Epstein, Charles L.
Overview
Works:  18 works in 80 publications in 1 language and 2,229 library holdings 

Roles:  Author 
Classifications:  RC78.7.D53, 515 
Publication Timeline
.
Most widely held works by
Charles L Epstein
Introduction to the mathematics of medical imaging by
Charles L Epstein(
Book
)
28 editions published between 2003 and 2008 in English and held by 365 WorldCat member libraries worldwide
"At the heart of every medical imaging technology is a sophisticated mathematical model of the measurement process and an algorithm to reconstruct an image from the measured data. This book provides a firm foundation in the mathematical tools used to model the measurements and derive the reconstruction algorithms used in most imaging modalities in current use. In the process, it also covers many important analytic concepts, as well as techniques used in Fourier analysis, integral equations, sampling theory, and noise analysis." "This text uses Xray computed tomography as a "pedagogical machine" to illustrate important ideas and incorporates extensive discussions of background material making the more advanced mathematical topics accessible to readers with a less formal mathematical education. The mathematical concepts are illuminated with over 200 illustrations and numerous exercises." "New to the second edition are a chapter on magnetic resonance imaging (MRI), a revised section on the relationship between the continuum and discrete Fourier transforms, a new section on Grangreat's formula, an improved description of the gridding method, and a new section on noise analysis in MRI." "The book is appropriate for one or twosemester courses at the advanced undergraduate or beginning graduate level on the mathematical foundations of modern medical imaging technologies. The text assumes an understanding of calculus, linear algebra, and basic mathematical analysis."Jacket
28 editions published between 2003 and 2008 in English and held by 365 WorldCat member libraries worldwide
"At the heart of every medical imaging technology is a sophisticated mathematical model of the measurement process and an algorithm to reconstruct an image from the measured data. This book provides a firm foundation in the mathematical tools used to model the measurements and derive the reconstruction algorithms used in most imaging modalities in current use. In the process, it also covers many important analytic concepts, as well as techniques used in Fourier analysis, integral equations, sampling theory, and noise analysis." "This text uses Xray computed tomography as a "pedagogical machine" to illustrate important ideas and incorporates extensive discussions of background material making the more advanced mathematical topics accessible to readers with a less formal mathematical education. The mathematical concepts are illuminated with over 200 illustrations and numerous exercises." "New to the second edition are a chapter on magnetic resonance imaging (MRI), a revised section on the relationship between the continuum and discrete Fourier transforms, a new section on Grangreat's formula, an improved description of the gridding method, and a new section on noise analysis in MRI." "The book is appropriate for one or twosemester courses at the advanced undergraduate or beginning graduate level on the mathematical foundations of modern medical imaging technologies. The text assumes an understanding of calculus, linear algebra, and basic mathematical analysis."Jacket
Degenerate diffusion operators arising in population biology by
Charles L Epstein(
Book
)
14 editions published in 2013 in English and held by 236 WorldCat member libraries worldwide
"This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process. Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hèolder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right halfplane. Epstein and Mazzeo also demonstrate precise asymptotic results for the longtime behavior of solutions to both the forward and backward Kolmogorov equations."
14 editions published in 2013 in English and held by 236 WorldCat member libraries worldwide
"This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process. Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hèolder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right halfplane. Epstein and Mazzeo also demonstrate precise asymptotic results for the longtime behavior of solutions to both the forward and backward Kolmogorov equations."
The spectral theory of geometrically periodic hyperbolic 3manifolds by
Charles L Epstein(
Book
)
18 editions published between 1983 and 1985 in English and held by 236 WorldCat member libraries worldwide
18 editions published between 1983 and 1985 in English and held by 236 WorldCat member libraries worldwide
 Gulfwar aftermath. Building up to a construction bonanza.ʹ A new world order for oil(
)
1 edition published in 1991 in English and held by 3 WorldCat member libraries worldwide
The first of these two articles deals with the task of rebuilding Kuwait after the Gulf War with Iraq. The size and award of contracts to Western companies and the issue of war reparations are discussed. The second article examines the implications for world oil markets of the war in the Gulf and its aftermath and discusses the measures for reducing oil consumption and dependance from OPEC oil.SCAD summary
1 edition published in 1991 in English and held by 3 WorldCat member libraries worldwide
The first of these two articles deals with the task of rebuilding Kuwait after the Gulf War with Iraq. The size and award of contracts to Western companies and the issue of war reparations are discussed. The second article examines the implications for world oil markets of the war in the Gulf and its aftermath and discusses the measures for reducing oil consumption and dependance from OPEC oil.SCAD summary
A relative index on the space of embeddable CRstructures by
Charles L Epstein(
Book
)
3 editions published in 1995 in English and held by 3 WorldCat member libraries worldwide
3 editions published in 1995 in English and held by 3 WorldCat member libraries worldwide
A stable manifold theorem for the curve shortening equation by
Charles L Epstein(
Book
)
1 edition published in 1986 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 1986 in English and held by 2 WorldCat member libraries worldwide
Introduction to the mathematics of medical by
Charles L Epstein(
Book
)
1 edition published in 2009 in English and held by 1 WorldCat member library worldwide
1 edition published in 2009 in English and held by 1 WorldCat member library worldwide
Optimal neural codes for natural stimuli by Zhuo Wang(
)
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
In terms of objective functions, most common models rely on information theoretic measures, whereas alternative formulations propose incorporating downstream decoding performance. We systematically evaluate different optimality criteria based upon the Lp reconstruction error of the maximum likelihood decoder. This parametric family of optimal criteria includes special cases such as the information maximization criterion and the mean squared loss minimization of decoding error. We analytically derive the optimal tuning curve of a single neuron in terms of the reconstruction error norm p to encode natural stimuli with an arbitrary input distribution
1 edition published in 2016 in English and held by 1 WorldCat member library worldwide
In terms of objective functions, most common models rely on information theoretic measures, whereas alternative formulations propose incorporating downstream decoding performance. We systematically evaluate different optimality criteria based upon the Lp reconstruction error of the maximum likelihood decoder. This parametric family of optimal criteria includes special cases such as the information maximization criterion and the mean squared loss minimization of decoding error. We analytically derive the optimal tuning curve of a single neuron in terms of the reconstruction error norm p to encode natural stimuli with an arbitrary input distribution
Gulfwar aftermath : building up to a construction bonanza(
)
1 edition published in 1991 in English and held by 1 WorldCat member library worldwide
1 edition published in 1991 in English and held by 1 WorldCat member library worldwide
Degenerate Diffusion Operators Arising in Population Biology (Annals of Mathematics Studies) by
Charles L Epstein(
Book
)
1 edition published in 2013 in Undetermined and held by 1 WorldCat member library worldwide
1 edition published in 2013 in Undetermined and held by 1 WorldCat member library worldwide
Genomic imprinting : a lasting memory of your parents(
Visual
)
1 edition published in 1992 in English and held by 1 WorldCat member library worldwide
1 edition published in 1992 in English and held by 1 WorldCat member library worldwide
CT CISM team response to Newtown, CT  December 14, 2012 by
Charles L Epstein(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
Advanced real analysis by
Anthony W Knapp(
Book
)
3 editions published in 2005 in English and held by 1 WorldCat member library worldwide
Advanced€Real Analysis systematically develops the concepts and tools that are vital to every mathematician, whether pure or applied, aspiring or established. This work presents a comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics. Key topics and features: . Early chapters treat the fundamentals of real variables, the theory of Fourier series for the Riemann integral, and the theoretical underpinnings of multivariable calculus and differential equations . Subsequent chapters develop measure theory, pointset topology, Fourier series for the Lebesgue integral, and the basics of Banach and Hilbert spaces . Later chapters provide a higherlevel view of the interaction between real analysis and algebra, including functional analysis, partial differential equations, and further topics in Fourier analysis . Throughout the text are problems that develop and illuminate aspects of the theory of probability . Includes many examples and hundreds of problems, and a chapter gives hints or complete solutions for most of the problems It requires of the reader only familiarity with some linear algebra and real variable theory, a few weeks' worth of group theory, and an acquaintance with proofs. Because it focuses on what every young mathematician needs to know about real analysis, this book is ideal both as a course text and for selfstudy, especially for graduate students preparing for qualifying examinations. Its scope and unique approach will appeal to instructors and professors in nearly all areas of pure mathematics, as well as applied mathematicians working in analytic areas such as statistics, math physics, and applied differential equations. Indeed, the clarity and breadth of Advanced Real Analysis make it a welcome addition to the personal library of every mathematician. TOC:Preface * Theory of calculus in one real variable * Metric spaces * Theory of differential calculus in several variables * Theory of ordinary differential equations and systems * Riemann integration in several variables * Abstract measure theory and Lebesgue measure * Measure theory for Euclidean space * Fourier transform in R^N * L^p spaces * Further topics in abstract measure theory * Topological spaces * Integration on locally compact spaces * Haar measure * Hilbert and Banach spaces * Distributions and their application to PDEs * References * Index
3 editions published in 2005 in English and held by 1 WorldCat member library worldwide
Advanced€Real Analysis systematically develops the concepts and tools that are vital to every mathematician, whether pure or applied, aspiring or established. This work presents a comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics. Key topics and features: . Early chapters treat the fundamentals of real variables, the theory of Fourier series for the Riemann integral, and the theoretical underpinnings of multivariable calculus and differential equations . Subsequent chapters develop measure theory, pointset topology, Fourier series for the Lebesgue integral, and the basics of Banach and Hilbert spaces . Later chapters provide a higherlevel view of the interaction between real analysis and algebra, including functional analysis, partial differential equations, and further topics in Fourier analysis . Throughout the text are problems that develop and illuminate aspects of the theory of probability . Includes many examples and hundreds of problems, and a chapter gives hints or complete solutions for most of the problems It requires of the reader only familiarity with some linear algebra and real variable theory, a few weeks' worth of group theory, and an acquaintance with proofs. Because it focuses on what every young mathematician needs to know about real analysis, this book is ideal both as a course text and for selfstudy, especially for graduate students preparing for qualifying examinations. Its scope and unique approach will appeal to instructors and professors in nearly all areas of pure mathematics, as well as applied mathematicians working in analytic areas such as statistics, math physics, and applied differential equations. Indeed, the clarity and breadth of Advanced Real Analysis make it a welcome addition to the personal library of every mathematician. TOC:Preface * Theory of calculus in one real variable * Metric spaces * Theory of differential calculus in several variables * Theory of ordinary differential equations and systems * Riemann integration in several variables * Abstract measure theory and Lebesgue measure * Measure theory for Euclidean space * Fourier transform in R^N * L^p spaces * Further topics in abstract measure theory * Topological spaces * Integration on locally compact spaces * Haar measure * Hilbert and Banach spaces * Distributions and their application to PDEs * References * Index
Mathematics of medical imaging by
Charles L Epstein(
Book
)
1 edition published in 2003 in English and held by 1 WorldCat member library worldwide
1 edition published in 2003 in English and held by 1 WorldCat member library worldwide
Magnetic resonance imaging of shortT₂ tissues with applications for quantifying cortical bone water and myelin by Cheng Li(
Book
)
1 edition published in 2014 in English and held by 0 WorldCat member libraries worldwide
The human body contains a variety of tissue species with short T₂ ranging from a few microseconds to hundreds of microseconds. Detection and quantification of these short T₂ species is of considerable clinical and scientific interest. Cortical bone water and myelin are two of the most important tissue constituents. Quantification of cortical bone water concentration allows for indirect estimation of bone pore volume and noninvasive assessment of bone quality. Myelin is essential for the proper functioning of the central nervous system (CNS). Direct assessment of myelin would reveal CNS abnormalities and enhance our understanding of neurological diseases. However, conventional MRI with echo times of several milliseconds or longer is unable to detect these shortlived MR signals. Recent advances in MRI technology and hardware have enabled development of a number of short T₂ imaging techniques, key among which are ultrashort echo time (UTE) imaging, zero echo time (ZTE) imaging, and sweep imaging with Fourier transform (SWIFT). While these pulse sequences are able to detect short T₂ species, they still suffer from signal interference between different T₂ tissue constituents, image artifacts and excessive scan time. These are primary technical hurdles for application to wholebody clinical scanners. In this thesis research, new MRI techniques for improving short T₂ tissue imaging have been developed to address these challenges with a focus on direct detection and quantification of cortical bone water and myelin on a clinical MRI scanner. The first focus of this research was to optimize long T₂ suppression in UTE imaging. Saturation and adiabatic RF pulses were designed to achieve maximum long T₂ suppression while maximizing the signal from short T₂ species. The imaging protocols were optimized by Bloch equation simulations and were validated using phantom and in vivo experiments. The results show excellent short T₂ contrast with these optimized pulse sequences. The problem of blurring artifacts resulting from the inhomogeneous excitation profile of the rectangular pulses in ZTE imaging was addressed. The proposed approach involves quadratic phasemodulated RF excitation and iterative solution of an inverse problem formulated from the signal model of ZTE imaging and is shown to effectively remove the image artifacts. Subsequently image acquisition efficiency was improved in order to attain clinicallyfeasible scan times. To accelerate the acquisition speed in UTE and ZTE imaging, compressed sensing was applied with a hybrid 3D UTE sequence. Further, the pulse sequence and reconstruction procedure were modified to enable anisotropic fieldofview shape conforming to the geometry of the elongated imaged object. These enhanced acquisition techniques were applied to the detection and quantification of cortical bone water. A new biomarker, the suppression ratio (a ratio image derived from two UTE images, one without and the other with long T₂ suppression), was conceived as a surrogate measure of cortical bone porosity. Experimental data suggest the suppression ratio may be a more direct measure of porosity than previously measured total bone water concentration. Lastly, the feasibility of directly detecting and quantifying spatiallyresolved myelin concentration with a clinical imager was explored, both theoretically and experimentally. Bloch equation simulations were conducted to investigate the intrinsic image resolution and the fraction of detectable myelin signal under current scanner hardware constraints. The feasibility of quantitative ZTE imaging of myelin extract and lamb spinal cord at 3T was demonstrated. The technological advances achieved in this dissertation research may facilitate translation of short T₂ MRI methods from the laboratory to the clinic
1 edition published in 2014 in English and held by 0 WorldCat member libraries worldwide
The human body contains a variety of tissue species with short T₂ ranging from a few microseconds to hundreds of microseconds. Detection and quantification of these short T₂ species is of considerable clinical and scientific interest. Cortical bone water and myelin are two of the most important tissue constituents. Quantification of cortical bone water concentration allows for indirect estimation of bone pore volume and noninvasive assessment of bone quality. Myelin is essential for the proper functioning of the central nervous system (CNS). Direct assessment of myelin would reveal CNS abnormalities and enhance our understanding of neurological diseases. However, conventional MRI with echo times of several milliseconds or longer is unable to detect these shortlived MR signals. Recent advances in MRI technology and hardware have enabled development of a number of short T₂ imaging techniques, key among which are ultrashort echo time (UTE) imaging, zero echo time (ZTE) imaging, and sweep imaging with Fourier transform (SWIFT). While these pulse sequences are able to detect short T₂ species, they still suffer from signal interference between different T₂ tissue constituents, image artifacts and excessive scan time. These are primary technical hurdles for application to wholebody clinical scanners. In this thesis research, new MRI techniques for improving short T₂ tissue imaging have been developed to address these challenges with a focus on direct detection and quantification of cortical bone water and myelin on a clinical MRI scanner. The first focus of this research was to optimize long T₂ suppression in UTE imaging. Saturation and adiabatic RF pulses were designed to achieve maximum long T₂ suppression while maximizing the signal from short T₂ species. The imaging protocols were optimized by Bloch equation simulations and were validated using phantom and in vivo experiments. The results show excellent short T₂ contrast with these optimized pulse sequences. The problem of blurring artifacts resulting from the inhomogeneous excitation profile of the rectangular pulses in ZTE imaging was addressed. The proposed approach involves quadratic phasemodulated RF excitation and iterative solution of an inverse problem formulated from the signal model of ZTE imaging and is shown to effectively remove the image artifacts. Subsequently image acquisition efficiency was improved in order to attain clinicallyfeasible scan times. To accelerate the acquisition speed in UTE and ZTE imaging, compressed sensing was applied with a hybrid 3D UTE sequence. Further, the pulse sequence and reconstruction procedure were modified to enable anisotropic fieldofview shape conforming to the geometry of the elongated imaged object. These enhanced acquisition techniques were applied to the detection and quantification of cortical bone water. A new biomarker, the suppression ratio (a ratio image derived from two UTE images, one without and the other with long T₂ suppression), was conceived as a surrogate measure of cortical bone porosity. Experimental data suggest the suppression ratio may be a more direct measure of porosity than previously measured total bone water concentration. Lastly, the feasibility of directly detecting and quantifying spatiallyresolved myelin concentration with a clinical imager was explored, both theoretically and experimentally. Bloch equation simulations were conducted to investigate the intrinsic image resolution and the fraction of detectable myelin signal under current scanner hardware constraints. The feasibility of quantitative ZTE imaging of myelin extract and lamb spinal cord at 3T was demonstrated. The technological advances achieved in this dissertation research may facilitate translation of short T₂ MRI methods from the laboratory to the clinic
The spectral theory of geometrically periodic hyperbolic manifolds by
Charles L Epstein(
Book
)
1 edition published in 1985 in Undetermined and held by 0 WorldCat member libraries worldwide
1 edition published in 1985 in Undetermined and held by 0 WorldCat member libraries worldwide
Seizures and pseudoseizures(
Visual
)
1 edition published in 1992 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 1992 in English and held by 0 WorldCat member libraries worldwide
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Diagnostic imagingMathematics Differential equations, Partial Distribution (Probability theory) Elliptic operators Fourier analysis Functional analysis Genomic imprinting Global analysis (Mathematics) Manifolds (Mathematics) Markov processes Mathematical analysis Mathematics Medical physicsMathematics Neurology Population biologyMathematical models Probabilities Searches and seizures Spectral theory (Mathematics) Threemanifolds (Topology)
Alternative Names
Charles Epstein Amerikaans wiskundige
Charles Epstein matemático estadounidense
Charles Epstein mathématicien américain, a professé à l'université de Pennsylvanie, Philadelphie.
Epstein, C. 1957
Epstein, C. L. 1957
Epstein, Charles 1957
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