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discussed the design of alternative finite impulse responseinterpolators based on prespecified bandpass and bandstopcharacteristics. More information on this can also be foundin the tutorial review by Crochiere and Rabiner [179].E. Cubic Convolution Interpolation in Image ProcessingThe early 1970s was also the time digital image processingreally started to develop. One of the first applicationsreported in the literature was the geometrical rectificationof digital images obtained from the first Earth ResourcesTechnology Satellite launched by the United States NationalAeronautics and Space Administration in 1972. The needfor more accurate interpolations than obtained by standardlinear interpolation in this application led to the developmentof a still very popular technique known as cubic convolution,which involves the use of a sinc-like kernel composed ofpiecewise cubic polynomials. Apart from being interpolating,the kernel was designed to be continuous and to havea continuous first derivative. Cubic convolution was firstmentioned by Rifman [180], discussed in some more detailby Simon [181], but the most general form of the kernelappeared first in a paper by Bernstein [182]ififif(23)where is a free parameter resulting from the fact that theinterpolation, continuity and continuous differentiability requirementsyield only seven equations, while the two cubicpolynomials defining the kernel make up a total of eight unknowncoefficients. 34 The explicit cubic convolution kernelgiven by Rifman and Bernstein in their respective papers isthe one corresponding to , which results from forcingthe first derivative of to be equal to that of the sinc functionat . Two alternative criteria for fixing were givenby Simon [181]. The first consists in requiring the secondderivative of the kernel to be continuous at , which resultsin. The second amounts to requiring thekernel to be capable of exact constant slope interpolation,which yields . Although not mentioned by Simon,the kernel corresponding to the latter choice for is not onlycapable of reproducing linear polynomials, but also quadraticpolynomials.F. Spline Interpolation in Image ProcessingThe use of splines for digital-image interpolation was firstinvestigated only a little later [183]–[186]. An importantpaper providing a detailed analysis was published in 1978 byHou and Andrews [186]. Their approach, mainly centeredaround the cubic B-spline, was based on matrix inversion34 Notice here that the application of convolution kernels to two-dimensional(2-D) data defined on Cartesian grids, as digital images are in mostpractical cases, has traditionally been done simply by extending (22) tof (x; y)= c '(x=T 0 k) '(y=T 0 l), where T andT denote the sampling interval in x and y direction, respectively. This approachcan, of course, be extended to any number of dimensions and allowsfor the definition and analysis of kernels in one dimension only.techniques for computing the coefficients in (22). Qualitativeexperiments involving magnification (enlargement)and minification (reduction) of image data demonstrated thesuperiority of cubic B-spline interpolation over techniquessuch as nearest neighbor or linear interpolation and eveninterpolation based on the truncated sinc function as kernel.The results of the magnification experiments also clearlyshowed the necessity for this type of interpolation to usethe coefficients rather than the original samples in(22) in order to preserve resolution and contrast as much aspossible.G. Cubic Convolution Interpolation RevisitedMeanwhile, research on cubic convolution interpolationcontinued. In 1981, Keys [187] published an important studythat provided new approximation-theoretic insights into thistechnique. He argued that the best choice for in (23) is thatthe Taylor series expansion of the interpolant resultingfrom cubic convolution interpolation of equidistant samplesof an original function agrees in as many terms as possiblewith that of the original function. By using this criterion, hefound that the optimal choice isin which case, for all . This implies that the interpolationerror goes to zero uniformly at a rate proportionalto the third power of the sample interval. In other words, forthis choice of , cubic convolution yields a third-order approximationof the original function. For all other choices of, he found that it yields only a first-order approximation,just like nearest neighbor interpolation. He also pointed atthe fact that cubic Lagrange and cubic spline interpolationboth yield a fourth-order approximation—the highest possiblewith piecewise cubics—and continued to derive a cubicconvolution kernel with the same property, at the cost of alarger spatial support.A second complementary study of the cubic convolutionkernel (23) was published a little later by Park andSchowengerdt [188]. Rather than studying the propertiesof the kernel in the spatial domain, they carried out afrequency-domain analysis. The Maclaurin series expansionof the Fourier transform of (23) can be derived as(24)where denotes radial frequency. Based on this fact, theyargued that the best choice for the free parameter is, since this maximizes the number of terms in which(24) agrees with the Fourier transform of the sinc kernel. Thatis to say, it provides the best low-frequency approximationto the “ideal” reconstruction filter. Park and Schowengerdt[188], [189] also studied the mean-square error or squarednormwith(25)the image obtained from interpolating the samplesof an original image using any interpolation328 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 3, MARCH 2002
kernel . They showed that if the original image is bandlimited,i.e., , for all larger than somein this one-dimensional analysis and if sampling is performedat a rate equal to or higher than the Nyquist rate, thiserror—which they called the “sampling and reconstructionblur”—is equal towhere(26)(27)with . In the case of undersampling, representsthe average error , where the averaging is over allpossible sets of samples , with . Theyargued that if the energy spectrum of is not known, theoptimal choice for is the one that yields the best low-frequencyapproximation to . Substituting for andcomputing the Maclaurin series expansion of the right-handside of (27), they found that this best approximation is obtainedby taking, again,. Notwithstanding theachievements of Keys and Park and Schowengerdt, it is interestingto note that cubic convolution interpolation correspondingto had been suggested in the literatureat least three times before these authors. Already mentionedis Simon’s paper [181], published in 1975. A little earlier, in1974, Catmull and Rom [190] had studied interpolation by“cardinal blending functions” of the type(28)where is the degree of polynomials resulting from theproduct on the right-hand side and is a weight functionor blending function centered around . Among theexamples they gave is the function corresponding toand the second-degree B-spline. This function can beshown to be equal to (23) with. In the fieldsof computer graphics and visualization, the third-ordercubic convolution kernel is therefore usually referred toas the Catmull–Rom spline. It has also been called the(modified or cardinal) cubic spline [191]–[197]. Finally, thiscubic convolution kernel is precisely the kernel implicitlyused in the previously mentioned osculatory interpolationscheme proposed around 1900 by Karup and King. Moredetails on this can be found in a recent paper [198], whichalso demonstrates the equivalence of Keys’ fourth-ordercubic convolution and Henderson’s osculatory interpolationscheme mentioned earlier.H. Cubic Convolution Versus Spline InterpolationA comparison of interpolation methods in medicalimaging was presented by Parker et al. [199] in 1983.Their study included the nearest neighbor kernel, the linearinterpolation kernel, the cubic B-spline, and two cubicconvolution kernels, 35 , the ones corresponding toand . Based on a frequency-domainanalysis they concluded that the cubic B-spline yields themost smoothing and that it is therefore better to use a cubicconvolution kernel. This conclusion, however, resulted froman incorrect use of the cubic B-spline for interpolation inthe sense that the kernel was applied directly to the originalsamples instead of the appropriate coefficients —anapproach that has been suggested (explicitly or implicitly)by many authors over the years [192], [200]–[205]. Thepoint was later discussed by Maeland [206] who derived thetrue spectrum of the cubic spline interpolator or cardinalcubic spline as the product of the spectrum of the requiredprefilter and that of the cubic B-spline. From a correctcomparison of the spectra, he concluded that cubic splineinterpolation is superior compared to cubic convolutioninterpolation—a conclusion that would later be confirmedrepeatedly by several evaluation studies (to be discussed inSection IV-M). 36I. Spline Interpolation RevisitedIn classical interpolation theory, it was already known thatit is better or even necessary in some cases to first apply sometransformation to the original data before applying a given interpolationformula. The general rule in such cases is to applytransformations that will make the interpolation as simple aspossible. The transformations themselves, of course, shouldpreferably also be as simple as possible. Stirling, in his 1730book [51] on finite differences, wrote: “As in common algebra,the whole art of the analyst does not consist in the resolutionof the equations, but in bringing the problems thereto.So likewise in this analysis: there is less dexterity required inthe performance of the process of interpolation than in thepreliminary determination of the sequences which are bestfitted for interpolation.” 37 It should be clear from the foregoingdiscussion that a similar statement applies to convolution-basedinterpolation using B-splines: the difficulty is notin the convolution, but in the preliminary determination ofthe coefficients . In order for B-spline interpolation to bea competitive technique, the computational cost of this preprocessingstep should be reduced to a minimum—in manysituations, the important issue is not just accuracy, but thetradeoff between accuracy and computational cost. Hou andAndrews [186], as many before and after them, solved theproblem by setting up a system of equations followed by matrixinversion. Even though there exist optimized techniques[215] for inverting the Toeplitz type of matrices occurring in35 Note that Parker et al. referred to them consistently as “high-resolutioncubic splines.” According to Schoenberg’s original definition, however, thecubic convolution kernel (23) is not a cubic spline, regardless of the valueof . Some people have called piecewise polynomial functions with lessthan maximum (nontrivial) smoothness “deficient splines.” See also de Boor[133], who adopted the definition of a spline function as a linear combinationof B-splines. When using the latter definition, the cubic convolution kernelmay indeed be called a spline. We will not do so, however, in this paper.36 It is, therefore, surprising that even though there are now textbooksthat acknowledge the superiority of spline interpolation [207]–[210], manybooks since the late 1980s [149], [211]–[214] give the impression that cubicconvolution is the state-of-the-art in image interpolation.37 The translation from Latin is as given by Whittaker and Robinson [23].MEIJERING: A CHRONOLOGY OF INTERPOLATION 329
Page 1: A Chronology of Interpolation: From
Page 4 and 5: nomial of given degree , i.e.,. It
Page 6 and 7: D. Studies on More General Interpol
Page 8 and 9: literature parallel to Shannon. In
Page 12 and 13: spline interpolation, this approach
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Page 18 and 19: Schoenberg’s theory of mathematic
Page 20 and 21: [82] M. Gasca and T. Sauer, “Poly
Page 22 and 23: [204] L. R. Watkins and J. Le Bihan
Page 24: [316] N. S.-N. Lam, “Spatial inte

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