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1) The kernel has th-order zeroes in the Fourier domain.More precisely.(43)2) The kernel is capable of reproducing all monomialsof degree. That is, for every, there exist coefficientssuch that(44)3) The first discrete moments of the kernel are constants.That is, for every, thereexists a such that(45)4) For each sufficiently smooth function , i.e., a functionwhose derivatives up to and including order are inthe space , there exists a constant , whichdoes not depend on , and a set of coefficientssuch that the norm of the difference between andits approximation obtained from (22) is bounded asas (46)This result holds for all compactly supported kernels [263],but also extends to noncompactly supported kernels havingsuitable inverse polynomial decay [264]–[267] or even lessstringent properties [268], [269]. Extensions have also beenmade to norms, [264], [265], [267], [270],[271]. Furthermore, although the classical Strang–Fix theoryapplies to the case of an orthogonal projection, alternative approximationmethods such as interpolation and quasiinterpolationalso yield an approximation error [270]–[273].A detailed treatment of the theory is outside the scope of thepresent paper and the reader is referred to mentioned papersfor more information.An interesting observation that follows from these equivalenceconditions is that even though the original functiondoes not at all have to be a polynomial, the capability of agiven kernel to let the approximation error go to zero aswhen (fourth condition) is determined by its ability toexactly reproduce all polynomials of maximum degree(second condition). In particular, it follows that in order forthe approximation to converge to the original function at allwhen , the kernel must have at least approximationorder , which implies that it must at least be able to reproducethe constant. If we take (first condition),which yields the usually desirable property of unit gain for,wehave , which implies that the kernel samplesmust sum to one regardless of the position of the kernelrelative to the sampling grid (third condition). This is generallyknown as the partition of unity condition—a conditionthat is not satisfied by virtually all so-called windowedsinc functions, which have frequently been proclaimed as themost appropriate alternative for the “ideal” interpolator.As can be appreciated from (46), the theoretical notion ofapproximation order is still rather qualitative and not suitablefor precise determination of the approximation error. Inmany applications, it would be very useful to have a morequantitative way of estimating the errorinduced by a given kernel and sampling step . In 1999, itwas shown by Blu and Unser [268], [274], [275] that for anyconvolution-based approximation scheme, this error is givenby the relation(47)where the first term is a Fourier-domain prediction of theerror, given by(48)and the second term goes to zero faster than . Here, denotesthe highest derivative of that still has finite energy.The error function in (48) is completely determined bythe prefiltering and reconstruction kernels involved in the approximationscheme. In the case where the scheme is actuallyan interpolation scheme, it follows that(49)If the original function is band-limited or otherwise sufficientlysmooth (which means that its intrinsic scale is largewith respect to the sampling step ), the prediction (48) isexact, that is to say,. In all other cases,represents the average of over all possible sets of samples, with . Note that if the kernel itselfis forced to possess the interpolation property ,with the Kronecker symbol, we have from the discreteFourier transform pairthat the denominator of (49) equals one, so that the errorfunction reduces to the previously mentioned function (27)proposed by Park and Schowengerdt [188], [189].It will be clear from (48) that the rate at which the approximationerror goes to zero as is determinedby the degree of flatness of the error function (49) near theorigin. This latter quantity follows from the Maclaurin seriesexpansion of the function. If all derivatives up to order2 at the origin are zero, this expansion can be written as, whereand where use has been made of the fact that all odd terms ofthe expansion are zero because of the symmetry of the function.Substitution into (48) then shows that the last conditionin the Strang–Fix theory can be reformulated more quantitativelyas follows: for sufficiently smooth functions , that isto say, functions for which is finite, the norm ofthe difference between and the approximation obtainedfrom (22) is given by [253], [274], [276], [277]as (50)In view of the practical need in many applications to optimizethe cost-performance tradeoff of interpolation, thisraises the questions of which kernels of given approximationorder have the smallest support and which of the latter kernelshave the smallest asymptotic constant . These questionswere recently answered by Blu et al. [278]. Concerning332 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 3, MARCH 2002
the first, they showed that these kernels are piecewise polynomialsof degree and that their support is of size. Moreover, the full class of these so-called maximal-orderminimum-support (MOMS) kernels is given by a linear combinationof an ( )th-degree B-spline and its derivatives 39(51)where and the remaining are free parametersthat can be tuned so as to let the kernel satisfy additionalcriteria. For example, if the kernel is supposed to have maximumsmoothness, it turns out that all of the latter arenecessarily zero, so that we are left with the ( )th-degreeB-spline itself. Alternatively, if the kernel is supposed to possessthe interpolation property, it follows that the are suchthat the kernel boils down to the ( )th-degree Lagrangecentral interpolation kernel. A more interesting design goal,however, is to minimize the asymptotic constant , the generalform of which for MOMS kernels is given by [278](52)whereis a polynomial of degree. It was shown by Blu et al. [278] that the whichminimizes (52) for any order can be obtained from the inductionrelation(53)which is initialized by. The resultingkernels were coined optimized MOMS (O-MOMS) kernels.From inspection of (53), it is clear that regardless of thevalue of , the corresponding polynomial will alwaysconsist of even terms only. Hence, if is even, the degree ofwill be and since an ( )th-degree B-spline isprecisely times continuously differentiable, it followsfrom (51) that the corresponding kernel is continuous, butthat its first derivative is discontinuous. If, on the other hand,is odd, the degree of the polynomial will be ,so that the corresponding kernel itself will be discontinuous.In order to have at least a continuous derivative, as may berequired for some applications, Blu et al. [278] also carriedout constrained minimizations of (52). The resulting kernelswere termed suboptimal MOMS (SO-MOMS).L. Development of Alternative Interpolation MethodsAlthough—as announced in the introduction—the mainconcern in this section is the transition from classical polynomialinterpolation approaches to modern convolution-basedapproaches and the many variants of the latter that have beenproposed in the signal and image processing literature, it maybe good to extend the perspectives and briefly discuss severalalternative methods that have been developed since the1980s. The goal here is not to be exhaustive, but to give animpression of the more specific interpolation problems and39 Blu et al.[278] point out that this fact had been published earlier by Ron[279] in a more mathematically abstract and general context: the theory ofexponential B-splines.their solutions as studied in mentioned fields of research overthe past two decades. Pointers to relevant literature are providedfor readers interested in more details.Deslauriers and Dubuc [280], [281], e.g., studied theproblem of extending known function values at theintegers to all integral multiples of , where the baseis also integer. In principle, any type of interpolation canbe used to compute theand the process can be iterated to find the value forany rational number whose denominator is an integralpower of . As pointed out by them, the main properties ofthis so-called -adic interpolation process are determinedby what they called the “fundamental function,” i.e., thekernel , which reveals itself when feeding the processwith a discrete impulse sequence. They showed that whenusing Waring–Lagrange interpolation 40 of any odd degree, the corresponding kernel has a support limited toand is capable of reproducing polynomialsof maximum degree . Ultimately, it satisfies the-scale relation(54)where . Taking , we have a dyadicinterpolation process, which has strong links with themultiresolution theory of wavelets. Indeed, for any , theDeslauriers-Dubuc kernel of order 2 is the autocorrelationof the Daubechies scaling function [284], [285]. Moredetails and further references on interpolating waveletsare given by, e.g., Mallat [285]. See Dai et al. [286] for arecent study on dyadic interpolation in the context of imageprocessing.Another approach that has received quite some attentionsince the 1980s is to consider the interpolation problem in theFourier domain. It is well known [287] that multiplication bya phase component in the frequency domain corresponds toa shift in the signal or image domain. Obvious applicationsof this property are translation and zooming of image data[288]–[292]. Interpolation based on the Fourier shift theoremis equivalent to another Fourier-based method, knownas zero-filled or zero-padded interpolation [293]–[297]. Inthe spatial domain, both techniques amount to assuming periodicityof the underlying signal or image and convolvingthe samples with the “periodized sinc,” or Dirichlet’s kernel[298]–[303]. Because of the assumed periodicity, the infiniteconvolution sum can be rewritten in finite form. Fourierbasedinterpolation is especially useful in situations wherethe data is acquired in Fourier space, such as in magnetic-resonanceimaging (MRI), but by use of the fast Fourier transformmay in principle be applied to any type of data. Variantsof this approach based on the fast Hartley transform[304]–[306] and discrete cosine transform [307]–[309] havealso been proposed. More details can be found in mentionedpapers and the references therein.An approach that has hitherto received relatively little attentionin the signal and image processing literature is to40 Subdivision algorithms for Hermite interpolation were later studied byMerrien and Dubuc [282], [283].MEIJERING: A CHRONOLOGY OF INTERPOLATION 333
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 10 and 11: discussed the design of alternative
Page 12 and 13: spline interpolation, this approach
Page 16 and 17: consider sampled data as realizatio
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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