Geoffrey D de Villiers (M Inst P. C.Phys., FIMA, C.Math) is currently an honorary senior research fellow in the School of Electronic, Electrical and Systems Engineering at the University of Birmingham. He is an applied mathematician with over 30 years of experience in signal processing. His specialty is linear inverse problems with particular emphasis on singular-function methods and resolution enhancement. He has worked on a wide variety of practical inverse problems in photon correlation spectroscopy, radar, sonar, communications, seismology, antenna array design, broadband array processing, computational imaging and, currently, gravitational imaging.E. Roy Pike FRS has been Clerk-Maxwell Professor for Theoretical Physics at King's College London, and head of its School of Physical Sciences and Engineering, and is currently Emeritus Professor of Physics.

"This is a unique text - a labor of love, I suspect - pulling together for the first time the remarkably large array of mathematical and statistical techniques used for analysis of resolution in many systems of importance today – optical, acoustical, radar, etc. The final two chapters provide the perspective of practical resolution challenges in a multitude of situations, from light scattering and particle sizing to microscopy and tomography. These chapters will be of great value to those concerned with the fundamental limits in these fields…. I believe it will find widespread use and value."– Dr. Robert G.W. Brown, Chief Executive Officer, American Institute of Physics"a tour-de-force covering aspects of history, mathematical theory and practical applications. The authors provide a penetrating insight into the often confused topic of resolution and in doing offer a unifying approach to the subject that is applicable not only to traditional optical systems but also modern day, computer-based systems such as radar and RF communications." – Prof. Ian Proudler, Loughborough University"This is a book on the ultimate limits of resolution in imaging when digital image processing is considered. Therefore it is also a book on linear inverse problems, a basic tool in computational imaging. It provides a thorough account of this topic showing that solving an inverse problem is a mathematical game which requires a deep understanding of its physical foundations. The mix of physics and mathematics is a unique feature of this book which can be basic not only for PhD students but also for researchers in the area of computational imaging." – Mario Bertero, Professor, University of Geneva"This beautiful book can be read as a novel presenting carefully our quest to get more and more information from our observations and measurements. Its authors are particularly good at relating it." – Pierre C. Sabatier "a ‘must have’ for anyone interested in imaging and the spatial resolution of images. This book provides detailed and very readable account of resolution in imaging and organizes the recent history of the subject in excellent fashion.… I strongly recommend this well written and highly informative book and thank the authors, both experts in this field, for the work they put into it." – Michael A. Fiddy, Professor, University of North Carolina at Charlotte"This unique book provides all the necessary background in physics and instrumentation as well as on the mathematical aspects of inverse problems." – Prof. Christine de Mol, Universite Libre de Bruxelles"With much attention directed towards achieving super-resolution in a number of research areas at present, this book proves that the fundamental concept of resolution — a prerequisite for understanding the challenges of super-resolution — offers a rich topic of scientific discussion…. As inversion methods play a central role in the extraction of details from raw data and in the subsequent (and undesired) noise amplification, the text provides an in-depth presentation of linear inverse problems." –Nature Photonics (Sep 2016)

• Early concepts of resolution. Modern concepts of resolution. Elementary functional analysis. Resolution and ill-posedness.Optimisation. Deterministic methods for linear inverse problems. Convolution equations and deterministic spectral analysis. Statisticalmethods and resolution. Statistical spectral analysis. Resolution in optical microscopy. Some further optical applications. Appendixes.The origin of spectacles. Set theory and mappings. Methods for finding the eigenvalue and singular-value decompositions. Topologicalspaces. Basic probability theory

Geoffrey D de Villiers (M Inst P. C.Phys., FIMA, C.Math) is currently an honorary senior research fellow in the School of Electronic, Electrical and Systems Engineering at the University of Birmingham. He is an applied mathematician with over 30 years of experience in signal processing. His specialty is linear inverse problems with particular emphasis on singular-function methods and resolution enhancement. He has worked on a wide variety of practical inverse problems in photon correlation spectroscopy, radar, sonar, communications, seismology, antenna array design, broadband array processing, computational imaging and, currently, gravitational imaging.E. Roy Pike FRS has been Clerk-Maxwell Professor for Theoretical Physics at King's College London, and head of its School of Physical Sciences and Engineering, and is currently Emeritus Professor of Physics.

"This is a unique text - a labor of love, I suspect - pulling together for the first time the remarkably large array of mathematical and statistical techniques used for analysis of resolution in many systems of importance today – optical, acoustical, radar, etc. The final two chapters provide the perspective of practical resolution challenges in a multitude of situations, from light scattering and particle sizing to microscopy and tomography. These chapters will be of great value to those concerned with the fundamental limits in these fields…. I believe it will find widespread use and value."– Dr. Robert G.W. Brown, Chief Executive Officer, American Institute of Physics"a tour-de-force covering aspects of history, mathematical theory and practical applications. The authors provide a penetrating insight into the often confused topic of resolution and in doing offer a unifying approach to the subject that is applicable not only to traditional optical systems but also modern day, computer-based systems such as radar and RF communications." – Prof. Ian Proudler, Loughborough University"This is a book on the ultimate limits of resolution in imaging when digital image processing is considered. Therefore it is also a book on linear inverse problems, a basic tool in computational imaging. It provides a thorough account of this topic showing that solving an inverse problem is a mathematical game which requires a deep understanding of its physical foundations. The mix of physics and mathematics is a unique feature of this book which can be basic not only for PhD students but also for researchers in the area of computational imaging." – Mario Bertero, Professor, University of Geneva"This beautiful book can be read as a novel presenting carefully our quest to get more and more information from our observations and measurements. Its authors are particularly good at relating it." – Pierre C. Sabatier "a ‘must have’ for anyone interested in imaging and the spatial resolution of images. This book provides detailed and very readable account of resolution in imaging and organizes the recent history of the subject in excellent fashion.… I strongly recommend this well written and highly informative book and thank the authors, both experts in this field, for the work they put into it." – Michael A. Fiddy, Professor, University of North Carolina at Charlotte"This unique book provides all the necessary background in physics and instrumentation as well as on the mathematical aspects of inverse problems." – Prof. Christine de Mol, Universite Libre de Bruxelles"With much attention directed towards achieving super-resolution in a number of research areas at present, this book proves that the fundamental concept of resolution — a prerequisite for understanding the challenges of super-resolution — offers a rich topic of scientific discussion…. As inversion methods play a central role in the extraction of details from raw data and in the subsequent (and undesired) noise amplification, the text provides an in-depth presentation of linear inverse problems." –Nature Photonics (Sep 2016)

• Early concepts of resolution. Modern concepts of resolution. Elementary functional analysis. Resolution and ill-posedness.Optimisation. Deterministic methods for linear inverse problems. Convolution equations and deterministic spectral analysis. Statisticalmethods and resolution. Statistical spectral analysis. Resolution in optical microscopy. Some further optical applications. Appendixes.The origin of spectacles. Set theory and mappings. Methods for finding the eigenvalue and singular-value decompositions. Topologicalspaces. Basic probability theory

Geoffrey D de Villiers (M Inst P. C.Phys., FIMA, C.Math) is currently an honorary senior research fellow in the School of Electronic, Electrical and Systems Engineering at the University of Birmingham. He is an applied mathematician with over 30 years of experience in signal processing. His specialty is linear inverse problems with particular emphasis on singular-function methods and resolution enhancement. He has worked on a wide variety of practical inverse problems in photon correlation spectroscopy, radar, sonar, communications, seismology, antenna array design, broadband array processing, computational imaging and, currently, gravitational imaging.E. Roy Pike FRS has been Clerk-Maxwell Professor for Theoretical Physics at King's College London, and head of its School of Physical Sciences and Engineering, and is currently Emeritus Professor of Physics.

"This is a unique text - a labor of love, I suspect - pulling together for the first time the remarkably large array of mathematical and statistical techniques used for analysis of resolution in many systems of importance today – optical, acoustical, radar, etc. The final two chapters provide the perspective of practical resolution challenges in a multitude of situations, from light scattering and particle sizing to microscopy and tomography. These chapters will be of great value to those concerned with the fundamental limits in these fields…. I believe it will find widespread use and value."– Dr. Robert G.W. Brown, Chief Executive Officer, American Institute of Physics"a tour-de-force covering aspects of history, mathematical theory and practical applications. The authors provide a penetrating insight into the often confused topic of resolution and in doing offer a unifying approach to the subject that is applicable not only to traditional optical systems but also modern day, computer-based systems such as radar and RF communications." – Prof. Ian Proudler, Loughborough University"This is a book on the ultimate limits of resolution in imaging when digital image processing is considered. Therefore it is also a book on linear inverse problems, a basic tool in computational imaging. It provides a thorough account of this topic showing that solving an inverse problem is a mathematical game which requires a deep understanding of its physical foundations. The mix of physics and mathematics is a unique feature of this book which can be basic not only for PhD students but also for researchers in the area of computational imaging." – Mario Bertero, Professor, University of Geneva"This beautiful book can be read as a novel presenting carefully our quest to get more and more information from our observations and measurements. Its authors are particularly good at relating it." – Pierre C. Sabatier "a ‘must have’ for anyone interested in imaging and the spatial resolution of images. This book provides detailed and very readable account of resolution in imaging and organizes the recent history of the subject in excellent fashion.… I strongly recommend this well written and highly informative book and thank the authors, both experts in this field, for the work they put into it." – Michael A. Fiddy, Professor, University of North Carolina at Charlotte"This unique book provides all the necessary background in physics and instrumentation as well as on the mathematical aspects of inverse problems." – Prof. Christine de Mol, Universite Libre de Bruxelles"With much attention directed towards achieving super-resolution in a number of research areas at present, this book proves that the fundamental concept of resolution — a prerequisite for understanding the challenges of super-resolution — offers a rich topic of scientific discussion…. As inversion methods play a central role in the extraction of details from raw data and in the subsequent (and undesired) noise amplification, the text provides an in-depth presentation of linear inverse problems." –Nature Photonics (Sep 2016)

• Early concepts of resolution. Modern concepts of resolution. Elementary functional analysis. Resolution and ill-posedness.Optimisation. Deterministic methods for linear inverse problems. Convolution equations and deterministic spectral analysis. Statisticalmethods and resolution. Statistical spectral analysis. Resolution in optical microscopy. Some further optical applications. Appendixes.The origin of spectacles. Set theory and mappings. Methods for finding the eigenvalue and singular-value decompositions. Topologicalspaces. Basic probability theory