A chronology of interpolation: from ancient astronomy to modern signal and image processing
Proceedings of the IEEEPublished 1 March 2002Open access
Erik Meijering
Citations649
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TL;DR
A chronological overview of the developments in interpolation theory, from the earliest times to the present date, brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective.
Abstract
This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation.
Keywords
Computer Science
Bell System Technical JournalA Mathematical Theory of Communication
80,675 Citations1948Claude E. Shannon
Elsevier eBooksA Wavelet Tour of Signal Processing
16,401 Citations1999Stéphane Mallat
An introduction to a Transient World and an Approximation Tour of Wavelet Packet and Local Cosine Bases.
Computers in PhysicsTen Lectures on Wavelets
15,386 Citations1992Ingrid Daubechies, Christopher Heil
This paper presents a meta-analyses of the wavelet transforms of Coxeter’s inequality and its applications to multiresolutional analysis and orthonormal bases.
Methods of Mathematical Physics
8,096 Citations1989R. Courant, David Hilbert
Proceedings of the IEEEOn the use of windows for harmonic analysis with the discrete Fourier transform
7,196 Citations1978F.J. Harris
A comprehensive catalog of data windows along with their significant performance parameters from which the different windows can be compared is included, and an example demonstrates the use and value of windows to resolve closely spaced harmonic signals characterized by large differences in amplitude.
Proceedings of the IRECommunication in the Presence of Noise
5,962 Citations1949Claude E. Shannon
A method is developed for representing any communication system geometrically and a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect.
American Journal of PhysicsThe Fourier Transform and Its Applications
5,455 Citations1966R. N. Bracewell, Peter B. Kahn
Journal of the Royal Statistical Society Series A (Statistics in Society)Statistics for Spatial Data.
5,160 Citations1993Mike Rees, Noel Cressie
Journal of the American Statistical AssociationSpline Models for Observational Data.
5,025 Citations1991Hans‐Georg Müller, Grace Wahba
Proceedings of the IEEECommunication In The Presence Of Noise
4,866 Citations1998Claude E. Shannon
A method is developed for representing any communication system geometrically and a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect.
Image Processing, Analysis and Machine Vision
3,719 Citations1993Milan Sonka, Václav Hlaváč +1 more
The digitized image and its properties are studied, including shape representation and description, and linear discrete image transforms, and texture analysis.
IEEE Transactions on Image ProcessingA pyramid approach to subpixel registration based on intensity
3,275 Citations1998P. Thévenaz, Urs E. Ruttimann +1 more
An automatic subpixel registration algorithm that minimizes the mean square intensity difference between a reference and a test data set, which can be either images (two-dimensional) or volumes (three-dimensional).
Transactions of the American Institute of Electrical EngineersCertain Topics in Telegraph Transmission Theory
3,006 Citations1928H. Nyquist
Journal of the American Statistical AssociationGeostatistics: Modeling Spatial Uncertainty
2,919 Citations2000Michael L. Stein, Jean‐Paul Chilès +1 more
Mathematics of ComputationSpline Functions: Basic Theory.
2,493 Citations1982Philip W. Smith, Larry L. Schumaker
The material covered provides the reader with the necessary tools for understanding the many applications of splines in such diverse areas as approximation theory, computer-aided geometric design, curve and surface design and fitting, image processing, numerical solution of differential equations, and increasingly in business and the biosciences.
Journal of the Southern African Institute of Mining and MetallurgyA statistical approach to some basic mine valuation problems on the Witwatersrand
2,443 Citations1951D.G. Krige
Digital Image Processing
2,398 Citations1991K.-D. Brinkmann
Remote sensing, models, and methods for image processing
2,147 Citations1997Robert A. Schowengerdt
The Nature of Remote Sensing: Introduction, Sensor Characteristics and Spectral Stastistics, and Spatial Transforms: Introduction.
IEEE Transactions on Image ProcessingNew edge-directed interpolation
2,012 Citations2001Xin Li, M.T. Orchard
Simulation results demonstrate that the new interpolation algorithm substantially improves the subjective quality of the interpolated images over conventional linear interpolation.
Introduction To Numerical Analysis
1,890 Citations1956F. B. Hildebrand
Proceedings of the IEEEThe Shannon sampling theorem—Its various extensions and applications: A tutorial review
1,701 Citations1977Abdul J. Jerri
This review paper will attempt to present the various contributions made for the sampling theorems with the necessary mathematical details to make it self-contained, including specific error bounds for the truncation, aliasing, jitter and parts of various other errors.
IEEE Signal Processing MagazineSplines: a perfect fit for signal and image processing
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Elsevier eBooksHandbook of Image and Video Processing
1,593 Citations2005Alan C. Bovik
The Handbook of Image and Video Processing contains a comprehensive and highly accessible presentation of all essential mathematics, techniques, and algorithms for every type of image and video processing used by scientists and engineers.
Numerical methods for scientists and engineers
1,529 Citations1962R. W. Hamming
Proceedings of the IEEESampling-50 years after Shannon
1,351 Citations2000Michaël Unser
The standard sampling paradigm is extended for a presentation of functions in the more general class of "shift-in-variant" function spaces, including splines and wavelets, and variations of sampling that can be understood from the same unifying perspective are reviewed.
IEEE Transactions on Acoustics Speech and Signal ProcessingCubic splines for image interpolation and digital filtering
1,262 Citations1978Hsieh S. Hou, Harry Andrews
Applications to image and signal processing include interpolation, smoothing, filtering, enlargement, and reduction, and experimental results are presented for illustrative purposes in two-dimensional image format.
Mathematical Thought from Ancient to Modern Times
1,186 Citations1990Morris Kline
Cambridge University Press eBooksMethods of Mathematical Physics
1,169 Citations1999Harold Jeffreys, Bertha Swirles
An Introduction to Splines for Use in Computer Graphics and Geometric Modeling
1,147 Citations1987Richard H. Bartels, John C. Beatty +1 more
A simple Approximation Technique - Uniform Cubic B-splines and an Explicity Formulation for Cubic Beta-splines are presented.
European Journal of Operational ResearchThe Fourier transform and its applications
1,147 Citations1980M. L. J. Hautus
Algorithms for Graphics and Image Processing
1,134 Citations1982Theodosios Pavlidis
This chapter discusses Graphics, Image Processing, and Pattern Recognition, and the Reconstruction techniques used in this program, as well as some of the problems faced in implementing this program.
IEEE Signal Processing MagazineSplitting the unit delay [FIR/all pass filters design]
1,121 Citations1996T.I. Laakso, Vesa Välimäki +2 more
This work presents a comprehensive review of FIR and allpass filter design techniques for bandlimited approximation of a fractional digital delay, focusing on simple and efficient methods that are well suited for fast coefficient update or continuous control of the delay value.
Mathematical Thought from Ancient to Modern Times
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Studies in the history of mathematics and physical sciencesA History of Ancient Mathematical Astronomy
971 Citations1975O. Neugebauer
IEEE Transactions on Signal ProcessingB-spline signal processing. I. Theory
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Physiological MeasurementHandbook of Image and Video Processing
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The subject is so mature, and the investment in both hardware and software so significant, that there must be little to be gained from research at the margins of any new approaches and this book is stupendously good value-for-money.
IEEE Transactions on Medical ImagingInterpolation revisited [medical images application]
821 Citations2000P. Thévenaz, Thierry Blu +1 more
Based on the theory of approximation, this paper presents a unified analysis of interpolation and resampling techniques and shows that, contrary to the common belief, those that perform best are not interpolating.
Proceedings of the Royal Society of EdinburghXVIII.—On the Functions which are represented by the Expansions of the Interpolation-Theory
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IEEE Transactions on Medical ImagingComparison of Interpolating Methods for Image Resampling
812 Citations1983J. Anthony Parker, Robert V. Kenyon +1 more
CERN Document Server (European Organization for Nuclear Research)Curve and surface fitting : an introduction
800 Citations1986Peter Lancaster, K. Šalkauskas
The purpose of this book is to reveal to the interested (but perhaps mathematically unsophisticated) user the foundations and major features of several basic methods for curve and surface fitting that are currently in use.
A continuously variable digital delay element
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Choice Reviews OnlineCurve and surface fitting with splines
746 Citations1993
Curve fitting is an introduction least-squares spline curve fitting smoothing spline Curve fitting more smoothingSpline curves fitting with convexity constraints.
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732 Citations1983II Leon W. Couch
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Mathematics of ComputationNumerical Methods Based on Sinc and Analytic Functions.
678 Citations1994Gerhard Schmeißer, Frank Stenger
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661 Citations1974Edwin Catmull, Raphael Rom
IEEE Transactions on Signal ProcessingB-spline signal processing. II. Efficiency design and applications
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It is proved that F is a continuous positive definite function and found almost precisely in which Lipschitz classes derivatives ofF belong.
Constructive ApproximationSymmetric iterative interpolation processes
570 Citations1989Gilles Deslauriers, Serge Dubuc
Proceedings of the IEEEA digital signal processing approach to interpolation
552 Citations1973Ronald W. Schafer, L. R. Rabiner
It is shown that linear interpolation and classical polynomial interpolation correspond to the use of the FIR interpolation filter, and a consideration of optimum FIR filters for interpolation that can be designed using linear programming techniques is considered.
IEEE Transactions on CommunicationsInterpolation in digital modems. II. Implementation and performance
546 Citations1993L. Erup, Floyd M. Gardner +1 more
The NCO-based control method presented in Part I is shown to be equivalent to a conventional phase locked loop and its operation is verified by simulation, demonstrating that simple interpolators give excellent performance.
Springer texts in electrical engineeringIntroduction to Shannon Sampling and Interpolation Theory
542 Citations1991Robert J. Marks
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Cambridge University Press eBooksImage Processing and Data Analysis
536 Citations1998Jean‐Luc Starck, Fionn Murtagh +1 more
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IEEE Transactions on Pattern Analysis and Machine IntelligenceFast B-spline transforms for continuous image representation and interpolation
527 Citations1991Michaël Unser, Akram Aldroubi +1 more
It is concluded that B-spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner.
Quarterly of Applied MathematicsContributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae
478 Citations1946I. J. Schoenberg
Proceedings of the IEEEInterpolation and decimation of digital signals—A tutorial review
475 Citations1981R. Crochiere, L. R. Rabiner
This paper presents a tutorial overview of multirate digital signal processing as applied to systems for decimation and interpolation and discusses a theoretical model for such systems (based on the sampling theorem), and shows how various structures can be derived to provide efficient implementations of these systems.
Journal of the American Statistical AssociationImage Processing and Data Analysis, The Multiscale Approach
472 Citations1999Lynne Seymour, Jean‐Luc Starck +2 more
B-SPLINE SIGNAL PROCESSING: PART II-EFFICIENT DESIGN AND APPLICATIONS
467 Citations1993Michaël Unser, Akram Aldroubi +1 more
This paper describes a class of recursive filtering algorithms for the efficient implementation of B-spline inter- polation and approximation techniques and shows how to con- struct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next.
Birkhäuser Boston eBooksContributions to the Problem of Approximation of Equidistant Data by Analytic Functions
465 Citations1988I. J. Schoenberg
IEEE Transactions on Medical ImagingShape-based interpolation of multidimensional objects
440 Citations1990S.P. Raya, Jayaram K. Udupa
A shape-based interpolation scheme for multidimensional images is presented that not only minimizes user involvement in interactive segmentation, but also leads to more accurate representation and depiction of dynamic as well as static objects.
IEEE Transactions on Image ProcessingSubpixel edge localization and the interpolation of still images
434 Citations1995K. Jensen, Dimitris Anastassiou
The algorithm is based on a source model emphasizing the visual integrity of detected edges and incorporates a novel edge fitting operator that has been developed for this application, and produces an image of increased resolution with noticeably sharper edges and lower mean-squared reconstruction error than that produced by linear techniques.
Sampling theory in Fourier and signal analysis : foundations
431 Citations1996John R. Higgins
Medical Entomology and ZoologyDigital Image Processing: Principles and Applications
426 Citations1994Gregory A. Baxes
The fundamental principles of Digital Image Processing are explained, as well as practical suggestions for improving the quality and efficiency of image processing.
Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and His System of the World
409 Citations1934Isaac Newton
ACM SIGGRAPH Computer GraphicsReconstruction filters in computer-graphics
385 Citations1988Don P. Mitchell, Arun N. Netravali
Digital Image Processing: Concepts, Algorithms, and Scientific Applications
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SIAM ReviewNumerical Methods Based on Whittaker Cardinal, or Sinc Functions
327 Citations1981Frank Stenger
Journal of Mathematical Analysis and ApplicationsInterpolation through an iterative scheme
312 Citations1986Serge Dubuc
Transactions of the American Mathematical SocietyApproximation from shift-invariant subspaces of 𝐿₂(𝐑^{𝐝})
302 Citations1994Carl de Boor, Ronald DeVore +1 more
IEEE Transactions on Image ProcessingRegularity-preserving image interpolation
285 Citations1999W.K. Carey, D.B. Chuang +1 more
IEEE Transactions on Medical ImagingOn the comparison of interpolation methods
256 Citations1988Einar Mæland
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SIAM ReviewSampling Theory for not Necessarily Band-Limited Functions: A Historical Overview
243 Citations1992P. L. Butzer, R. L. Stens
The aim of this overview is to show that one of its roots is a basic paper of de la Vallee Poussin of 1908, and the historical development of sampling theory from 1908 to the present, especially the matter dealing with not necessarily band-limited functions.
A Fourier Analysis of the Finite Element Variational Method
240 Citations2011Gilbert Strang, George J. Fix
Journal of Magnetic Resonance ImagingReduction of partial‐volume artifacts with zero‐filled interpolation in three‐dimensional MR angiography
222 Citations1994Yiping P. Du, Dennis L. Parker +2 more
A substantial improvement in vessel contrast and vessel continuity was observed in all cases and it is demonstrated that zero‐filled interpolation can also be implemented by means of voxel shifting in real space.
IEEE Transactions on Image ProcessingQuadratic interpolation for image resampling
206 Citations1997Neil A. Dodgson
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IEEE Transactions on Image ProcessingConvolution-based interpolation for fast, high-quality rotation of images
195 Citations1995Michaël Unser, P. Thévenaz +1 more
The results indicate that the present algorithm in its higher-order versions outperforms all standard high-accuracy methods of which it is aware, both in terms of speed and quality.
IEEE Transactions on Image ProcessingEnlargement or reduction of digital images with minimum loss of information
195 Citations1995Michaël Unser, Akram Aldroubi +1 more
This paper derives optimal spline algorithms for the enlargement or reduction of digital images by arbitrary (noninteger) scaling factors and discusses some properties of this approach and its connection with the classical technique of bandlimiting a signal, which provides the asymptotic limit of the algorithm as the order of the spline tends to infinity.
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191 Citations1993Borislav Bojanov, Hakop Hakopian +1 more
IEEE Transactions on Medical ImagingUnwarping of unidirectionally distorted EPI images
186 Citations2000Jan Kybic, P. Thévenaz +2 more
This paper suggests a new approach using an algorithm specifically developed for the automatic registration of distorted EPI images with corresponding anatomically correct MRI images, which is faster, more reliable, and more precise than the manual method.
IEEE Transactions on Medical ImagingAn objective comparison of 3-D image interpolation methods
179 Citations1998George J. Grevera, Jayaram K. Udupa
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IBM Journal of Research and DevelopmentDigital Image Processing of Earth Observation Sensor Data
178 Citations1976R. Bernstein
Digital image processing techniques that were developed to precisely correct Landsat multispectral Earth observation data are described and illustrations of the results achieved are given.
CERN Document Server (European Organization for Nuclear Research)Practical Handbook on Image Processing for Scientific Applications
169 Citations1997Bernd Jähne, B. Jaehne
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Journal of Computational and Applied MathematicsOn the history of multivariate polynomial interpolation
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IEEE Transactions on Visualization and Computer GraphicsEvaluation and design of filters using a Taylor series expansion
156 Citations1997Torsten Möller, Raghu Machiraju +2 more
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An evaluation of reconstruction filters for volume rendering
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This paper is concerned with interpolation methods that are equivalent to convolving the samples with a reconstruction filter; this covers all commonly used schemes, including trilinear and cubic interpolation.
Journal of Mathematical Analysis and ApplicationsOn Hermite-Birkhoff interpolation
145 Citations1966I. J. Schoenberg
IEEE Transactions on Image ProcessingWarped distance for space-variant linear image interpolation
143 Citations1999Giovanni Ramponi
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