... Full text Cite

Lossless image compression using integer to integer wavelet transforms

Journal Article IEEE International Conference on Image Processing · December 1, 1997 Invertible wavelet transforms that map integers to integers are important for lossless representations. In this paper, we present an approach to build integer to integer wavelet transforms based upon the idea of factoring wavelet transforms into lifting st ... Cite

From the Original Framer to Present-Day Time-Freuency and Time-Scale Frames

Journal Article Journal of Fourier Analysis and Applications · December 1, 1997 Cite

Differential reassignment

Journal Article IEEE Signal Processing Letters · December 1, 1997 A geometrical description is given for reassignment vector fields of spectrograms. These vector fields are shown to be connected with both an intrinsic phase characterization and a scalar potential. This allows for the generalization of the original reassi ... Full text Cite

On the approximation power of convolution-based least squares versus interpolation

Journal Article IEEE Transactions on Signal Processing · December 1, 1997 There are many signal processing tasks for which convolution-based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine-based methods. The coefficients of the correspon ... Full text Cite

Regularity of Refinable Function Vectors

Journal Article Journal of Fourier Analysis and Applications · January 1, 1997 We study the existence and regularity of compactly supported solutions φ = (φν)ν=0r- 1 of vector refinement equations. The space spanned by the translates of φν can only provide approximation order if the refinement mask P has certain particular factorizat ... Full text Cite

Where do wavelets come from? - a personal point of view

Journal Article Proceedings of the IEEE · April 1, 1996 The subject area of wavelets is connected to older ideas in many other fields, including pure and applied mathematics, physics, computer science and engineering. The rapid development of wavelet tools led to the application of wavelets as a standard part o ... Full text Cite

A new technique to estimate the regularity of refinable functions

Journal Article Revista Matematica Iberoamericana · January 1, 1996 We study the regularity of refinable functions by analyzing the spectral properties of special operators associated to the refinement equation; in particular, we use the Fredholm determinant theory to derive numerical estimates for the spectral radius of t ... Full text Cite

How smooth is the smoothest function in a given refinable space?

Journal Article Applied and Computational Harmonic Analysis · January 1, 1996 Full text Cite

How does truncation of the mask affect a refinable function?

Journal Article Constructive Approximation · September 1, 1995 If the mask of a refinable function has infinitely many coefficients, or if the coefficients are irrational, then it is often replaced by a finite mask with coefficients with terminating decimal expansions when it comes to applications. This note studies h ... Full text Cite

A celebration or women in mathematics

Journal Article Notices of the American Mathematical Society · January 1, 1995 Cite

Gabor Time-Frequency Lattices and the Wexler-Raz Identity

Journal Article Journal of Fourier Analysis and Applications · January 1, 1994 Gabor time-frequency lattices are sets of functions of the form (Formula presented.) generated from a given function (Formula presented.) by discrete translations in time and frequency. They are potential tools for the decomposition and handling of signals ... Full text Cite

Two Recent Results on Wavelets: Wavelet Bases for the Interval, and Biorthogonal Wavelets Diagonalizing the Derivative Operator

Journal Article Wavelet Analysis and Its Applications · January 1, 1994 The following two questions are often asked by researchers interested in applying wavelet bases to concrete numerical problems: 1) how does one adapt a wavelet basis on IR to a wavelet basis on an interval without terrible edge effects? 2) how does the wav ... Full text Cite

A decay theorem for refinable functions

Journal Article Applied Mathematics Letters · January 1, 1994 We show that a refinable function with absolutely summable mask cannot have exponential decay in both time and frequency. © 1994. ... Full text Cite

Wavelets on the interval and fast wavelet transforms

Journal Article Applied and Computational Harmonic Analysis · January 1, 1993 We discuss several constructions of orthonormal wavelet bases on the interval, and we introduce a new construction that avoids some of the disadvantages of earlier constructions. © 1993 Academic Press Inc. ... Full text Cite

Two Theorems on Lattice Expansions

Journal Article IEEE Transactions on Information Theory · January 1, 1993 It is shown that there is a trade-off between the smoothness and decay properties of the dual functions, occurring in the lattice expansion problem. More precisely, it is shown that if g and [formula Omitted] are dual, then 1) at least one of H1/2 Full text Cite

Sets of matrices all infinite products of which converge

Journal Article Linear Algebra and Its Applications · January 15, 1992 An infinite product ∏∞i=1Mi of matrices converges (on the right) if limi→∞ M1 ... Mi exists. A set ∑={Ai:i≥1}of n x n matrices is called an RCP set (right- convergent product set) if all infinite products with each element drawn from ∑ converge. Such sets ... Full text Cite

Biorthogonal bases of compactly supported wavelets

Journal Article Communications on Pure and Applied Mathematics · January 1, 1992 Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with ... Full text Cite

Wavelet transform image coding using trellis coded vector quantization

Conference ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings · January 1, 1992 A combination of trellis coded quantization (TCQ) and its vector alphabet generalization TCVQ is used to code the coefficients resulting from a biorthogonal wavelet transform in an image. TCVQ is a vector trellis coder with fixed rate, very good rate-disto ... Full text Cite

Image coding using wavelet transform.

Journal Article IEEE transactions on image processing : a publication of the IEEE Signal Processing Society · January 1992 A scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed. This method involves two steps. First, a wavelet transform used in order to obtain a set of biorthogonal subclasses of images: ... Full text Cite

Image coding using vector quantization in the wavelet transform domain

Journal Article ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings · December 1, 1990 A two-step scheme for image compression that takes into account psychovisual features in space and frequency domains is proposed. A wavelet transform is first used in order to obtain a set of orthonormal subclasses of images; the original image is decompos ... Cite

The Wavelet Transform, Time-Frequency Localization and Signal Analysis

Journal Article IEEE Transactions on Information Theory · January 1, 1990 Two different procedures are studied by which a frequency analysis of a time-dependent signal can be effected, locally in time. The first procedure is the short-time or windowed Fourier transform, the second is the “wavelet transform,” in which high freque ... Full text Cite

Wavelets: A tool for time-frequency analysis

Journal Article · December 1, 1989 Summary form only given. In the simplest case, a family wavelets is generated by dilating and translating a single function of one variable: ha,b(x) = |a|-1/2h (x-b/a). The parameters a and b may vary continuously, or be restricted to a discrete lattice of ... Cite

Wavelet transform, time-frequency localization and signal analysis

Journal Article · December 1, 1988 Summary form only given, as follows. Two different procedures are studied by which a frequency analysis of a time-dependent signal can be effected, locally in time. The first procedure is the short-time or windowed Fourier transform; the second is the wave ... Cite

Time-frequency localisation operators-a geometric phase space approach: II. The use of dilations

Journal Article Inverse Problems · December 1, 1988 Operators which localise both in time and frequency are constructed. They restrict to a finite time interval and cut off low as well as high frequencies (band-pass filters). Explicit expressions for eigenvalues and eigenfunctions (Laguerre functions) are g ... Full text Cite

Frames in the bargmann space of entire functions

Journal Article Communications on Pure and Applied Mathematics · January 1, 1988 We look at the decomposition of arbitrary f in L2(R) in terms of the family of functions φmn(x) = π−1/4exp{ − 1/2imnab + i max − 1/2(x − nb)2}, with a, b > 0. We derive bounds and explicit formulas for the minimal expansion coefficients in the case where a ... Full text Cite

Orthonormal bases of compactly supported wavelets

Journal Article Communications on Pure and Applied Mathematics · January 1, 1988 We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms ... Full text Cite

Time-Frequency Localization Operators: A Geometric Phase Space Approach

Journal Article IEEE Transactions on Information Theory · January 1, 1988 We define a set of operators which localize in both time and frequency. These operators are similar to but different from the low-pass time-limiting operators, the singular functions of which are the prolate spheroidal wave functions. Our construction diff ... Full text Cite

Wiener measures for path integrals with affine kinematic variables

Journal Article Journal of Mathematical Physics · January 1, 1987 The results obtained earlier have been generalized to show that the path integral for the affine coherent state matrix element of a unitary evolution operator exp(-iTH) can be written as a well-defined Wiener integral, involving Wiener measure on the Lobac ... Full text Cite

Painless nonorthogonal expansions

Journal Article Journal of Mathematical Physics · January 1, 1986 In a Hilbert spaced ℋ, discrete families of vectors {hj} with the property that f = ∑j〈j|f〉h j for every f in ℋ are considered. This expansion formula is obviously true if the family is an orthonormal basis of ℋ, but also can hold in situations where the h ... Full text Cite

Quantum-mechanical path integrals with Wiener measure for all polynomial Hamiltonians. II

Journal Article Journal of Mathematical Physics · January 1, 1985 The coherent-state representation of quantum-mechanical propagators as well-defined phase-space path integrals involving Wiener measure on continuous phase-space paths in the limit that the diffusion constant diverges is formulated and proved. This constru ... Full text Cite

One electron molecules with relativistic kinetic energy: Properties of the discrete spectrum

Journal Article Communications in Mathematical Physics · December 1, 1984 We discuss the discrete spectrum of the operator {Mathematical expression}. More specifically, we study 1) the behaviour of the eigenvalues when the internuclear distances contract, 2) the existence of a c-independent lower bound for HK(c)-mc2, 3) the nonr ... Full text Cite

Relativistic Molecules with Coulomb Interaction

Journal Article North-Holland Mathematics Studies · January 1, 1984 As an approximation to a relativistic one-electron molecule, we study the operator, with Zj≥ 0 for all j. H is bounded below iff for all j. Under this condition, we show that 1) The system is stable when the nuclear repulsion is taken into account, i.e., w ... Full text Cite

Quantum Mechanical Path Integrals with Wiener Measures for all Polynomial Hamiltonians

Journal Article Physical Review Letters · January 1, 1984 We construct arbitrary matrix elements of the quantum evolution operator for a wide class of self-adjoint canonical Hamiltonians, including those which are polynomial in the Heisenberg operators, as the limit of well defined path integrals involving Wiener ... Full text Cite

One-electron relativistic molecules with Coulomb interaction

Journal Article Communications in Mathematical Physics · December 1, 1983 As an approximation to a relativistic one-electron molecule, we study the operator {Mathematical expression} with Zj≧0, e-2=137.04. H is bounded below if and only if e2Zj≦2/π all j. Assuming this condition, the system is unstable when e2∑Zj>2/π in the sens ... Full text Cite

An uncertainty principle for fermions with generalized kinetic energy

Journal Article Communications in Mathematical Physics · December 1, 1983 We derive semiclassical upper bounds for the number of bound states and the sum of negative eigenvalues of the one-particle Hamiltonians h=f(-i∇)+V(x) acting on L2(ℝn). These bounds are then used to derive a lower bound on the kinetic energy {Mathematical ... Full text Cite

Measures for more quadratic path integrals

Journal Article Letters in Mathematical Physics · May 1, 1983 We show that the coherent state matrix elements of the quantum mechanical propagator for all quadratic Hamiltonians may be represented as the limit of path integrals with respect to appropriately modified Wiener measures as the associated diffusion constan ... Full text Cite

Measures for path integrals

Journal Article Physical Review Letters · January 1, 1982 By exploitation of the overcompleteness of coherent states expressions are presented for path integrals in terms of genuine (Wiener) path-space measures for driven harmonic oscillators which when projected onto the subspace spanned by coherent-state matrix ... Full text Cite

An integral transform related to quantization. II. Some mathematical properties

Journal Article Journal of Mathematical Physics · January 1, 1982 We study in more detail the mathematical properties of the integral transform relating the matrix elements between coherent states of a quantum operator to the corresponding classical function. Explicit families of Hilbert spaces are constructed between wh ... Full text Cite

Continuity statements and counterintuitive examples in connection with Weyl quantization

Journal Article Journal of Mathematical Physics · January 1, 1982 We use the properties of an integral transform relating a classical function f with the matrix elements between coherent states of its quantal counterpart Q f, to derive continuity properties of the Weyl transform from classes of distributions to classes o ... Full text Cite

Constructing measures for path integrals

Journal Article Journal of Mathematical Physics · January 1, 1981 The overcompleteness of the coherent states for the Heisenberg-Weyl group implies that many different integral kernels can be used to represent the same operator. Within such an equivalence class we construct an integral kernel to represent the quantum-mec ... Full text Cite

On the distributions corresponding to bounded operators in the Weyl quantization

Journal Article Communications in Mathematical Physics · October 1, 1980 Using properties of an integral transform giving directly the matrix elements of a quantum mechanical operator from the corresponding classical function, we restrict the class of distributions corresponding to bounded operators. As a consequence, we can ex ... Full text Cite

An integral transform related to quantization

Journal Article Journal of Mathematical Physics · January 1, 1979 We study in some detail the correspondence between a function f on phase space and the matrix elements (Qf)(a, b) of its quantized Q f between the coherent states |a< and |b<. It is an integral transform: Qf(a, b) = ∫{a, b |v} f(v) dv which resembles in ma ... Full text Cite

A mathematical condition for a sublattice of a propositional system to represent a physical subsystem, with a physical interpretation

Journal Article Letters in Mathematical Physics · January 1, 1979 We display three equivalent conditions for a sublattice, isomorphic to a P {Mathematical expression}, of the propositional system P(ℋ) of a quantum system to be the representation of a physical subsystem (see [1]). These conditions are valid for dim {Mathe ... Full text Cite

A characterization of subsystems in physics

Journal Article Letters in Mathematical Physics · January 1, 1979 Working within the framework of the propositional system formalism, we use a previous study [1] of the description of two independent physical systems as one big physical system to derive a characterization of a (non-interacting) physical subsystem. We dis ... Full text Cite

An application of hyperdifferential operators to holomorphic quantization

Journal Article Letters in Mathematical Physics · November 1, 1978 We use a hyperdifferential operator approach to study holomorphic quantization. We explicitly construct the Hilbert space operator which corresponds to a given holomorphic function. We further construct the adjoint and products of such operators and we dis ... Full text Cite

Foreword

Chapter Cite

Removal of Canvas Patterns in Digital Acquisitions of Paintings

Journal Article IEEE Transactions on Image Processing Open Access Cite