from ancient astronomy to modern signal and image processing

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