## A chronology of interpolation: From ancient astronomy to modern signal and image processing (2002)

Venue: | Proceedings of the IEEE |

Citations: | 75 - 0 self |

### BibTeX

@INPROCEEDINGS{Meijering02achronology,

author = {Erik Meijering},

title = {A chronology of interpolation: From ancient astronomy to modern signal and image processing},

booktitle = {Proceedings of the IEEE},

year = {2002},

pages = {319--342}

}

### Years of Citing Articles

### OpenURL

### 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—Approximation, convolution-based interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.

### Citations

7146 |
A mathematical theory of communication
- Shannon
- 1948
(Show Context)
Citation Context ...s [76], [108], [109] on the cardinal series that Shannon recognized their evident importance to the field of communication. He formulated the now well-known sampling theorem, which he first published =-=[110]-=- without proof in 1948 and the subsequent year with full proof in a paper [111] apparently written already in 1940: “If a function contains no frequencies higher than cps [cycles per second], it is co... |

2339 |
A wavelet tour of signal processing
- Mallat
- 1998
(Show Context)
Citation Context ...ss, which has strong links with the multiresolution theory of wavelets. Indeed, for any ,, the Deslauriers-Dubuc kernel of order 2'n is the autocorrelation of the Daubechies a scaling function [284], =-=[285]-=-. More details and further references on interpolating wavelets are given by, e.g., Mallat [285]. See Dai et al. [286] for a recent study on dyadic interpolation in the context of image processing. An... |

1942 |
lectures on wavelets
- Daubechies, Ten
- 1992
(Show Context)
Citation Context ...n process, which has strong links with the multiresolution theory of wavelets. Indeed, for any ,, the Deslauriers-Dubuc kernel of order 2'n is the autocorrelation of the Daubechies a scaling function =-=[284]-=-, [285]. More details and further references on interpolating wavelets are given by, e.g., Mallat [285]. See Dai et al. [286] for a recent study on dyadic interpolation in the context of image process... |

1418 |
Spline Models for Observational Data
- Wahba
- 1990
(Show Context)
Citation Context ...decades of further study by Schoenberg [155] [157], found their way into approximation theory [158] [164], mono- and multivariate interpolation [81], [165] [167], numerical analysis [168], statistics =-=[140]-=-, and other branches of mathematics [169]. With the advent of digital computers, splines had a major impact on geometrical modeling and computer-aided geometric design [170] [174], computer graphics [... |

1288 |
A practical guide to splines
- Boor
- 1978
(Show Context)
Citation Context ...el (23) is not a cubic spline, regardless of the value of . Some people have called piecewise polynomial functions with less than maximum (nontrivial) smoothness “deficient splines.” See also de Boor =-=[133]-=-, who adopted the definition of a spline function as a linear combination of B-splines. When using the latter definition, the cubic convolution kernel may indeed be called a spline. We will not do so,... |

972 | Curves and Surfaces for Computer Aided Geometric Design, fifth ed - Farin - 2001 |

751 | Constructive Approximation
- DeVore, Lorentz
- 1993
(Show Context)
Citation Context ...e degree also numerical analysis [127, 206, 304, 305]. Splines, on the other hand, and after some two decades of further study by Schoenberg [278, 279, 281], found their way into approximation theory =-=[64, 69, 77, 86, 140, 239, 284]-=-, mono- and multivariate interpolation [22, 52, 298, 299], numerical analysis [256], statistics [340], and other branches of mathematics [5]. With the advent of digital computers, splines had a major ... |

605 |
Communication in the presence of noise
- Shannon
- 1949
(Show Context)
Citation Context ...a given set of known function values, irrespective of their number.sPP-12 A Chronology of Interpolation first published [288] without proof in 1948, and the subsequent year with full proof in a paper =-=[289]-=- apparently written already in 1940: If a function f contains no frequencies higher than W cycles per second, it is completely determined by giving its ordinates at a series of points spaced 1/2W seco... |

452 |
zur Gathen and
- von
- 2003
(Show Context)
Citation Context ...ok Shùshū Jiŭzhāng (“Mathematical Treatise in Nine Sections”, 1247 AD). In the West it was rediscovered by Horner [143] in 1819, and it can be found under his name in many books on numerical analysis =-=[131, 138, 171, 285, 310, 337]-=-. Fifteen years earlier, however, it had also been proposed by Ruffini [265] (see also Cajori [36]). And even earlier, around 1669, it was used by Newton [234].sPP-22 A Chronology of Interpolation to ... |

418 | Spline Functions: Basic Theory - Schumaker - 1981 |

412 |
Digital Image Warping
- Wolberg
- 1990
(Show Context)
Citation Context ...cts for the blurring effects of the corresponding B-spline convolution 36 It is therefore surprising that, even though there are now textbooks that acknowledge the superiority of spline interpolation =-=[27, 150, 151, 354]-=-, many books since the late eighties [11, 115, 282, 297, 300] give the impression that cubic convolution is the state-of-the-art in image interpolation. 37 The translation from Latin is as given by Wh... |

381 |
On the use of windows for harmonic analysis with the discrete Fourier transform
- Harris
- 1978
(Show Context)
Citation Context ... and cubic convolution kernels [187], [188], [192], [355], as well as the cubic B-spline interpolator, various Lagrange- [255] and Gaussian-based [356] interpolators and truncated and Blackman-Harris =-=[357]-=- windowed-sinc kernels of different spatial support. From the results of computational-cost analyses and forward -backward transformation experiments carried out on CCD-photographs, MRI sections, and ... |

350 |
A mathematical theory of communication,” Bell System Tech
- Shannon
- 1948
(Show Context)
Citation Context ...al observation, since all classical schemes yield the exact same polynomial for a given set of known function values, irrespective of their number.sPP-12 A Chronology of Interpolation first published =-=[288]-=- without proof in 1948, and the subsequent year with full proof in a paper [289] apparently written already in 1940: If a function f contains no frequencies higher than W cycles per second, it is comp... |

328 |
Interpolation and Approximation
- DAVIS
- 1975
(Show Context)
Citation Context ... with those in the previous or next section. For a more detailed treatment of these and other formulae, including such aspects as accuracy and implementation, see several early works on interpolation =-=[63,105,157,238,302,349]-=- and the calculus of finite differences [23,102,160,224], as well as more general books on numerical analysis [112, 138, 152, 232, 285, 310], most of which also discuss inverse interpolation and the r... |

311 |
Interpolation of spatial data: some theory for kriging
- Stein
- 1001
(Show Context)
Citation Context ...oped empirical methods for determining true ore-grade distributions from distributions based on sampled ore grades [312], [313]. Kriging has been studied in the context of geostatistics [312], [314], =-=[315]-=-, cartography [316], and meteorology [317] and is closely related to interpolation by thin-plate splines or radial basis functions [318], [319]. In medical imaging, the technique seems to have been ap... |

285 |
An Introduction to Splines for Use in Computer Graphics and Geometric Modeling
- Bartels, Beatty, et al.
- 1987
(Show Context)
Citation Context ...other branches of mathematics [169]. With the advent of digital computers, splines had a major impact on geometrical modeling and computer-aided geometric design [170] [174], computer graphics [175], =-=[176]-=-, and even font design [177] to mention but a few practical applications. In the remainder of this section, we will focus primarily on the further developments in signal and image processing. D. Convo... |

282 | Cubic Convolution Interpolation for Digital Image Processing
- Keys, G
- 1981
(Show Context)
Citation Context ...n (22) in order to preserve resolution and contrast as much as possible. IV.G Cubic Convolution Interpolation Revisited Meanwhile, research on cubic convolution interpolation continued. In 1981, Keys =-=[166]-=- published an important study that provided new, approximation-theoretic insights into this technique. He argued that the best choice for α in (23) is that for which the Taylor series expansion of the... |

280 | Splines: A perfect fit for signal and image processing
- Unser
- 1999
(Show Context)
Citation Context ...e output samples resulfing from the second anticaus filter. For the initialization of the caus filter, we may use or-symmetfic bound conditions, i.e., .s. .st, for (k +l)mod(2K 2) 0, which results in =-=[222]-=- 2] 3 is/ (36) 9I O i'l' c} -- 1 - z[ ' In most practical cases, K- will be sufficiently large to justify taking 1/(1 2,, 2, z i ) 1 and terminating the summation much earlier. An initial value for th... |

279 |
Geostatistics: Modeling Spatial Uncertainty, 695 p (Wiley
- Chilès, Delfiner
- 1999
(Show Context)
Citation Context ...engineer who developed empirical methods for determining true ore-grade distributions from distributions based on sampled ore grades [55, 56]. Kriging has been studied in the context of geostatistics =-=[50,56,303]-=-, cartography [181], and meteorology [106], and is closely related to interpolation by thin-plate splines or radial basis functions [83,229]. In medical imaging, the technique seems to have been appli... |

233 | Sampling-50 years after Shannon
- Unser
- 2000
(Show Context)
Citation Context ...ure mapping [250], online signal interpolation [251], and fast spline transformation [252]. For more detailed information, the reader is referred to mentioned papers as well as several reviews [222], =-=[253]-=-. J. Development of Alternative Piecewise Polynomial Kernels Independent of the just mentioned developments, research on alternative piecewise polynomial interpolation kernels continued. Mitchell and ... |

199 |
Certain Topics in Telegraph Transmission Theory
- Nyquist
- 1928
(Show Context)
Citation Context ...nts spaced 1/2W seconds apart. Later on in the paper, he referred to the critical sampling interval T =1/2W as the Nyquist interval corresponding to the band W , in recognition of Nyquist’s discovery =-=[240]-=- of the fundamental importance of this interval in connection with telegraphy. In describing the reconstruction process, he pointed out that There is one and only one function whose spectrum is limite... |

184 |
Symmetric iterative interpolation processes
- Deslauriers, Dubuc
- 1989
(Show Context)
Citation Context ...ms and their solutions as studied in mentioned fields of research over the past two decades. Pointers to relevant literature are provided for readers interested in more details. Deslauriers and Dubuc =-=[76, 84]-=-, for example, studied the problem of extending known function values f(k) at the integers to all integral multiples of 1/b, wherethebasebis also integer. In principle, any type of interpolation can b... |

180 |
Contributions 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
- Schoenberg
- 1946
(Show Context)
Citation Context ...t an interpolant. The need for practically applicable methods for interpolation or smoothing of empirical data also formed the impetus to Schoenberg’s study of the subject. In his 1946 landmark paper =-=[274, 275]-=- he noted that for every osculatory interpolation formula applied to equidistant data, where he assumed the distance to be unity, there exists an even function Φ : R → R, in terms of which the formula... |

170 |
Algorithms for Graphics and Image Processing
- Pavlidis
- 1982
(Show Context)
Citation Context ...], and other branches of mathematics [169]. With the advent of digital computers, splines had a major impact on geometrical modeling and computer-aided geometric design [170] [174], computer graphics =-=[175]-=-, [176], and even font design [177] to mention but a few practical applications. In the remainder of this section, we will focus primarily on the further developments in signal and image processing. D... |

167 |
Mathematical Thought from Ancient to Modern Times
- Kline
- 1972
(Show Context)
Citation Context ...n fact f(xo + T)I .... that f(Xo) = f(xu + T) -- f(xo) and simil for the higher order differences.) The spial version for x0 = 0 was laler used by Maclatin [37] as a funmemal tool. See also e.g., ine =-=[38]. 1egory w-=-as also awe of a later method by Mercator, for in his leller he refers to his own melhod as ing "both more easie and universal than either Briggs or Mercalor's" [39].sFrance, it was Mouton w... |

163 |
A Fourier analysis of the finite element variational method
- Strang, Fix
- 1971
(Show Context)
Citation Context ... obtained in this area of mathematics were not noticed by researchers in signal and image processing until relatively recently [323, 333]. An important example is the theory developed by Strang & Fix =-=[308]-=- in the early 1970s, and further studied by many others, which relates the approximation error to properties of the kernel involved. Specifically, it implies that the following conditions are equivale... |

162 | Remote sensing: Models and methods for image processing - Schowengerdt - 1997 |

157 | New edge-directed interpolation
- Li, Orchard
- 2001
(Show Context)
Citation Context ...aches have also been reported. Examples of specific applications where the potential advantages of adaptive interpolation approaches have been demonstrated are image resolution enhancement or zooming =-=[38, 61, 107, 142, 154, 191, 259, 317]-=-, spatial and temporal coding or compression of images and image sequences [320, 321], texture mapping [147], and volume rendering [269]. Clearly, the results of such adaptive methods depend on the pe... |

155 | Curve and Surface Fitting with Splines - Dierckx - 1993 |

154 |
Introduction to numerical analysis
- Hildebrand
- 1974
(Show Context)
Citation Context ...ccuracy and implementation, see several early works on interpolation [ 19], [21], [23] [26] and the calculus of finite differences [27] [30], as well as more general books on numerical analysis [20], =-=[31]-=- [35], most of which also discuss inverse interpolation and the role of interpolation in numerical differentiation and integration. MEIJERING: A CHRONOLOGY OF INTERPOLATION 321 nomial f:s-sof given de... |

154 | An evaluation of reconstruction filters for volume rendering
- Marschner, Lobb
- 1994
(Show Context)
Citation Context ... the cubic convolution kernel (23) corresponding tos: -1/2 and the cubic B-spline. The whole family, sometimes also referred to as BC-splines [196], was later studied by several authors [195], [196], =-=[254]-=- in the fields of visualization and computer graphics. An extension of the results of Park and Schowengerdt [188] concerning the previously discussed frequency-domain error analysis was presented by S... |

147 | Interpolation revisited
- Thévenaz, Blu, et al.
- 2000
(Show Context)
Citation Context ... follows: for sufficiently smooth functions f, that is to say functions for which �f (L) �L2 is finite, the L2-norm of the difference between f and the approximation fT obtained from (22) is given by =-=[20, 313, 314, 325]-=- �f − fT �L2 = Cϕ · T L ·�f (L) �L2 as T → 0. (50) In view of the practical need in many applications to optimize the cost-performance trade-off of interpolation, this raises the questions of which of... |

144 |
The Theory of Splines and Their Applications
- Ahlberg, Nilson, et al.
- 1967
(Show Context)
Citation Context ...55] [157], found their way into approximation theory [158] [164], mono- and multivariate interpolation [81], [165] [167], numerical analysis [168], statistics [140], and other branches of mathematics =-=[169]-=-. With the advent of digital computers, splines had a major impact on geometrical modeling and computer-aided geometric design [170] [174], computer graphics [175], [176], and even font design [177] t... |

142 |
Principles of geostatistics
- Matheron
- 1963
(Show Context)
Citation Context ...ean-squared prediction error, which requires a model of the covariance between the data points. This approach to interpolation is generally known as kriging—a term coined by the statistician Matheron =-=[215]-=- in honor of D. G. Krige [175], a South-African mining engineer who developed empirical methods for determining true ore-grade distributions from distributions based on sampled ore grades [55, 56]. Kr... |

140 | Structural Pattern Recognition - Pavlidis - 1977 |

139 | Approximation from shift-invariant subspaces of L2(Rd
- Boor, DeVore, et al.
- 1994
(Show Context)
Citation Context ...s result holds for 1 compactly supported kernels [263], but Mso extends to noncompactly supported kernels having suitable inverse polynomial decay [264] [267] or even less stringent properties [268], =-=[269]-=-. Extensions have Mso been mMe to Lp norms, isps[264], [265], [267], [270], [271]. Ftheore, Mthough the classicM SangWix theow applies to the case of an orthogonal projection, alternative approximatio... |

134 |
Numerical Methods for Scientists and Engineers
- Hamming
- 1962
(Show Context)
Citation Context .... The former had great impact on communication engineering [144] [147], numerous signal processing and analysis applications [98], [124], [125], [148] [150] and to some degree also numerical analysis =-=[151]-=- [154]. Splines, on the other hand and after some two decades of further study by Schoenberg [155] [157], found their way into approximation theory [158] [164], mono- and multivariate interpolation [8... |

134 | Weierstrass and approximation theory
- PINKUS
- 2000
(Show Context)
Citation Context ...he Newton-Gauss formula, (9), and set out to answer the question of 24 For more detailed information on the development of approximation theory, see the recently published historical review by Pinkus =-=[253]-=-.sIV The Information and Communication Era PP-11 which one of the cotabular functions is represented by it. The answer, he proved, is that under certain conditions it represents the cardinal function ... |

129 |
Kestutis Salkauskas, Curve and Surface Fitting: an Introduction
- Lancaster
- 1986
(Show Context)
Citation Context ...analysis [168], statistics [140], and other branches of mathematics [169]. With the advent of digital computers, splines had a major impact on geometrical modeling and computer-aided geometric design =-=[170]-=- [174], computer graphics [175], [176], and even font design [177] to mention but a few practical applications. In the remainder of this section, we will focus primarily on the further developments in... |

128 | B-Spline Signal Processing: Part I—Theory
- Unser, Aldroubi, et al.
- 1993
(Show Context)
Citation Context ...3] for inverting the Toeplitz type of matrices occurring in spline interpolation, this approach is unnecessarily complex and computationally expensive. In the early 1990s it was shown by Unser et al. =-=[326, 328, 329]-=- that the B-spline interpolation problem can be solved much more efficiently by using a digital filtering approach. Writing βn rather than Mn+1 for a B-spline of degree n, we obtain the following from... |

127 |
Multivariate Splines
- Chui
- 1988
(Show Context)
Citation Context ...4]. Splines, on the other hand and after some two decades of further study by Schoenberg [155] [157], found their way into approximation theory [158] [164], mono- and multivariate interpolation [81], =-=[165]-=- [167], numerical analysis [168], statistics [140], and other branches of mathematics [169]. With the advent of digital computers, splines had a major impact on geometrical modeling and computer-aided... |

124 | Seminumerical Algorithms. The Art of Computer Programming, Anonymous Addison-Wesley - Knuth - 1969 |

123 |
Interpolation through an iterative scheme
- Dubuc
- 1986
(Show Context)
Citation Context ...ms and their solutions as studied in mentioned fields of research over the past two decades. Pointers to relevant literature are provided for readers interested in more details. Deslauriers and Dubuc =-=[76, 84]-=-, for example, studied the problem of extending known function values f(k) at the integers to all integral multiples of 1/b, wherethebasebis also integer. In principle, any type of interpolation can b... |

123 |
Cubic Splines for Image Interpolation and Digital Filtering
- Hou, Andrews
- 1978
(Show Context)
Citation Context ...oducing linear polynomials, but also quadratic polynomials. IV.F Spline Interpolation in Image Processing The use of splines for digital image interpolation was first investigated only a little later =-=[7,144,145,249]-=-. An important paper providing a detailed analysis was published in 1978 34 Notice here that the application of convolution kernels to two-dimensional data defined on Cartesian grids, as digital image... |

122 | A pyramid approach to subpixel registration based on intensity
- Thévenaz, Ruttimann, et al.
- 1998
(Show Context)
Citation Context ...fficient algorithms for such purposes as image rotation [240], the enlargement or reduction of images [241], [242], the construction of multiresolution image pyramids [243], [244], image registration =-=[245]-=-, [246], wavelet transformation [247] [249], texture mapping [250], online signal interpolation [251], and fast spline transformation [252]. For more detailed information, the reader is referred to me... |

120 |
Numerical methods based on Sinc and analytic functions
- Stenger
- 1993
(Show Context)
Citation Context ...former had great impact on communication engineering [144] [147], numerous signal processing and analysis applications [98], [124], [125], [148] [150] and to some degree also numerical analysis [151] =-=[154]-=-. Splines, on the other hand and after some two decades of further study by Schoenberg [155] [157], found their way into approximation theory [158] [164], mono- and multivariate interpolation [81], [1... |

118 |
Probability and information theory, with application to radar
- Woodward
- 1953
(Show Context)
Citation Context ...lis [ 125]. Although it has become well known in connection with the sampling theorem, it was not used by Shannon in his original papers, but appears to have been introduced first in 1953 by Woodward =-=[143]-=-. functions f are piecewise polynomials of degree ,t, which join smoothly according to the definition of a spline, the coefficients cj are computed from the samples .s and the kernel is the ,th degree... |

115 |
On calculation with B-splines
- Boor
- 1972
(Show Context)
Citation Context ...50] and to some degree also numerical analysis [151]–[154]. Splines, on the other hand and after some two decades of further study by Schoenberg [155]–[157], found their way into approximation theory =-=[158]-=-–[164], mono- and multivariate interpolation [81], [165]–[167], numerical analysis [168], statistics [140], and other branches of mathematics [169]. With the advent of digital computers, splines had a... |

115 |
B-Spline signal processing: Part II– Efficient design and applications
- Unser, Aldroubi, et al.
- 1993
(Show Context)
Citation Context ...3] for inverting the Toeplitz type of matrices occurring in spline interpolation, this approach is unnecessarily complex and computationally expensive. In the early 1990s it was shown by Unser et al. =-=[326, 328, 329]-=- that the B-spline interpolation problem can be solved much more efficiently by using a digital filtering approach. Writing βn rather than Mn+1 for a B-spline of degree n, we obtain the following from... |

112 |
Image Processing and data analysis: the multiscale approach
- Starck, Murtagh, et al.
- 1998
(Show Context)
Citation Context ...ne convolution 36 It is therefore surprising that, even though there are now textbooks that acknowledge the superiority of spline interpolation [27, 150, 151, 354], many books since the late eighties =-=[11, 115, 282, 297, 300]-=- give the impression that cubic convolution is the state-of-the-art in image interpolation. 37 The translation from Latin is as given by Whittaker & Robinson [349].sPP-20 A Chronology of Interpolation... |