A Chronology of Interpolation - · PDF fileFrom his study of ephemerides found on ancient astro- ... In his book S yu an y uji an ... PP-4 A Chronology of Interpolation

A Chronology of InterpolationFrom Ancient Astronomy to Modern Signal and Image Processing

Erik Meijering

Proceedings of the IEEE, vol. 90, no. 3, March 2002, pp. 319–342

Abstract—This paper presents a chronological overview of the developments in interpolationtheory, from the earliest times to the present date. It brings out the connections between theresults obtained in different ages, thereby putting the techniques currently used in signal and imageprocessing into historical perspective. A summary of the insights and recommendations that followfrom relatively recent theoretical as well as experimental studies concludes the presentation.

Keywords—Approximation, convolution-based interpolation, history, image processing, polyno-mial interpolation, signal processing, splines.

It is an extremely useful thing to have knowledge of the true origins of memorablediscoveries, especially those that have been found not by accident but by dint ofmeditation. It is not so much that thereby history may attribute to each man his owndiscoveries and others should be encouraged to earn like commendation, as that the artof making discoveries should be extended by considering noteworthy examples of it.1

I Introduction

The problem of constructing a continuously-defined function from given discrete data isunavoidable whenever one wishes to manipulate the data in a way that requires informationnot included explicitly in the data. In this age of ever increasing digitization in the storage,processing, analysis, and communication of information, it is not difficult to find examplesof applications where this problem occurs. The relatively easiest and in many applicationsoften most desired approach to solve the problem is interpolation, where an approximatingfunction is constructed in such a way as to agree perfectly with the usually unknownoriginal function at the given measurement points.2 In view of its increasing relevance, itis only natural that the subject of interpolation is receiving more and more attention thesedays.3 However, in times where all efforts are directed towards the future, the past may

1Leibniz, in the opening paragraph of his Historia et Origo Calculi Differentialis [188]. The translationgiven here was taken from a paper by Child [49].

2The word “interpolation” originates from the Latin verb interpolare, a contraction of “inter”, meaning“between”, and “polare”, meaning “to polish”. That is to say, to smooth in between given pieces ofinformation. It seems that the word was introduced in the English literature for the first time around1612, and was then used in the sense of “to alter or enlarge [texts] by insertion of new matter” [293]. Theoriginal Latin word appears [10] to have been used first in a mathematical sense by Wallis in his 1655 bookon infinitesimal arithmetic [341].

3A search in the multidisciplinary databases of bibliographic information collected by the Institute forScientific Information in the Web of Science will reveal that the number of publications containing theword “interpolation” in the title, list of keywords, or the abstract, has dramatically increased over thepast decade, even when taking into account the intrinsic (and likewise dramatic) increase in the numberof publications as a function of time.

PP-2 A Chronology of Interpolation

easily be forgotten. It is no sinecure, scanning the literature, to get a clear picture of thedevelopment of the subject through the ages. This is quite unfortunate, since it implies arisk of researchers going over grounds covered earlier by others. History has shown manyexamples of this, and several new examples are revealed here. The goal of the present paperis to provide a systematic overview of the developments in interpolation theory, from theearliest times to the present date, and to put the most well-known techniques currentlyused in signal and image processing applications into historical perspective. The paper isintended to serve as a tutorial and a useful source of links to the appropriate literaturefor anyone interested in interpolation, whether it be its history, theory, or applications.

As already suggested by the title, the organization of the paper is largely chronological.Section II presents an overview of the earliest-known uses of interpolation in antiquity anddescribes the more sophisticated interpolation methods developed in different parts of theworld during the Middle Ages. Next, Section III discusses the origins of the most importanttechniques developed in Europe during the period of Scientific Revolution, which in thepresent context lasted from the early 17th until the late 19th century. A discussion of thedevelopments in what could be called the Information and Communication Era, coveringroughly the past century, is provided in Section IV. Here, the focus of attention is on theresults that have had the largest impact on the advancement of the subject in signal andimage processing, in particular on the development of techniques for the manipulation ofintensity data defined on uniform grids. Although recently developed alternative methodsfor specific interpolation tasks in this area are also mentioned briefly, the discussion in thispart of the paper is restricted mainly to convolution-based methods, which is justified bythe fact that these are the most frequently used interpolation methods, probably becauseof their versatility and relatively low complexity. Finally, summarizing and concludingremarks are made in Section V.

II Ancient Times and the Middle Ages

In his 1909 book on interpolation [316], Thiele characterized the subject as “the art ofreading between the lines in a [numerical] table”. Examples of fields in which this problemarises naturally and inevitably are astronomy and, related to this, calendar computation.And because man has been interested in these since day one, it should not surprise usthat it is in these fields that the first interpolation methods were conceived. This sectiondiscusses the earliest-known contributions to interpolation theory.

II.A Interpolation in Ancient Babylon and Greece

In antiquity, astronomy was all about time keeping and making predictions concerningastronomical events. This served important practical needs: farmers, for example, wouldbase their planting strategies on these predictions. To this end it was of great importanceto keep up lists—so-called ephemerides—of the positions of the sun, moon, and the knownplanets for regular time intervals. Obviously these lists would contain gaps, due to eitheratmospherical conditions hampering observation or the fact that celestial bodies may notbe visible during certain periods. From his study of ephemerides found on ancient astro-nomical cuneiform tablets originating from Uruk and Babylon in the Seleucid period (thelast three centuries BC), the historian-mathematician Neugebauer [230, 231] concludedthat interpolation was used in order to fill these gaps. Apart from linear interpolation, thetablets also revealed the use of more complex interpolation methods. Precise formulationsof the latter methods have not survived, however.

II Ancient Times and the Middle Ages PP-3

An early example of the use of interpolation methods in ancient Greece dates fromabout the same period. Toomer [318] believes that Hipparchus of Rhodes (190–120 BC)used linear interpolation in the construction of tables of the so-called “chord function” (re-lated to the sine function) for the purpose of computing the positions of celestial bodies.Later examples are found in the Almagest (“The Mathematical Compilation”, ca. 140 AD)of Claudius Ptolemy, the Egypt-born Greek astronomer-mathematician who propoundedthe geocentric view of the universe which prevailed until the 16th century. Apart fromtheory, this influential work also contains numerical tables of a wide variety of trigonomet-ric functions defined for astronomical purposes. To avoid the tedious calculations involvedin the construction of tables of functions of more than one variable, Ptolemy used an ap-proach that amounts to tabulating the function only for the variable for which the functionvaries most, given two bounding values of the other variable, and to provide a table ofcoefficients to be used in an “adaptive” linear interpolation scheme for computation of thefunction for intermediate values of this latter variable [335].

II.B Interpolation in Early-Medieval China and India

Analysis of the computational techniques on which early-medieval Chinese ephemeridesare based often reveals the use of higher-order interpolation formulae.4 The first personto use second-order interpolation for computing the positions of the sun and the moon inconstructing a calendar is said to be the astronomer Liu Zhuo. Around 600 AD he usedthis technique in producing the so-called Huang jı lı, or “Imperial Standard Calendar”.According to Lı Yan & Du Shıran [204], the formula involved in his computations reads,in modern notation:5

f(x0 + ξT ) = f(x0) +ξ

2(∆1 + ∆2) + ξ(∆1 − ∆2) − ξ2

2(∆1 − ∆2), (1)

with 0 6 ξ < 1, T > 0, ∆1 = f(x0+T )−f(x0), and ∆2 = f(x0+2T )−f(x0+T ), and withf(x0), f(x0 + T ), and f(x0 + 2T ) the observed results at times x0, x0 + T , and x0 + 2T ,respectively. This formula is closely related to later Western interpolation formulae, to bediscussed in the next section. Methods for second-order interpolation of unequal-intervalobservations were later used by the astronomer Monk Yı Xıng in producing the so-called“Da Yan Calendar” (727 AD) and by Xu Ang in producing the “Xuan Mıng Calendar”(822 AD). The latter also used a second-order formula for interpolation of equal-intervalobservations equivalent to the formula used by Liu Zhuo.

Accurate computation of the motion of celestial bodies, however, requires more sophis-ticated interpolation techniques than just second order. More complex techniques werelater developed by Guo Shoujıng and others. In 1280 AD they produced the so-calledShou shı lı, or “Works and Days Calendar”, for which they used third-order interpolation.Although they did not write down explicitly third-order interpolation formulae, it followsfrom their computations recorded in tables that they had grasped the principle.

Important contributions in the area of finite-difference computation were made by theChinese mathematician Zhu Shıjie. In his book Sıyuan yujian (“Jade Mirror of the FourOrigins”, 1303 AD), he gave the following problem (quoted freely from Martzloff [214]):

4The paragraphs on Chinese contributions to interpolation theory are based on the information providedin the books by Martzloff [214] and Lı Yan & Du Shıran [204]. For a more elaborate treatment of thetechniques and formulae mentioned here, see the latter. This reference was brought to the author’s attentionby Phillips in the brief historical notes on interpolation in his recent book [252].

5Note that, although supposedly unintended, the actual formula given by Lı Yan and Du Shıran is onlyvalid in cases where the time interval equals one, since the variable is not normalized. The formula givenhere is identical to theirs, except that we use a normalized variable.


Soldiers are recruited in cubes. On the first day, the side of the cube is three. On thefollowing days, it is one more per day. At present, it is fifteen. Each soldier receivestwo hundred and fifty guan per day. What is the number of soldiers recruited andwhat is the total amount paid out?

In explaining the answer to the first question, Zhu Shıjie gives a sequence of verbalinstructions (a “resolutory rule”) for finding the solution, which, when cast in modernalgebraic notation, reveals the suggestion to use the following formula:

f(n) = n∆0 +12!n(n− 1)∆2

0

+13!n(n− 1)(n − 2)∆3

0

+14!n(n− 1)(n − 2)(n − 3)∆4

0,

(2)

where f(n) is the total number of soldiers recruited in n days and the differences are definedby ∆j = f(j + 1) − f(j) and ∆i

j = ∆i−1j+1 −∆i−1

j , with i > 1 and j > 0 integers. Althoughthe specific problem requires only differences up to fourth order, the proposed formula tosolve it can easily be generalized to any arbitrary degree, and has close connections withlater Western interpolation formulae to be discussed in the next section.

In India, work on higher-order interpolation started around the same time as in China.6

In his work Dhyanagraha (ca. 625 AD), the astronomer-mathematician Brahmagupta in-cluded a passage in which he proposed a method for second-order interpolation of the sineand versed sine functions. Rephrasing the original Sanskrit text in algebraic language,Gupta [124] arrived at the following formula:

f(x0 + ξT ) = f(x0) +ξ

2

{∆f(x0 − T ) + ∆f(x0)

}

+ξ2

2

{∆f(x0) − ∆f(x0 − T )

},

(3)

with ∆f(x0) = f(x0 + T ) − f(x0). In a later work, Khandakhadyaka (665 AD), Brah-magupta also described a more general method that allowed for interpolation of unequal-interval data. In the case of equal intervals, this method reduces to (3).

Another rule for making second-order interpolations can be found in a commentary onthe seventh-century work Mahabhaskarıya by Bhaskara I, ascribed to Govindasvami (ca.800–850 AD). Expressed in algebraic notation, it reads [124]:

f(x0 + ξT ) = f(x0) + ξ∆f(x0) +ξ(ξ − 1)

2

{∆f(x0) − ∆f(x0 − T )

}. (4)

It is not difficult to see that this formula is equivalent to (3). According to Gupta [124],it is also found in two early 15th-century commentaries by Paramesvara.

II.C Late Medieval Sources on Interpolation

Use of the just described second-order interpolation formulae amounts to fitting a parabolathrough three consecutive tabular values. Kennedy [164] mentions that parabolic inter-polation schemes are also found in several Arabic and Persian sources. Noteworthy are

6Martzloff [214], referring to Cullen [58], conjectures that this may not be a coincidence, since it wasthe time when Indian and Chinese astronomers were working together at the court of the Tang.

III The Age of Scientific Revolution PP-5

the works al-Qanun’l-Mas’udi (“Canon Masudicus”, 11th century) by al-Bırunı and Zıj-i-Khaqanı (early 15th century) by al-Kashı. Concerning the parabolic interpolation methodsdescribed therein, Gupta [124] and later Rashed [260] have pointed at possible Indian in-fluences, since the important works of Brahmagupta were translated into Arabic as earlyas the eighth century AD. Not to mention the fact that al-Bırunı himself travelled throughand resided in several parts of India, studied Indian literature in the original, wrote a bookabout India, and translated several Sanskrit texts into Arabic [163].

III The Age of Scientific Revolution

Apparently, totally unaware of the important results obtained much earlier in other partsof the world, interpolation theory in Western countries started to develop only after agreat revolution in scientific thinking. Especially the new developments in astronomy andphysics, initiated by Copernicus, continued by Kepler and Galileo, and culminating inthe theories of Newton, gave strong impetus to the further advancement of mathematics,including what is now called “classical” interpolation theory.7 This section highlightsthe most important contributions to interpolation theory in Western countries until thebeginning of the 20th century.

III.A A General Interpolation Formula for Equidistant Data

Before reviewing the different classical interpolation formulae, we first study one of thebetter known.8 Suppose that we are given measurements of some quantity at x0, x0 ± T ,x0±2T , . . . , and that in order to obtain its value at any intermediate point x0+ξT, ξ ∈ R,we locally model it as a polynomial f : R → R of given degree n ∈ N, that is, f(x0 +ξT ) =a0 + a1ξ + a2ξ

2 + . . . + anξn. It is easy to show [349] that any such polynomial can be

written in terms of factorials [ξ]k = ξ(ξ − 1)(ξ − 2) · · · (ξ − k + 1), with k > 0 integer, asf(x0 + ξT ) = c0 + c1[ξ] + c2[ξ]2 + . . . + cn[ξ]n. If we now define the first-order differenceof any function φ : R → R at any ξ as ∆φ(ξ) = φ(ξ + 1) − φ(ξ), and similarly thehigher-order differences as ∆pφ(ξ) = ∆p−1φ(ξ + 1) − ∆p−1φ(ξ), for all p > 1 integer, itfollows that ∆[ξ]k = k[ξ]k−1. By repeated application of the difference operator ∆ to thefactorial representation of f(x0 + ξT ), and taking ξ = 0, we find that the coefficients ck,k = 0, 1, . . . , n, can be expressed as ck = ∆kf(x0)/k!, so that if n could be made arbitrarilylarge, we would have:

f(x0 + ξT ) = f(x0) + ξ∆f(x0) +12!ξ(ξ − 1)∆2f(x0)

+13!ξ(ξ − 1)(ξ − 2)∆3f(x0)

+14!ξ(ξ − 1)(ξ − 2)(ξ − 3)∆4f(x0) + . . .

(5)

7In constructing the chronology of classical interpolation formulae presented in this section, theinteresting—though individually incomplete—accounts given by Fraser [105], Goldstine [112], Joffe [157],and Turnbull [322] have been most helpful.

8This section includes explicit formulae only insofar as necessary to demonstrate the link with those inthe previous or next section. For a more detailed treatment of these and other formulae, including suchaspects 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 role of interpolationin numerical differentiation and integration.


This general formula9 was first written down in 1670 by Gregory and can be foundin a letter by him to Collins [118]. Particular cases of it, however, had been publishedseveral decades earlier by Briggs,10 the man who brought to fruition the work of Napier onlogarithms. In the introductory chapters to his major works [29,30] he described the preciserules by which he carried out his computations, including interpolations, in constructingthe tables contained therein. In the first, for example, he described a subtabulation rulethat, when written in algebraic notation, amounts to (5) for the case when third- andhigher-order differences are negligible [112, 322]. It is known [112, 200] that still earlier,around 1611, Harriot used a formula equivalent to (5) up to fifth-order differences.11 Buteven he was not the first in the world to write down such rules. It is not difficult to seethat the right-hand side of the second-order interpolation formula used by Liu Zhuo, (1),can be rewritten so that it equals the first three terms of (5). And if we replace the integerargument n by the real variable x in Zhu Shıjie’s formula, (2), we obtain at once (5) forthe case when x0 = 0, f(0) = 0, and T = 1.

III.B Newton’s General Interpolation Formulae

Notwithstanding these facts, it is justified to say that “there is no single person who didso much for this field, as for so many others, as Newton” [112]. His enthusiasm becomesclear in a letter he wrote to Oldenburg [237], where he first describes a method by whichcertain functions may be expressed in series of powers of x, and then goes on to say:12

But I attach little importance to this method because when simple series are notobtainable with sufficient ease, I have another method not yet published by which theproblem is easily dealt with. It is based upon a convenient, ready and general solutionof this problem. To describe a geometrical curve which shall pass through any givenpoints. (...) And although it may seem to be intractable at first sight, it is neverthelessquite the contrary. Perhaps indeed it is one of the prettiest problems that I can everhope to solve.

The contributions of Newton to the subject are contained in (i) a letter [236] to Smith in1675, (ii) a manuscript entitled Methodus Differentialis [235], published in 1711, althoughearlier versions were probably written in the mid-1670s, (iii) a manuscript entitled RegulaDifferentiarum, written in 1676 but first discovered and published in the 20th century[104,105], and (iv) Lemma V in Book III of his celebrated Principia [233], which appearedin 1687.13 The latter was published first and contains two formulae. The first dealswith equal-interval data and is precisely (5), which Newton seems to have discovered

9It is interesting to note that Taylor [311] obtained his now well-known series as a simple corollary to(5). It follows, indeed, by substituting ξ = h/T and taking T → 0. (It is important to realize here thatf(x0) is in fact f(x0 + ξT )|ξ=0, so that ∆f(x0) = f(x0 + T ) − f(x0), and similar for the higher-orderdifferences.) The special version for x0 = 0 was later used by Maclaurin [208] as a fundamental tool. Seealso e.g. Kline [169].

10Gregory was also aware of a later method by Mercator, for in his letter he refers to his own method asbeing “both more easie and universal than either Briggs or Mercator’s” [118]. In France, it was Moutonwho used a similar method around that time [228].

11Goldstine [112] speculates that Briggs was aware of Harriot’s work on the subject, and is inclined torefer to the formula as the Harriot-Briggs relation. Neither Harriot nor Briggs, however, ever explainedhow they obtained their respective rules, and it has remained unclear up till today.

12The somewhat free translation from the original Latin is from Fraser [105] and differs, although notfundamentally, from that given by Turnbull [237].

13All of these are reproduced (whether or not translated) and discussed in a booklet by Fraser [105].


independently of Gregory.14 The second formula deals with the more general case ofarbitrary-interval data and may be derived as follows.

Suppose that the values of the aforementioned quantity are given at x0, x±1, x±2, . . . ,which may be arbitrary, and that in order to obtain its value at intermediate points wemodel it again as a polynomial function f : R → R. If we then define the first-order divideddifference of any function φ : R → R for any two ξ0 6= ξ1 as φ(ξ0, ξ1) =

(φ(ξ0)−φ(ξ1)

)/(ξ0−

ξ1), it follows that the value of f at any x ∈ R could be written as f(x) = f(x0) + (x −x0)f(x, x0). And if we define the higher-order divided differences15 as φ(ξ0, . . . , ξp) =(φ(ξ0, . . . , ξp−1) − φ(ξ1, . . . , ξp)

)/(ξ0 − ξp), for all p > 1 integer, we can substitute for

f(x, x0) the expression that follows from the definition of f(x, x0, x1), and subsequentlyfor f(x, x0, x1) the expression that follows from the definition of f(x, x0, x1, x2), etc., sothat if we could go on, we would have

f(x) = f(x0) + (x− x0)f(x0, x1)

+ (x− x0)(x− x1)f(x0, x1, x2)

+ (x− x0)(x− x1)(x− x2)f(x0, x1, x2, x3) + . . .

(6)

It is this formula that can be considered the most general of all classical interpolationformulae. As we will see in the sequel, all later formulae can easily be derived from it.16

III.C Variations of Newton’s General Interpolation Formulae

The presentation of the two interpolation formulae in the Principia is heavily condensedand contains no proofs. Newton’s Methodus Differentialis contains a more elaborate treat-ment, including proofs and several alternative formulae. Three of those formulae forequal-interval data were discussed a few years later by Stirling [306].17 These are theGregory-Newton formula, and two central-difference formulae, the first of which is nowknown as the Newton-Stirling formula:18

f(x0 + ξT ) = f(x0) + ξ∆f(x0) + ∆f(x0 − T )

2

+12!ξ2∆2f(x0 − T )

+13!ξ(ξ2 − 12

)∆3f(x0 − T ) + ∆3f(x0 − 2T )2

+14!ξ2

(ξ2 − 12

)∆4f(x0 − 2T ) + . . .

(7)

It is interesting to note that Brahmagupta’s formula, (3), is in fact the Newton-Stirlingformula for the case when the third- and higher-order differences are zero.

14This is probably why it is nowadays usually referred to as the Gregory-Newton formula. There isreason to suspect, however, that Newton must have been familiar with Briggs’ works [105].

15Although Newton appears to have been the first to use these for interpolation, he did not call them“divided differences”. It has been said [349] that it was De Morgan [75] who first used the term.

16Equation (5), for example, follows by substituting x1 = x0 + T , x2 = x0 + 2T , x3 = x0 + 3T ,. . . , andx = x0 + ξT , and rewriting the divided differences f(x0, . . . , xk) in terms of finite differences ∆kf(x0).

17Newton’s general formula was treated by him in his 1730 booklet [307] on the subject.18The formula may be derived from (6) by substituting x1 = x0 + T , x2 = x0 − T , x3 = x0 + 2T ,

x4 = x0 − 2T ,. . . , and x = x0 + ξT , rewriting the divided differences f(x0, . . . , xk) in terms of finitedifferences ∆kf(x0), and rearranging the terms [349].


A very elegant alternative representation of Newton’s general formula (6) that doesnot require the computation of finite or divided differences was published in 1779 byWaring [344], and reads:

f(x) = f(x0)(x− x1)(x− x2)(x− x3) · · ·

(x0 − x1)(x0 − x2)(x0 − x3) · · · +

f(x1)(x− x0)(x− x2)(x− x3) · · ·

(x1 − x0)(x1 − x2)(x1 − x3) · · · +

f(x2)(x− x0)(x− x1)(x− x3) · · ·

(x2 − x0)(x2 − x1)(x2 − x3) · · · + . . .

(8)

It is nowadays usually attributed to Lagrange, who, in apparent ignorance of Waring’spaper, published it 16 years later [180]. The formula may also be obtained from a closelyrelated representation of Newton’s formula due to Euler [90]. According to Joffe [157],it was Gauss who first noticed the logical connection and proved the equivalence of theformulae by Newton, Euler, and Waring-Lagrange, as appears from his posthumous works[110], although Gauss did not refer to his predecessors.

In 1812, Gauss delivered a lecture on interpolation, the substance of which was recordedby his then student, Encke, who first published it not until almost two decades later [88].Apart from other formulae, he also derived the one which is now known as the Newton-Gauss formula:

f(x0 + ξT ) = f(x0) + ξ∆f(x0) +12!ξ(ξ − 1)∆2f(x0 − T )

+13!

(ξ + 1)ξ(ξ − 1)∆3f(x0 − T )

+14!

(ξ + 1)ξ(ξ − 1)(ξ − 2)∆4f(x0 − 2T ) + . . .

(9)

It is this formula19 that formed the basis for later theories on sampling and reconstruction,as will be discussed in the next section. Note that this formula too had its precursor, inthe form of Govindasvami’s rule, (4).

In the course of the 19th century, two more formulae closely related to (9) were devel-oped. The first appeared in a paper by Bessel [14] on computing the motion of the moon,and was published by him because, in his own words, he could “not recollect having seenit anywhere”. The formula is, however, equivalent to one of Newton’s in his MethodusDifferentialis, which is the second central-difference formula discussed by Stirling [306],and has therefore been called the Newton-Bessel formula. The second formula, whichhas frequently been used by statisticians and actuaries, was developed by Everett [91,92]around 1900, and reads:

f(x0 + ξT ) = F (ξ, δ)f(x0 + T ) + F (1 − ξ, δ)f(x0), (10)

where

F (ξ, δ) = ξ +13!ξ(ξ2 − 12)δ2 +

15!ξ(ξ2 − 12)(ξ2 − 22)δ4 + . . . , (11)

19Even more easily than the Newton-Stirling formula, this formula follows from (6) by proceeding in asimilar fashion as in the previous footnote.


and use has been made of Sheppard’s central-difference operator δ, defined by δφ(ξ) =φ(ξ + 1

2) − φ(ξ − 12 ) and δpφ(ξ) = δp−1φ(ξ + 1

2 ) − δp−1φ(ξ − 12), p > 1 integer, for any

function φ : R → R at any ξ. The elegance of this formula lies in the fact that, incontrast with the earlier mentioned formulae, it involves only the even-order differencesof the two table-entries between which to interpolate.20 It was later noted by Joffe [157]and Lidstone [193], respectively, that the formulae of Bessel and Everett had alternativelybeen proven by Laplace by means of his method of generating functions [73,74].

III.D Studies on More General Interpolation Problems

By the beginning of the 20th century, the problem of interpolation by finite or divided dif-ferences had been studied by astronomers, mathematicians, statisticians, and actuaries,21

and most of the now well-known variants of Newton’s original formulae had been workedout. This is not to say, however, that there are no more advanced developments to reporton. Quite to the contrary. Already in 1821, Cauchy [43] studied interpolation by meansof a ratio of two polynomials and showed that the solution to this problem is unique, theWaring-Lagrange formula being the special case for the second polynomial equal to one.22

Generalizations for solving the problem of multivariate interpolation in the case of fairlyarbitrary point configurations began to appear in the second half of the 19th century, inthe works of Borchardt and Kronecker [24,108,176].

A generalization of a different nature was published in 1878 by Hermite [134], whostudied and solved the problem of finding a polynomial of which also the first few deriva-tives assume prespecified values at given points, where the order of the highest derivativemay differ from point to point. And in a paper [15] published in 1906, Birkhoff studied theeven more general problem: given any set of points, find a polynomial function that satis-fies prespecified criteria concerning its value and/or the value of any of its derivatives foreach individual point.23 Hermite and Birkhoff type of interpolation problems—and their

20This is achieved by expanding the odd-order differences in the Newton-Gauss formula according totheir definition and rearranging the terms after simple transformations of the binomial coefficients [349].Alternatively, we could expand the even-order differences so as to end up with only odd-order differences.The resulting formula appears to have been described first by Steffensen [302], and is therefore sometimesreferred to as such [138,268], although he himself calls it Everett’s second interpolation formula.

21Many of them introduced their own system of notation and terminology, leading to confusion andresearchers reformulating existing results. The point was discussed by Joffe [157], who also made anattempt to standardize yet another system. It is, however, Sheppard’s notation [290] for central and meandifferences that has survived in most later publications.

22It was Cauchy, also, who in 1840 found an expression for the error caused by truncating finite-differenceinterpolation series [42]. The absolute value of this so-called Cauchy remainder term can be minimizedby choosing the abscissae as the zeroes of the polynomials introduced later by Tchebychef [312]. See e.g.Davis [63], Hildebrand [138], or Schwarz [285] for more details.

23Birkhoff interpolation, also known as lacunary interpolation, initially received little attention, untilSchoenberg [276] revived interest in the subject. The problem has since usually been stated in terms ofthe pair (E, X), where X = {xi}m

i=0 is the set of points, or nodes, and E = [ei,j ]mi=0,

kj=0 is the so-called

incidence matrix, or interpolation matrix, with ei,j = 1 for those i and j for which the interpolatingpolynomial P is to satisfy a given criterion P (j)(xi) = ci,j , and ei,j = 0 otherwise. Several special caseshad been studied earlier and carry their own name: if E is an (m+1)×1 column matrix with ei,0 = 1 for alli = 0, . . . , m, we have the Waring-Lagrange interpolation problem; if, on the other hand, it is a 1× (k + 1)row matrix with e0,j = 1 for all j = 0, . . . , k, we may speak of a Taylor interpolation problem [63]; if Eis an (m + 1) × (k + 1) matrix with ei,j = 1 for all i = 0, . . . , m and j = 0, . . . , ki, with ki 6 k, we haveHermite’s interpolation problem, where the case ki = k for all i = 0, . . . , m, with usually k = 1, is alsocalled osculatory or osculating interpolation [63,138,302]; the problem corresponding to E = I , with I the(m + 1)× (m + 1) unit matrix, was studied by Abel [1], and later by Gontcharoff and others [63,114,353];and finally we mention the two-point interpolation problem studied by Lidstone [63,194,352,353], for whichE is a 2 × (k + 1) matrix with ei,j = 1 for all j even.


multivariate versions, not necessarily on Cartesian grids—have received much attention inthe past decades. A more detailed treatment is outside the scope of this paper, however,and the reader is referred to relevant books and reviews [22,108,109,201–203].

III.E Approximation versus Interpolation

Another important development from the late 1800s is the rise of approximation the-ory. For a long time, one of the main reasons for the use of polynomials had been thefact that they are simply easy to manipulate, e.g. to differentiate or integrate. In 1885,Weierstrass [346] also justified their use for approximation by establishing the so-calledapproximation theorem, which states that every continuous function on a closed intervalcan be approximated uniformly to any prescribed accuracy by a polynomial.24 The the-orem does not provide any means of obtaining such a polynomial, however, and it soonbecame clear that it does not necessarily apply if the polynomial is forced to agree with thefunction at given points within the interval, i.e., in the case of an interpolating polynomial.

Examples of meromorphic functions for which the Waring-Lagrange interpolator doesnot converge uniformly were given by Meray [220, 221] and later Runge [266]—especiallythe latter has become well known and can be found in most modern books on the topic.A more general result is due to Faber [93], who in 1914 showed that for any prescribedtriangular system of interpolation points there exists a continuous function for which thecorresponding Waring-Lagrange interpolation process carried out on these points doesnot converge uniformly to this function. Although it has later been proven possible toconstruct interpolating polynomials that do converge properly for all continuous functions,for example by using the Hermite type of interpolation scheme proposed by Fejer [96] in1916, these findings clearly revealed the “inflexibility” of algebraic polynomials and theirlimited applicability to interpolation.

IV The Information and Communication Era

When Fraser [105], in 1927, summed up the state of affairs in classical interpolation theory,he also expressed his expectations concerning the future, and speculated: “The 20th cen-tury will no doubt see extensions and developments of the subject of interpolation beyondthe boundaries marked by Newton 250 years ago”. This section gives an overview of theadvances in interpolation theory in the past century, proving that Fraser was right. In fact,he was so right that our rendition of this part of history is necessarily of a more limitednature than the expositions in the previous sections. After having given an overview of thedevelopments that led to the two most important theorems on which modern interpolationtheory rests, we focus primarily on their later impact on signal and image processing.

IV.A From Cardinal Function to Sampling Theory

In his celebrated 1915 paper [348], E. T. Whittaker noted that given the values of afunction f corresponding to an infinite number of equidistant values of its argument: x0,x0 ± T , x0 ± 2T , . . ., from which we can construct a table of differences for interpolation,there exist many other functions that give rise to exactly the same difference-table. Hethen considered the Newton-Gauss formula, (9), and set out to answer the question of

24For more detailed information on the development of approximation theory, see the recently publishedhistorical review by Pinkus [253].

IV The Information and Communication Era PP-11

which one of the cotabular functions is represented by it. The answer, he proved, is thatunder certain conditions it represents the cardinal function

C(x) =+∞∑

k=−∞f(x0 + kT )

sinπ

T(x− x0 − kT )

π

T(x− x0 − kT )

, (12)

which he observed to have the remarkable properties that, apart from the fact that it iscotabular with the original function since C(x0 + kT ) = f(x0 + kT ) for all k ∈ Z, it hasno singularities and all “constituents” of period less than 2T are absent.

Although Whittaker did not refer to any earlier works, it is now known25 that the series(12) with x0 = 0 and T = 1 had essentially been given as early as 1899 by Borel [26],who had obtained it as the limiting case of the Waring-Lagrange interpolation formula.26

Borel, however, did not establish the “bandlimited” nature of the resulting function. Nordid Steffensen in a paper [301] published in 1914, in which he gave the same formula asBorel, though he referred to Hadamard [125] as his source. De la Vallee Poussin [71], in1908, studied the closely related case where the summation is over a finite interval, butthe number of known function values in that interval goes to infinity by taking T = π/mand m → ∞. In contrast with the Waring-Lagrange polynomial interpolator, which maydiverge as we have seen in the previous section, he found that the resulting interpolatingfunction converges to the original function at any point in the interval where that functionis continuous and of bounded variation.

The issue of convergence was an important one in subsequent studies of the cardinalfunction. In establishing the equivalence of the cardinal function and the function obtainedby the Newton-Gauss interpolation formula, Whittaker had assumed convergence of bothseries expansions. Later authors showed, by particular examples, that the former maydiverge when the latter converges. The precise relationship was studied by Ferrar [98],who showed that when the series of type (12) converges, (9) also converges and has the samesum; if, on the other hand, (9) is convergent, (12) is either convergent or has a generalizedsum in the sense used by de la Vallee Poussin for Fourier series [72,358]. Concerning theconvergence of the cardinal series, Ferrar [97–99], and later J. M. Whittaker [350,351,353],studied several criteria. Perhaps the most important is that of

∑k 6=0 |sk/k| < ∞, where

sk = f(x0 + kT ), being a sufficient condition for having absolute convergence—a criterionthat had also been given by Borel [26]. It was J. M. Whittaker [350,353], also, who gavemore refined statements as to the relation between the cardinal series and the truncatedFourier integral representation of a function in the case of convergence—results that alsorelate to the property called by Ferrar the “consistency” of the series [97, 99, 350, 353],which implies the possibility of reproducing the cardinal function as given in (12) by usingits values C(x′0 + kT ′), with 0 < T ′ < T .

It must have been only shortly after publication of J. M. Whittaker’s works [350,351, 353] on the cardinal series that Shannon recognized their evident importance to thefield of communication. He formulated the now well-known sampling theorem, which he

25For more information on the development of sampling theory, the reader is referred to the historicalaccounts given by Higgins [135] and Butzer & Stens [35].

26Since Borel [26], the equivalence of the Waring-Lagrange interpolation formula and (12) in the caseof infinitely many known function values between which to interpolate has been pointed out and provenby many authors [31, 35, 97, 135, 139, 149, 211, 353]. Apart from the Newton-Gauss formula, as shown byWhittaker [348], equivalence also holds for other classical interpolation formulae, such as Newton’s divideddifference formula [301,353], or the formulae by Everett, Stirling, and Bessel [216], discussed in the previoussection. Given Borel’s result, this is an almost trivial observation, since all classical schemes yield the exactsame polynomial for a given set of known function values, irrespective of their number.


first published [288] without proof in 1948, and the subsequent year with full proof in apaper [289] apparently written already in 1940:

If a function f contains no frequencies higher thanW cycles per second, it is completelydetermined by giving its ordinates at a series of points spaced 1/2W seconds apart.

Later on in the paper, he referred to the critical sampling interval T = 1/2W as theNyquist 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 describingthe reconstruction process, he pointed out that

There is one and only one function whose spectrum is limited to a band W , and whichpasses through given values at sampling points separated 1/2W seconds apart. Thefunction can be simply reconstructed from the samples by using a pulse of the typesin(2πWx)/2πWx. (...) Mathematically, this process can be described as follows. Letsk be the kth sample. Then the function f is represented by

f(x) =∞∑

k=−∞sk

sinπ(2Wx− k)π(2Wx− k)

. (13)

As pointed out by Higgins [135], the sampling theorem should really be considered intwo parts, as done above: the first stating the fact that a bandlimited function is completelydetermined by its samples, the second describing how to reconstruct the function usingits samples. Both parts of the sampling theorem were given in a somewhat differentform by J. M. Whittaker [350, 351, 353] and before him also by Ogura [241, 242]. Theywere probably not aware of the fact that the first part of the theorem had been statedas early as 1897 by Borel [25].27 As we have seen, Borel also used around that timewhat became known as the cardinal series. However, he appears not to have made thelink [135]. In later years it became known that the sampling theorem had been presentedbefore Shannon to the Russian communication community by Kotel’nikov [173]. In moreimplicit, verbal form, it had also been described in the German literature by Raabe [257].Several authors [33,205] have mentioned that Someya [296] introduced the theorem in theJapanese literature parallel to Shannon. In the English literature, Weston [347] introducedit independently of Shannon around the same time.28

IV.B From Osculatory Interpolation Problems to Splines

Having arrived at this point, we go back again more than half a century to follow aparallel development of quite different nature. It is clear that practical application of anyof the classical polynomial interpolation formulae discussed in the previous section impliestaking into account only the first few of the infinitely many terms; in most situationsit will be computationally prohibitive to consider all, or even a large number of knownfunction values when computing an interpolated value. Keeping the number of terms fixed

27Several authors, following Black [16], have claimed that this first part of the sampling theorem wasstated even earlier by Cauchy, in a paper [41] published in 1841. However, the paper of Cauchy does notcontain such a statement, as has been pointed out by Higgins [135].

28As a consequence of the discovery of the several independent introductions of the sampling theorem,people started to refer to the theorem by including the names of the aforementioned authors, resulting insuch catchphrases as “the Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem” [155] or even “theWhittaker-Kotel’nikov-Raabe-Shannon-Someya sampling theorem” [33]. To avoid confusion, perhaps thebest thing to do is to refer to it as the sampling theorem, “rather than trying to find a title that doesjustice to all claimants” [136].


implies fixing the degree of the polynomial curves resulting in each interpolation interval.Irrespective of the degree, however, the composite piecewise polynomial interpolant willgenerally not be continuously differentiable at the transition points.

The need for smoother interpolants in some applications led in the late 1800s to thedevelopment of so-called osculatory interpolation techniques, most of which appeared inthe actuarial literature [122, 286, 291]. A well-known example of this is the formula pro-posed in 1899 by Karup [162] and independently described by King [168] a few years later,which may be obtained from Everett’s general formula (10) by taking

F (ξ, δ) = ξ +12ξ2(ξ − 1)δ2, (14)

and results in a piecewise third-degree polynomial interpolant that is continuous and, incontrast with Everett’s third-degree interpolant, is also continuously differentiable every-where.29 By using this formula, it is possible to reproduce polynomials up to seconddegree. Another example is the formula proposed in 1906 by Henderson [130], which maybe obtained from (10) by substituting

F (ξ, δ) = ξ +16ξ(ξ2 − 1)δ2 − 1

12ξ2(ξ − 1)δ4, (15)

and also yields a continuously differentiable, piecewise third-degree polynomial interpolant,but is capable of reproducing polynomials up to third degree. A third example is theformula published in 1927 by Jenkins [153], obtained from (10) by taking

F (ξ, δ) = ξ +16ξ(ξ2 − 1)δ2 − 1

36ξ3δ4. (16)

The first and second term of this function are equal to those of Henderson’s and Everett’sfunction, but the third term is chosen in such a way that the resulting composite curve isa piecewise third-degree polynomial that is twice continuously differentiable. The price topay, however, is that this curve is not an interpolant.

The need for practically applicable methods for interpolation or smoothing of em-pirical data also formed the impetus to Schoenberg’s study of the subject. In his 1946landmark paper [274, 275] he noted that for every osculatory interpolation formula ap-plied to equidistant data, where he assumed the distance to be unity, there exists an evenfunction Φ : R → R, in terms of which the formula may be written as

f(x) =∞∑

k=−∞yk Φ(x− k), (17)

where Φ, which he termed the basic function of the formula, completely determines theproperties of the resulting interpolant and reveals itself when applying the initial formula tothe impulse sequence defined by y0 = 1 and yk = 0, ∀k 6= 0. By analogy with Whittaker’scardinal series, (12), Schoenberg referred to the general expression (17) as a formula of

29The word “osculatory” originates from the Latin verb osculari, which literally means “to kiss” andcan be translated here as “joining smoothly”. Notice that the meaning of the word in this context is moregeneral than in Footnote 23: Hermite interpolation may be considered that type of osculatory interpolationwhere the derivatives up to some degree are not only supposed to be continuous everywhere but are alsorequired to assume prespecified values at the sample points. It is especially this latter type of interpolationproblem to which the adjective “osculatory” has been attached in later publications [32,65,161,192, 267].For a more elaborate discussion of osculatory interpolation in the original sense of the word, see severalsurvey papers [122,286,291].


the cardinal type, but noted that the basic function Φ(x) = sin(πx)/πx is inadequate fornumerical purposes due to its excessively low damping rate. The basic functions involved inWaring-Lagrange interpolation, on the other hand, possess the limiting property of beingat most continuous, but not continuously differentiable. He then pointed at the smoothcurves obtained by the use of a mechanical spline,30 argued that these are piecewise cubicarcs with a continuous first- and second-order derivative, and continued to introduce thenotion of the mathematical spline:

A real function f defined for all real x is called a spline curve of order L and denotedby SL if it enjoys the following properties: 1) it is composed of polynomial arcs ofdegree at most L−1; 2) it is of class CL−2, i.e., f has L−2 continuous derivatives; 3)the only possible function points of the various polynomial arcs are the integer pointsx = L if L is even, or else the points x = L+ 1

2 if L is odd.

Notice that these requirements are satisfied by the curves resulting from the aforemen-tioned smoothing formula proposed by Jenkins, and also studied by Schoenberg [274],which constitutes one of the earliest examples of a spline generating formula.

After having given the definition of a spline curve, Schoenberg continued to prove that

Any spline curve SL may be represented in one and only one way in the form

SL(x) =∞∑

k=−∞yk ML(x− k) (18)

for appropriate values of the coefficients yk. There are no convergence difficultiessince ML(x) vanishes for |x| > L/2. Thus (18) represents an SL for arbitrary {yk}and represents the most general one.

Here, ML : R → R denotes the so-called B-spline of degree n = L − 1, which he haddefined earlier in the paper as the inverse Fourier integral

ML(x) =12π

∫ ∞

−∞

(sin(ω/2)ω/2

)L

eiωxdω, (19)

and, equivalently, also as31

ML(x) =1

(L− 1)!δLxL−1

+ , (20)

where δp is again the pth-order central difference operator and xn+ denotes the one-sided

power function defined as

xn+ ,

{xn, if x > 0

0, if x < 0.(21)

30The word “spline” can be traced back to the 18th century but by the end of the 19th century wasused to refer to “a flexible strip of wood or hard rubber used by draftsmen in laying out broad sweepingcurves” [293]. Such mechanical splines were used e.g. to draw curves needed in the fabrication of crosssections of ships’ hulls. Drucks or weights were placed on the strip to force it to go through given points,and the free portion of the strip would assume a position in space that minimized the bending energy [340].

31In his original 1946 paper [274, 275], Schoenberg referred to the ML strictly as “basic functions” or“basic spline curves”. The abbreviation “B-splines” was first coined by him twenty years later [277]. It isalso interesting here to point at the connection with probability theory: as is clear from (19), ML(x) can bewritten as the n-fold convolution of the indicator function M1(x) with itself, from which it follows that aB-spline of degree n represents the probability density function of the sum of L = n+1 independent randomvariables with uniform distribution in the interval [− 1

2, 1

2]. The explicit formula of this function was known

as early as 1820 by Laplace [73], and is essentially (20), as also acknowledged by Schoenberg [274]. Forfurther details on the history of B-splines, see e.g. Butzer et al. [34].


IV.C Convolution-Based Function Representation

Although there are certainly differences, it is interesting to look at the similarities betweenthe theorems described by Shannon and Schoenberg: both of them involve the definitionof a class of functions f : R → R satisfying certain properties, and both involve the repre-sentation of these functions by a mixed convolution of a set of coefficients ck with somebasic function, or kernel, ϕ : R → R, according to the formula

fT (x) =∑k∈Z

ck ϕ(x/T − k), (22)

where the subscript T , the sampling interval, is now added to stress the fact that we aredealing with a representation—or, depending on ϕ, perhaps only an approximation—of anysuch original function f based on its T -equidistant samples. In the case of Shannon, the fare bandlimited functions, the coefficients ck are simply the samples sk = f(kT ), and thekernel is the sinc function,32 defined as sinc(x) , sin(πx)/πx. In Schoenberg’s theorem,the functions f are piecewise polynomials of degree n that join smoothly according to thedefinition of a spline, the coefficients ck are computed from the samples sk, and the kernelis the nth-degree B-spline.33

In the decades to follow, both Shannon’s and Schoenberg’s paper would prove mostfruitful, but largely in different fields. The former had great impact on communicationengineering [17, 54, 159, 287], numerous signal processing and analysis applications [39,115, 136, 155, 211, 243], and to some 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 impact on geometrical modeling and computer-aided geometric design[78, 94, 182, 213, 223], computer graphics [9, 250], even font design [172], to mention but afew practical applications. In the remainder of this section, we will focus primarily on thefurther developments in signal and image processing.

IV.D Convolution-Based Interpolation in Signal Processing

When using Waring-Lagrange interpolation, the choice for the degree n of the resultingpolynomial pieces fixes the number of samples to be used in any interpolation interval ton+1. There is still freedom, however, to choose the position of the interpolation intervals,the end points of which constitute the transition points of the polynomial pieces of theinterpolant, relative to the sample intervals. It is easy to see that if they are chosen tocoincide with the sample intervals, there are n possibilities of choosing the position—in thesequence of samples to be used—of the two samples making up the interpolation interval,and that each of these possibilities gives rise to a different impulse response, or kernel.

In their 1973 study [271] of these kernels for use in digital signal processing, Schafer& Rabiner concluded that the only ones that are symmetrical, and thus do not introducephase distortions, are those corresponding to the cases where n is odd and the number

32The term “sinc” is usually held to be short for the Latin sinus cardinalis [136]. Although it has becomewell 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 [355].

33Later, in Section IV.I, it will become clear that the kinship between both theorems goes even further,in the sense that the sampling theorem for bandlimited functions is the limiting case of the samplingtheorem for splines.


of constituent samples on either side of the interpolation interval is the same. It must bepointed out, however, that their conclusion does not hold in general but is a consequenceof letting the interpolation and sample intervals coincide. If the interpolation intervals arechosen according to the parity of n, as in the aforementioned definition of the mathematicalspline, then the kernels corresponding to Waring-Lagrange central interpolation for even nwill also be symmetrical. Examples of these had already been given by Schoenberg [274].Schafer & Rabiner also studied the spectral properties of the odd-degree kernels, concludedthat the higher-order kernels possess considerably better lowpass properties than linearinterpolation, and discussed the design of alternative finite impulse response interpolatorsbased on prespecified bandpass and bandstop characteristics. More information on thiscan also be found in the tutorial review by Crochiere & Rabiner [57].

IV.E Cubic Convolution Interpolation in Image Processing

The early 1970s was also the time digital image processing really started to develop. One ofthe first applications reported in the literature was the geometrical rectification of digitalimages obtained from the first Earth Resources Technology Satellite (ERTS) launchedby the United States National Aeronautics and Space Administration (NASA) in 1972.The need for more accurate interpolations than obtained by standard linear interpolationin this application led to the development of a still very popular technique known ascubic convolution, which involves the use of a sinc-like kernel composed of piecewise cubicpolynomials. Apart from being interpolating, the kernel was designed to be continuous andto have a continuous first derivative. Cubic convolution was first mentioned by Rifman[262], discussed in some more detail by Simon [292], but the most general form of thekernel appeared first in a paper by Bernstein [13]:

ψ(x) =

(α+ 2)|x|3 − (α+ 3)|x|2 + 1, if 0 6 |x| < 1

α|x|3 − 5α|x|2 + 8α|x| − 4α, if 1 6 |x| < 2

0, if 2 6 |x|,(23)

where α is a free parameter resulting from the fact that the interpolation, continuity, andcontinuous differentiability requirements yield only seven equations, while the two cubicpolynomials defining the kernel make up a total of eight unknown coefficients.34 Theexplicit cubic convolution kernel given by Rifman and Bernstein in their respective papersis the one corresponding to α = −1, which results from forcing the first derivative of ψ tobe equal to that of the sinc function at x = 1. Two alternative criteria for fixing α weregiven by Simon [292]. The first consists in requiring the second derivative of the kernel tobe continuous at x = 1, which results in α = −3

4 . The second amounts to requiring thekernel to be capable of exact constant slope interpolation, which yields α = −1

2 . Althoughnot mentioned by Simon, the kernel corresponding to the latter choice for α is not onlycapable of reproducing 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

34Notice here that the application of convolution kernels to two-dimensional data defined on Cartesiangrids, as digital images are in most practical cases, has traditionally been done simply by extending (22)to fT (x, y) =

∑k∈Z

∑l∈Z

ck,l ϕ(x/Tx − k) ϕ(y/Ty − l), where Tx and Ty denote the sampling interval in xand y direction, respectively. This approach can of course be extended to any number of dimensions andallows for the definition and analysis of kernels in one dimension only.


by Hou & Andrews [145]. Their approach, mainly centered around the cubic B-spline, wasbased on matrix inversion techniques for computing the coefficients ck in (22). Qualitativeexperiments involving magnification (enlargement) and minification (reduction) of imagedata demonstrated the superiority of cubic B-Spline interpolation over techniques such asnearest-neighbor or linear interpolation, and even interpolation based on the truncatedsinc function as kernel. The results of the magnification experiments also clearly showedthe necessity for this type of interpolation to use the coefficients ck rather than the originalsamples sk in (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 intothis technique. He argued that the best choice for α in (23) is that for which the Taylorseries expansion of the interpolant fT resulting from cubic convolution interpolation ofequidistant samples of an original function f , agrees in as many terms as possible withthat of the original function. By using this criterion, he found that the optimal choice isα = −1

2 , in which case fT (x)−f(x) = O(T 3) for all x. This implies that the interpolationerror goes to zero uniformly at a rate proportional to the third power of the sample interval.In other words, for this choice of α cubic convolution yields a third-order approximationof the original function. For all other choices of α, he found that it yields only a first-orderapproximation, just like nearest-neighbor interpolation. He also pointed at the fact thatcubic Lagrange and cubic spline interpolation both yield a fourth-order approximation—the highest possible with piecewise cubics—and continued to derive a cubic convolutionkernel with the same property, at the cost of a larger spatial support.

A second, complementary study of the cubic convolution kernel (23) was publisheda little later by Park & Schowengerdt [246]. Rather than studying the properties of thekernel in the spatial domain, they carried out a frequency-domain analysis. The Maclaurinseries expansion of the Fourier transform of (23) can be derived to be

ψ(ω) = 1 − 415

(2α + 1)(ω/2)2 +135

(16α + 1)(ω/2)4 + . . . , (24)

where, as usual, ω denotes radial frequency. Based on this fact, they argued that thebest choice for the free parameter is α = −1

2 , since this maximizes the number of termsin which (24) agrees with the Fourier transform of the sinc kernel. That is to say, itprovides the best low-frequency approximation to the “ideal” reconstruction filter. Park& Schowengerdt [245,246] also studied the mean-square error, or squared L2-norm

ε2 =∫ ∞

−∞|f1(x) − f(x)|2dx, (25)

with f1 the image obtained from interpolating the samples sk = f(k) of an original imagef using any interpolation kernel ϕ. They showed that if the original image is bandlimited,i.e., f(ω) = 0 for all |ω| larger than some ωmax > 0 in this one-dimensional analysis, and ifsampling is performed at a rate equal to or higher than the Nyquist rate, this error—whichthey called the “sampling and reconstruction blur”—is equal to

η2 =12π

∫ ωmax

−ωmax

|f(ω)|2E(ω)dω, (26)

where


E(ω) = |1 − ϕ(ω)|2 +∑k∈Z∗

|ϕ(ω + 2πk)|2, (27)

with Z∗ = Z\{0}. In the case of undersampling, η2 represents the average error 〈ε2〉,where the averaging is over all possible sets of samples f(k + τ), with τ ∈ [0, 1]. Theyargued that if the energy spectrum of f is not known, the optimal choice for α is the onethat yields the best low-frequency approximation to E(ω) = 0. Substituting ψ for ϕ andcomputing the Maclaurin series expansion of the right-hand side of (27), they found thatthis best approximation is obtained by taking, again, α = −1

2 .Notwithstanding the achievements of Keys and Park & Schowengerdt, it is interesting

to note that cubic convolution interpolation corresponding to α = −12 had been suggested

in the literature at least three times before these authors. Already mentioned is Simon’spaper [292], published in 1975. A little earlier, in 1974, Catmull & Rom [40] had studiedinterpolation by “cardinal blending functions” of the type

Cn(x) =n∑

i=0

w(x+ i)i∏

j=i−nj 6=0

(x

j+ 1

)(28)

where n is the degree of polynomials resulting from the product on the right-hand sideand w is a weight function, or blending function, centered around x = n/2. Among theexamples they gave is the function corresponding to n = 1 and w the second-degree B-spline. This function can be shown to be equal to (23) with α = −1

2 . In the fields ofcomputer graphics and visualization, the third-order cubic convolution kernel is thereforeusually referred to as the Catmull-Rom spline. It has also been called the (modified orcardinal) cubic spline [12,120,132,167,207,225,226]. Finally, this cubic convolution kernelis precisely the kernel implicitly used in the previously mentioned osculatory interpolationscheme proposed around 1900 by Karup and King. More details on this can be found ina recent paper [217], which also demonstrates the equivalence of Keys’ fourth-order cubicconvolution and Henderson’s osculatory interpolation scheme mentioned earlier.

IV.H Cubic Convolution versus Spline Interpolation

A comparison of interpolation methods in medical imaging was presented by Parker etal. [247] in 1983. Their study included the nearest-neighbor kernel, the linear interpolationkernel, the cubic B-spline, and two cubic convolution kernels,35 viz. the ones correspondingto α = −1

2 and α = −1. Based on a frequency-domain analysis they concluded that thecubic B-spline yields the most smoothing and that it is therefore better to use a cubicconvolution kernel. This conclusion, however, resulted from an incorrect use of the cubicB-spline for interpolation, in the sense that the kernel was applied directly to the originalsamples sk instead of the appropriate coefficients ck—an approach that has been suggested(explicitly or implicitly) by many authors over the years [61, 101, 225, 227, 255, 270, 345].The point was later discussed by Maeland [209], who derived the true spectrum of thecubic spline interpolator, or cardinal cubic spline, as the product of the spectrum of the

35Note that Parker et al. referred to them consistently as “high-resolution cubic splines”. According toSchoenberg’s original definition, however, the cubic convolution kernel (23) is not a cubic spline, regardlessof the value of α. Some people have called piecewise polynomial functions with less than maximum(nontrivial) smoothness “deficient splines”. See also de Boor [65], who adopted the definition of a splinefunction as a linear combination of B-splines. When using the latter definition, the cubic convolutionkernel may indeed be called a spline. We will not do so, however, in this paper.


required prefilter and that of the cubic B-spline. From a correct comparison of the spectra,he concluded that cubic spline interpolation is superior compared to cubic convolutioninterpolation—a conclusion that would later be confirmed repeatedly by several evaluationstudies (to be discussed in Section IV.M).36

IV.I Spline Interpolation Revisited

In classical interpolation theory it was already known that it is better or even necessary insome cases to first apply some transformation to the original data before applying a giveninterpolation formula. The general rule in such cases is to apply transformations that willmake the interpolation as simple as possible. The transformations themselves, of course,should preferably also be as simple as possible. Stirling, in his 1730 book [307] on finitedifferences, wrote about this:37

As in common algebra, the whole art of the analyst does not consist in the resolution ofthe equations, but in bringing the problems thereto. So likewise in this analysis: thereis less dexterity required in the performance of the process of interpolation than in thepreliminary determination of the sequences which are best fitted for interpolation.

It should be clear from the foregoing discussion that a similar statement applies toconvolution-based interpolation using B-splines: the difficulty is not in the convolution, butin the preliminary determination of the coefficients ck. In order for B-spline interpolationto be a competitive technique, the computational cost of this preprocessing step should bereduced to a minimum—in many situations the important issue is not just accuracy, butthe trade-off between accuracy and computational cost. Hou & Andrews [145], as manybefore and after them, solved the problem by setting up a system of equations followedby matrix inversion. Even though there exist optimized techniques [113] for inverting theToeplitz type of matrices occurring in spline interpolation, this approach is unnecessarilycomplex and computationally expensive.

In the early 1990s it was shown by Unser et al. [326,328,329] that the B-spline inter-polation 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 fromapplying the interpolation requirement to (22):∑

l∈Z

cl βn(k − l) = sk, ∀k ∈ Z. (29)

Recalling that the z-transform of a convolution of two discrete sequences is equal to theproduct of the individual z-transforms, the z-transform of (29) reads C(z)Bn(z) = S(z).Consequently, the B-spline coefficients can be obtained as

C(z) =(Bn(z)

)−1S(z). (30)

Since, by definition, Bn(z) =∑

k∈Zβn(k)z−k, it follows from insertion of the explicit

form of βn that (Bn(z))−1 = 1 for n = 0 and n = 1, which implies that in these casesC(z) = S(z), that is to say, ck = sk. For any n > 2, however, (Bn(z))−1 is a digital “high-boost” filter that corrects for the blurring effects of the corresponding B-spline convolution

36It is therefore surprising that, even though there are now textbooks that acknowledge the superiorityof spline interpolation [27,150,151,354], many books since the late eighties [11,115,282,297, 300] give theimpression that cubic convolution is the state-of-the-art in image interpolation.

37The translation from Latin is as given by Whittaker & Robinson [349].


kernel. Although this was known to Hou & Andrews [145] and later authors [116,184,319],they did not realize that this filter can be implemented recursively.

Since βn is even for any n ∈ N, we have (Bn(z))−1 = (Bn(z−1))−1, which implies thatthe poles of the filter come in reciprocal pairs, so that the filter can be factorized as

(Bn(z)

)−1 = γ

bn/2c∏i=1

H(z; zi), (31)

where γ = 1/βn(bn/2c) is a constant factor and

H(z; zi) =−zi

(1 − ziz−1)(1 − ziz)(32)

is the factor corresponding to the pole pair {zi, z−1i }, with |zi| < 1. Since the poles of

(Bn(z))−1 are the zeroes of Bn(z), they are obtained by solving Bn(z) = 0. By a furtherfactorization of (32) into H(z; zi) = H−(z; zi)H+(z; zi), with

H+(z; zi) =1

(1 − ziz−1), H−(z; zi) =

−zi(1 − ziz)

, (33)

and by using the shift property of the z-transform, it is not difficult to show that in thespatial domain, application of H+(z; zi) followed by H−(z; zi) to given samples sk, k =0, 1, 2, . . . ,K − 1, amounts to applying the recursive filters:

c+k = sk + zi c+k−1, k = 1, 2, . . . ,K − 1, (34)

c−k = zi(c−k+1 − c+k

), k = K − 2, . . . , 1, 0, (35)

where the c+k are intermediate output samples resulting from the first, causal filter, andthe c−k are the output samples resulting from the second, anti-causal filter. For the initial-ization of the causal filter we may use mirror-symmetric boundary conditions, i.e., sk = sl

for (k + l)mod(2K − 2) = 0, which can be shown to result in [324]

c+0 =1

1 − z2K−2i

2K−3∑l=0

zlisl. (36)

In most practical cases, K will be sufficiently large to justify taking 1/(1−z2K−2i ) = 1 and

terminating the summation much earlier. An initial value for the anti-causal filter may beobtained from a partial-fraction expansion of (32), resulting in [329]

c−K−1 =−zi

(1 − z2i )

(2c+K−1 − sK−1

). (37)

Summarizing, the prefilter (Bn(z))−1 corresponding to a B-spline of degree n hasbn/2c pole pairs {zi, z−1

i }, |zi| < 1, and (34) and (35), with initial conditions (36) and(37), respectively, need to be applied successively for each zi, where the input to the nextiteration is formed by the output of the previous, and the input to the first causal filter bythe original samples sk. The coefficients ck to be used in (22) are precisely the c−k of thefinal anti-causal filter, after scaling by the constant factor γ. Notice, furthermore, that inthe subsequent evaluation of the convolution (22) for any x, the polynomial pieces of thekernel are computed most efficiently by using “nested multiplication” by x. In other words,by considering each of the polynomials in the form x(· · · (x(xan + an−1) + an−2) . . . ) + a0


rather than anxn + an−1x

n−1 + . . . + a1x + a0. It can easily be seen that the nestedform requires only 2n floating-point operations (multiplications and additions) comparedto 3n− 1 in the case of direct evaluation of the polynomial.38

It is interesting to have a look at the interpolation kernel implicitly used in (22) inthe case of B-spline interpolation. Writing (bn)−1 for the spatial-domain version of theprefilter (Bn)−1 corresponding to an nth-degree B-spline, it follows from (30) that thecoefficients are given by ck = ((bn)−1 ∗ s)k, where “∗” denotes convolution. Substitutingthis expression into (22), together with ϕ = βn, we obtain

fT (x) =∑k∈Z

((bn)−1 ∗ s)

kβn(x/T − k), (38)

which can be rewritten in cardinal form as

fT (x) =∑k∈Z

sk ϑn(x/T − k), (39)

where ϑn is the so-called cardinal spline of degree n, given by

ϑn(x) =∑k∈Z

(bn)−1k βn(x− k). (40)

Similar to the sinc function, this kernel satisfies the interpolation property: it vanishesfor integer values of its argument, except at the origin, where it assumes unit value.Furthermore, for all n > 2, it has infinite support. And as n goes to infinity, ϑn convergesto the sinc function. Although this result was already known to Schoenberg [280], it didnot reach the signal and image processing community until recently [6, 327].

In the years to follow, the described digital filtering approach to spline interpolationwould be used in the design of efficient algorithms for such purposes as image rotation [334],the enlargement or reduction of images [183,331], the construction of multiresolution imagepyramids [330, 342], image registration [178, 315], wavelet transformation [332, 338, 339],texture mapping [147], on-line signal interpolation [197], and fast spline transformation[100]. For more detailed information, the reader is referred to mentioned papers as wellas several reviews [324,325].

IV.J Development of Alternative Piecewise Polynomial Kernels

Independent of the just mentioned developments, research on alternative piecewise polyno-mial interpolation kernels continued. Mitchell & Netravali [225] derived a two-parametercubic kernel by imposing the requirements of continuity and continuous differentiability,but by replacing the interpolation condition by the requirement of first-order approxima-tion, i.e., the ability to reproduce the constant. By means of an analysis in the spirit ofKeys [166], they also obtained a criterion to be satisfied by the two parameters in order

38It is precisely this trick that is implemented in the digital filter structure described by Farrow [95] in1988 and that later authors [81,89,179,258,336] have referred to as the “Farrow structure”. It was alreadyknown, however, to medieval Chinese mathematicians. Jia Xian (middle 11th century) appears to have usedit for solving cubic equations. A generalization to polynomials of arbitrary degrees was described first byQın Jiushao [204,214] (also written as Ch’in Chiu-shao [251]), in his book Shushu Jiuzhang (“MathematicalTreatise in Nine Sections”, 1247 AD). In the West it was rediscovered by Horner [143] in 1819, and it canbe found under his name in many books on numerical analysis [131, 138, 171, 285, 310, 337]. Fifteen yearsearlier, however, it had also been proposed by Ruffini [265] (see also Cajori [36]). And even earlier, around1669, it was used by Newton [234].


to have at least second-order approximation. Special instances of their kernel include thecubic convolution kernel (23) corresponding to α = −1

2 , and the cubic B-spline. Thewhole family, sometimes also referred to as BC-splines [226], was later studied by severalauthors [207,212,226] in the fields of visualization and computer graphics.

An extension of the results of Park & Schowengerdt [246] concerning the previouslydiscussed frequency-domain error analysis was presented by Schaum [273]. Instead of theL2-norm, (25), he studied the performance metric

ε2s =∑k∈Z

|f1(k + s) − f(k + s)|2, (41)

which summarizes the total interpolation error at a given shift s with respect to the originalsampling grid. He found that in the case of oversampling, this error too is given by (26),where η2 and E may now be written as η2

s and Es, respectively, with

Es(ω) =

∣∣∣∣∣1 −∑k∈Z

ei2πksϕ(ω + 2πk)

∣∣∣∣∣2

. (42)

Also, in the case of undersampling, η2s represents the error ε2s averaged over all possible

grid placements. He then pointed out that interpolation kernels are optimally designed ifas many derivatives as possible of their corresponding error function Es are zero at ω = 0.By further analyzing Es, he showed that for L-point interpolation kernels, i.e., kernelsthat extend over L samples in computing (22) at any x with ck = sk, this requirementimplies that the kernel must be able to reproduce all monomials of degree n 6 L− 1, andthat this is the case for the Lagrange central interpolation kernels. Schaum also derivedoptimal interpolators for specific power spectra |f(ω)|2.

Several authors have developed interpolation kernels defined explicitly as finite linearcombinations of B-splines. Chen et al. [46], for example, described a kernel which theytermed the “local interpolatory cardinal spline” and is composed of cubic B-splines only.When applying a scaling factor of two to its argument, this function very closely resemblesKeys’ fourth-order cubic convolution kernel, except that it is twice rather than once con-tinuously differentiable. Knockaert & Olyslager [170] discovered a class of what they called“modified B-splines”. To each integral approximation order L > 0 corresponds exactlyone kernel of this class. For L = 1 and L = 2, these are (scaled versions of) the zeroth-degree and first-degree B-spline, respectively. For any L > 2, the corresponding kernel isa finite linear combination of B-splines of different degrees, such that the composite kernelis interpolating and that its degree is as low as possible.

In recent years, the design methodologies originally used by Keys [166] have been em-ployed more than once again in developing alternative interpolation kernels. Dodgson [79],defying the earlier mentioned claim of Schafer & Rabiner concerning even-degree piecewisepolynomial interpolators, used them to derive a symmetric second-degree interpolationkernel. In contrast with the quadratic Lagrange interpolator, this kernel is continuous. Itsorder of approximation, however, is one less. German [111] used Keys’ ideas to develop acontinuously-differentiable quartic interpolator with fifth order of approximation. And thepresent author [219] combined them with Park & Schowengerdt’s frequency-domain erroranalysis to derive a class of odd-degree piecewise polynomial interpolation kernels withincreasing regularity. All of these kernels, however, have the same order of approximationas the optimal cubic convolution kernel.


IV.K The Impact of Approximation Theory

The notion of approximation order, defined as the rate at which the error of an approxi-mation goes to zero when the distance between the samples goes to zero, has already beenused at several points in the previous subsections. In general, an approximating functionis computed as a linear combination of basis functions, whose coefficients are based on thesamples of the original function. This is the case e.g. with approximations obtained from(22), where the basis functions are translates of a single kernel, ϕ.

The concept of convolution-based approximation has been studied intensively overthe past decades, primarily in approximation theory, but the interesting results that hadbeen obtained in this area of mathematics were not noticed by researchers in signal andimage processing until relatively recently [323, 333]. An important example is the theorydeveloped 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, itimplies that the following conditions are equivalent:

i) The kernel ϕ has Lth-order zeroes in the Fourier domain. More precisely,{ϕ(0) 6= 0,

ϕ(n)(2πk) = 0 ∀k ∈ Z∗, n ∈ {0, 1, . . . , L− 1}.(43)

ii) The kernel ϕ is capable of reproducing all monomials of degree n 6 L− 1. That is,for every n ∈ {0, 1, . . . , L− 1} there exist coefficients ck,n ∈ R such that

∑k∈Z

ck,n ϕ(x− k) = xn. (44)

iii) The first L discrete moments of the kernel ϕ are constants. That is, for everyn ∈ {0, 1, . . . , L− 1}, there exists a µn ∈ R such that∑

k∈Z

(x− k)nϕ(x− k) = µn. (45)

iv) For each sufficiently smooth function f , that is a function whose derivatives up toand including order L are in the space L2, there exists a constant C ∈ R, whichdoes not depend on f , and a set of coefficients ck ∈ R, such that the L2-norm of thedifference between f and its approximation fT obtained from (22) is bounded as

‖f − fT‖L2 6 C · TL · ‖f (L)‖L2 as T → 0. (46)

This result holds for all compactly supported kernels [308], but also extends to non-compactly supported kernels having suitable inverse polynomial decay [70, 156, 195, 196]or even less stringent properties [19, 68]. Extensions have also been made to Lp-norms,1 6 p 6 ∞ [67, 70, 156, 187, 195]. Furthermore, although the classical Strang-Fix theoryapplies to the case of an orthogonal projection, alternative approximation methods suchas interpolation and quasi-interpolation also yield an O(TL) approximation error [53, 66,67,187]. A detailed treatment of the theory is outside the scope of the present paper andthe reader is referred to mentioned papers for more information.

An interesting observation that follows from these equivalence conditions is that eventhough the original function does not at all have to be a polynomial, the capability of a


given kernel to let the approximation error go to zero as TL when T → 0 (fourth condition)is determined by its ability to exactly reproduce all polynomials of maximum degree L−1(second condition). In particular it follows that in order for the approximation to convergeto the original function at all when T → 0, the kernel must have at least approximationorder L = 1, which implies that it must at least be able to reproduce the constant. And ifwe take ϕ(0) = 1 (first condition), which yields the usually desirable property of unit DCgain, we have µ0 = 1, which implies that the kernel samples must sum to one regardless ofthe position of the kernel relative to the sampling grid (third condition). This is generallyknown as the partition of unity condition—a condition that is not satisfied by virtuallyall so-called windowed sinc functions, which have frequently been proclaimed as the mostappropriate alternative for the “ideal” interpolator.

As can be appreciated from (46), the theoretical notion of approximation order is stillrather qualitative and not suitable for precise determination of the approximation error.In many applications it would be very useful to have a more quantitative way of estimatingthe error ε(T ) = ‖f − fT‖L2 induced by a given kernel ϕ and sampling step T . In 1999 itwas shown by Blu & Unser [19–21] that for any convolution-based approximation scheme,this error is given by the relation

ε(T ) = η(T ) + o(T r), (47)

where the first term is a Fourier-domain prediction of the error, given by

η(T ) =

√12π

∫ ∞

−∞|f(ω)|2 E(Tω)dω, (48)

and the second term goes to zero faster than T r. Here, r denotes the highest derivativeof f that still has finite energy. The error function E in (48) is completely determined bythe prefiltering and reconstruction kernels involved in the approximation scheme. In thecase where the scheme is actually an interpolation scheme, it follows that

E(ω) =

∣∣∣∣∣∣∑k∈Z∗

ϕ(ω + 2πk)

∣∣∣∣∣∣2

+∑k∈Z∗

|ϕ(ω + 2πk)|2

∣∣∣∣∣∑k∈Z

ϕ(ω + 2πk)

∣∣∣∣∣2 . (49)

If the original function f is bandlimited, or otherwise sufficiently smooth (which meansthat its intrinsic scale is large with respect to the sampling step T ), the prediction (48) isexact, that is to say, ε(T ) = η(T ). In all other cases, η2(T ) represents the average of ε2(T )over all possible sets of samples f(kT + τ), with τ ∈ [0, T ]. Note that if the kernel ϕ itselfis forced to possess the interpolation property ϕ(k) = δk, with δk the Kronecker symbol,we have from the discrete Fourier transform pair ϕ(k) = δk ⇔ ∑

k∈Zϕ(ω + 2πk) = 1 that

the denominator of (49) equals one, so that the error function reduces to the previouslymentioned function (27) proposed by Park & Schowengerdt [245,246].

It will be clear from (48) that the rate at which the approximation error goes to zeroas T → 0 is determined by the degree of flatness of the error function (49) near theorigin. This latter quantity follows from the Maclaurin series expansion of the function.If all derivatives up to order 2L at the origin are zero, this expansion can be written asE(ω) = C2

ϕ ω2L + O(ω2L+2), where C2

ϕ = E(2L)(0)/(2L)!, and where use has been madeof the fact that all odd terms of the expansion are zero because of the symmetry of the


function. Substitution into (48) then shows that the last condition in the Strang-Fix theorycan be reformulated more quantitatively as follows: for sufficiently smooth functions f ,that is to say functions for which ‖f (L)‖L2 is finite, the L2-norm of the difference betweenf and the approximation fT obtained from (22) is given by [20,313,314,325]

‖f − fT ‖L2 = Cϕ · TL · ‖f (L)‖L2 as T → 0. (50)

In view of the practical need in many applications to optimize the cost-performancetrade-off of interpolation, this raises the questions of which of the kernels of given ap-proximation order L have the smallest support, and which of the latter kernels havethe smallest asymptotic constant Cϕ. These questions were recently answered by Blu etal. [18]. Concerning the first, they showed that these kernels are piecewise polynomials ofdegree L− 1 and that their support is of size L. Moreover, the full class of these so-calledmaximal order, minimum support (MOMS) kernels is given by a linear combination of an(L− 1)th-degree B-spline and its derivatives:39

ϕ(x) =L−1∑n=0

λndn

dxnβL−1(x), (51)

where λ0 = 1, and the remaining λn are free parameters that can be tuned so as to let thekernel satisfy additional criteria. For example, if the kernel is supposed to have maximumsmoothness, it turns out that all of the latter λn are necessarily zero, so that we are leftwith the (L−1)th-degree B-spline itself. Alternatively, if the kernel is supposed to possessthe interpolation property, it follows that the λn are such that the kernel boils down to the(L − 1)th-degree Lagrange central interpolation kernel. A more interesting design goal,however, is to minimize the asymptotic constant Cϕ, the general form of which for MOMSkernels is given by [18]

Cϕ =

√√√√ ∑n∈Z∗

∣∣∣∣ΛL(i2πn)(i2πn)L

∣∣∣∣2

, (52)

where ΛL(x) =∑L−1

n=0 λnxn is a polynomial of degree L−1. It was shown by Blu et al. [18]

that the ΛL that minimizes (52) for any order L can be obtained from the inductionrelation

Λn+1(x) = Λn(x) +x2

4(4L2 − 1)Λn−1(x), (53)

which is initialized by Λ1(x) = Λ2(x) = 1. The resulting kernels were coined optimized,maximal order, minimum support kernels (or O-MOMS for short).

From inspection of (53) it is clear that regardless of the value of L, the correspondingpolynomial ΛL will always consist of even terms only. Hence, if L is even, the degree of ΛL

will be L− 2, and since an (L− 1)th-degree B-spline is precisely L− 2 times continuouslydifferentiable, it follows from (51) that the corresponding kernel is continuous, but thatits first derivative is discontinuous. If, on the other hand, L is odd, the degree of thepolynomial ΛL will be L− 1, so that the corresponding kernel itself will be discontinuous.In order to have at least a continuous derivative, as may be required for some applications,Blu et al. [18] also carried out constrained minimizations of (52). The resulting kernelswere termed suboptimal MOMS (or SO-MOMS for short).

39Blu et al. [18] point out that this fact had been published earlier by Ron [263] in a more mathematicallyabstract and general context: the theory of exponential B-splines.


IV.L Development of Alternative Interpolation Methods

Although—as announced in the introduction—the main concern in this section is thetransition from classical polynomial interpolation approaches to modern convolution-basedapproaches and the many variants of the latter that have been proposed in the signal andimage processing literature, it may be good to extend the perspectives and briefly discussseveral alternative methods that have been developed since the 1980s. The goal here isnot to be exhaustive, but to give an impression of the more specific interpolation problemsand 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 knownfunction values f(k) at the integers to all integral multiples of 1/b, where the base b is alsointeger. In principle, any type of interpolation can be used to compute the f(k+r/b), r =0, 1, . . . , b − 1, and the process can be iterated to find the value f(x) for any rationalnumber x whose denominator is an integral power of b. As pointed out by them, themain properties of this so-called b-adic interpolation process are determined by what theycalled the “fundamental function”, i.e. the kernel, ϕ, which reveals itself when feeding theprocess with a discrete impulse sequence. They showed that when using Waring-Lagrangeinterpolation40 of any odd degree 2n − 1, the corresponding kernel has a support limitedto [−2n+1, 2n− 1] and is capable of reproducing polynomials of maximum degree 2n− 1.Ultimately, it satisfies the b-scale relation

ϕ( x

bm

)=

∑k∈Z

ck ϕ(x− k), (54)

where ck = ϕ(k/bm). Taking b = 2, we have a dyadic interpolation process, which hasstrong links with the multiresolution theory of wavelets. Indeed, for any n, the Deslauriers-Dubuc kernel of order 2n is the autocorrelation of the Daubechies n scaling function[62, 210]. More details and further references on interpolating wavelets are given by e.g.Mallat [210]. See Dai et al. [59] for a recent study on dyadic interpolation in the contextof image processing.

Another approach that has received quite some attention since the 1980s is to considerthe interpolation problem in the Fourier domain. It is well known [28] that multiplicationby a phase component in the frequency domain, corresponds to a shift in the signal orimage domain. Obvious applications of this property are translation and zooming of imagedata [47, 82, 87, 148, 174]. Interpolation based on the Fourier shift theorem is equivalentto another Fourier-based method, known as zero-filled or zero-padded interpolation [103,141,254,294,295]. In the spatial domain, both techniques amount to assuming periodicityof the underlying signal or image and convolving the samples with the “periodized sinc”,or Dirichlet’s kernel [37, 44, 80, 272, 356, 357]. Because of the assumed periodicity, theinfinite convolution sum can be rewritten in finite form. Fourier-based interpolation isespecially useful in situations where the data is acquired in Fourier space, such as inmagnetic resonance imaging (MRI), but by use of the fast Fourier transform (FFT) may inprinciple be applied to any type of data. Variants of this approach based on the fast Hartleytransform (FHT) [2,146,199] and discrete cosine transform (DCT) [3,4,343] have also beenproposed. More details can be found in mentioned papers and the references therein.

An approach that has hitherto received relatively little attention in the signal and imageprocessing literature is to consider sampled data as realizations of a random process atgiven spatial locations, and to make inferences on the unobserved values of the process

40Subdivision algorithms for Hermite interpolation were later studied by Merrien and Dubuc [85,222].


by means of a statistically optimal predictor. Here, “optimal” refers to minimizing themean-squared prediction error, which requires a model of the covariance between the datapoints. This approach to interpolation is generally known as kriging—a term coined bythe statistician Matheron [215] in honor of D. G. Krige [175], a South-African miningengineer who developed empirical methods for determining true ore-grade distributionsfrom distributions based on sampled ore grades [55, 56]. Kriging has been studied in thecontext of geostatistics [50,56,303], cartography [181], and meteorology [106], and is closelyrelated to interpolation by thin-plate splines or radial basis functions [83,229]. In medicalimaging, the technique seems to have been applied first by Stytz & Parrot [248,309]. Morerecent studies related to kriging in signal and image processing include those of Kerwin &Prince [165] and Leung et al. [189].

A special type of interpolation problem arises when dealing with binary data, such ase.g. segmented images. It is not difficult to see that in that case, the use of any of theaforementioned convolution-based interpolation methods followed by requantization prac-tically boils down to nearest-neighbor interpolation. In order to cope with this problem,it is necessary to consider the shape of the objects—that is to say their contours, ratherthan their “grey-level” distributions. For the interpolation of slices in a three-dimensional(3D) data set, for example, this may be accomplished by extracting the contours of in-terest and to apply elastic matching to estimate intermediate contours [48, 198]. An al-ternative is to apply any of the described convolution-based interpolation methods to thedistance transform of the binary data. This latter approach to shape-based interpolationwas originally proposed in the numerical analysis literature [190], and was first adapted tomedical image processing by Raya & Udupa [261]. Later authors have proposed variantsof their method by using alternative distance transforms [133], and morphological opera-tors [45, 123, 158, 185]. Extensions to the interpolation of tree-like image structures [137]and even ordinary grey-level images [51,119] have also been made.

In a way, kriging and shape-based interpolation may be considered the precursorsof more recent image interpolation techniques, which attempt to incorporate knowledgeabout the image content. The idea with most of these techniques is to adapt or choosebetween existing interpolation techniques, depending on the outcome of an initial imageanalysis phase. Since edges are often the more prevalent image features, most researchershave focussed on gradient-based schemes for the analysis phase, although region-basedapproaches have also been reported. Examples of specific applications where the potentialadvantages of adaptive interpolation approaches have been demonstrated are image res-olution enhancement or zooming [38, 61, 107, 142, 154, 191, 259, 317], spatial and temporalcoding 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 theperformance of the employed analysis scheme as much as on that of the eventual (oftenconvolution-based) interpolators.

IV.M Evaluation Studies and Their Conclusions

To return to our main topic: apart from the study by Parker et al. [247] discussed inSection IV.H, many more comparative evaluation studies of interpolation methods havebeen published over the years. Most of these appeared in the medical-imaging relatedliterature. Perhaps this can be explained from the fact that especially in medical appli-cations, the issues of accuracy, quality, and also speed can be of vital importance. Theloss of information and the introduction of distortions and artifacts caused by any ma-nipulation of image data should be minimized in order to minimize their influence on theclinicians’ judgements [314].


Schreiner et al. [283] studied the performance of nearest-neighbor, linear, and cubicconvolution interpolation in generating maximum intensity projections (MIPs) of 3D mag-netic resonance angiography (MRA) data for the purpose of detection and quantification ofvascular anomalies. From the results of experiments involving both a computer-generatedvessel model and clinical MRA data they concluded that the choice for an interpola-tion method can have a dramatic effect on the information contained in MIPs. However,whereas the improvement of linear over nearest-neighbor interpolation was considerable,the further improvement of cubic convolution interpolation was found to be negligible inthis application. Similar observations had been made earlier by Herman et al. [132] in thecontext of image reconstruction from projections.

Ostuni et al. [244] analyzed the effects of linear and cubic spline interpolation, aswell as truncated and Hann-windowed sinc interpolation on the reslicing of functionalmagnetic resonance imaging (fMRI) data. From the results of forward-backward geometrictransformation experiments on clinical fMRI data they concluded that the interpolationerrors caused by cubic spline interpolation are much smaller than those due to linearand truncated-sinc interpolation. In fact, the errors produced by the latter two typesof interpolation were found to be similar in magnitude, even with a spatial support ofeight sample intervals for the truncated-sinc kernel compared to only two for the linearinterpolation kernel. The Hann-windowed sinc kernel with a spatial support extendingover six to eight sample intervals performed comparably to cubic spline interpolation intheir experiments, but it required much more computation time.

Using similar reorientation experiments, Haddad & Porenta [126] found that the choicefor an interpolation technique significantly affects the outcome of quantitative measure-ments in myocardial perfusion imaging based on single photon emission computed tomog-raphy (SPECT). They concluded that cubic convolution is superior to several alternativemethods, such as local averaging, linear interpolation, and what they called “hybrid”interpolation, which combines in-plane 2D linear interpolation with through-plane cubicLagrange interpolation—an approach that had been shown earlier [177] to yield better re-sults than linear interpolation only. Here we could also mention several studies in the fieldof remote sensing [128, 167, 264], which also showed the superiority of cubic convolutionover linear and nearest-neighbor interpolation.

Grevera & Udupa [120] compared interpolation methods for the very specific taskof doubling the number of slices of 3D medical data sets. Their study included notonly convolution-based methods, but also several of the shape-based interpolation meth-ods [117, 119] mentioned earlier. The experiments consisted in subsampling a numberof magnetic resonance (MR) and computed tomography (CT) data sets, followed by in-terpolation to restore the original resolutions. Based on the results they concluded thatthere is evidence that shape-based interpolation is the most accurate method for this task.Note, however, that concerning the convolution-based methods, the study was limitedto nearest-neighbor, linear, and two forms of cubic convolution interpolation (althoughthey referred to the latter as cubic spline interpolation). In a later, more task-specificstudy [121], which led to the same conclusion, shape-based interpolation was comparedonly to linear interpolation.

A number of independent, large-scale evaluations of convolution-based interpolationmethods for the purpose of geometrical transformation of medical image data have recentlybeen carried out. Lehmann et al. [186], for example, compared a total of 31 kernels,including the nearest-neighbor, linear, and several quadratic [79] and cubic convolutionkernels [60,166,225,246], as well as the cubic B-spline interpolator, various Lagrange- [273]and Gaussian-based [8] interpolators, and truncated and Blackman-Harris [129] windowed-

V Summary and Conclusions PP-29

sinc kernels of different spatial support. From the results of computational-cost analysesand forward-backward transformation experiments carried out on CCD-photographs, MRIsections, and X-ray images, it followed that overall, cubic B-spline interpolation providesthe best cost-performance trade-off.

An even more elaborate study was presented by the present author [218], who carriedout a cost-performance analysis of a total of 126 kernels with spatial support ranging fromtwo to ten grid intervals. Apart from most of the kernels studied by Lehmann et al., thisstudy also included higher-degree generalizations of the cubic convolution kernel [219],cardinal spline and Lagrange central interpolation kernels up to ninth degree, as well aswindowed-sinc kernels using over a dozen different window functions well known from theliterature on harmonic analysis of signals [129]. The experiments involved the rotationand subpixel translation of medical data sets from many different modalities, includingCT, three different types of MRI, PET, SPECT, as well as 3D rotational and X-rayangiography. The results revealed that of all mentioned types of interpolation, splineinterpolation generally performs statistically significantly better.

Finally we mention the studies by Thevenaz et al. [313,314], who carried out theoreticalas well as experimental comparisons of many different convolution-based interpolationschemes. Concerning the former, they discussed the approximation-theoretical aspects ofsuch schemes and pointed at the importance of having a high approximation order, ratherthan a high regularity, and a small value for the asymptotic constant, as discussed in theprevious subsection. Indeed, the results of their experiments, which included all of theaforementioned piecewise polynomial kernels, as well as the quartic convolution kernel byGerman [111], several types of windowed-sinc kernels, and the O-MOMS [18], confirmedthe theoretical predictions and clearly showed the superiority of kernels with optimizedproperties in these terms.

V Summary and Conclusions

The goal in this paper was to give an overview of the developments in interpolation the-ory of all ages and to put the important techniques currently used in signal and imageprocessing into historical perspective. We pointed at relatively recent research into thehistory of science, in particular of mathematical astronomy, which has revealed that rudi-mentary solutions to the interpolation problem date back to early antiquity. We gaveexamples of interpolation techniques originally conceived by ancient Babylonian as wellas early-medieval Chinese, Indian, and Arabic astronomers and mathematicians, and webriefly discussed the links with the classical interpolation techniques developed in Westerncountries from the 17th until the 19th century.

The available historical material has not yet given reason to suspect that the earliest-known contributors to classical interpolation theory were influenced in any way by men-tioned ancient and medieval Eastern works. Among these early contributors were Harriotand Briggs, who in the first half of the 17th century developed higher-order interpolationschemes for the purpose of subtabulation. A generalization of their rules for equidistantdata was given independently by Gregory and Newton. We saw, however, that it is Newtonwho deserves the credit for having put classical interpolation theory on a firm foundation.He invented the concept of divided differences, allowing for a general interpolation formulaapplicable to data at arbitrary intervals, and gave several special formulae that follow fromit. In the course of the 18th and 19th century, these formulae were further studied by manyothers, including Stirling, Gauss, Waring, Euler, Lagrange, Bessel, Laplace, and Everett,


whose names are nowadays inextricably bound up with formulae that can easily be derivedfrom Newton’s regula generalis.

Whereas the developments until the end of 19th century had been impressive, the de-velopments in the past century have been explosive. We briefly discussed early results inapproximation theory, which revealed the limitations of interpolation by algebraic polyno-mials. We then discussed two major extensions of classical interpolation theory introducedin the first half of the 20th century: firstly the concept of the cardinal function, mainlydue to E. T. Whittaker, but also studied before him by Borel and others, and eventuallyleading to the sampling theorem for bandlimited functions as found in the works of J. M.Whittaker, Kotel’nikov, Shannon, and several others, and secondly the concept of oscu-latory interpolation, researched by many and eventually resulting in Schoenberg’s theoryof mathematical splines. We pointed at the important consequence of these extensions: aformulation of the interpolation problem in terms of a convolution of a set of coefficientswith some fixed kernel.

The remainder of the paper was focussed on the further development of convolution-based interpolation in signal and image processing. The earliest and probably most im-portant techniques studied in this context were cubic convolution interpolation and splineinterpolation, and we have discussed in some detail the contributions of various researchersin improving these techniques. Concerning cubic convolution, we highlighted the work ofKeys and also Park & Schowengerdt, who presented different techniques for deriving themathematically most optimal value for the free parameter involved in this scheme. Con-cerning spline interpolation, we discussed the work of Unser and his coworkers, who in-vented a fast recursive algorithm for carrying out the prefiltering required for this scheme,thereby making cubic spline interpolation as computationally cheap as cubic convolutioninterpolation. As a curiosity we remarked that not only spline interpolation, but cubicconvolution interpolation too can be traced back to osculatory interpolation techniquesknown from the beginning of the 20th century—a fact that, to the author’s knowledge,has not been pointed out before.

After a summary of the development of many alternative piecewise polynomial kernels,we discussed some of the interesting results known in approximation theory for some timenow, but brought to the attention of signal and image processors only recently. Theseare, in particular, the equivalence conditions due to Strang & Fix, which link the behaviorof the approximation error as a function of the sampling step to specific spatial andFourier domain properties of the employed convolution kernel. We discussed the extensionsby Blu & Unser, who showed how to obtain more precise, quantitative estimates of theapproximation error, based on an error function that is completely determined by thekernel. We also pointed at their recent efforts towards minimization of the interpolationerror, which has resulted in the development of kernels with minimal support and optimalapproximation properties.

After a brief discussion of several alternative methods proposed over the past twodecades for specific interpolation problems, we finally summarized the results of quite anumber of evaluation studies carried out recently, primarily in medical imaging. All ofthese have clearly shown the dramatic effect the wrong choice for an interpolation methodcan have on the information content of the data. From this it can be concluded that theissue of interpolation deserves more attention than it has received so far in some signal andimage processing applications. The results of these studies also strongly suggest that inorder to reduce the errors caused by convolution-based interpolation, it is more importantfor a kernel to have good approximation theoretical properties, in terms of approximationorder and asymptotic behavior, than to have a high degree of regularity. In applications

References PP-31

where the latter is an important issue, it follows that the most suitable kernels are B-splines, since they combine a maximal order of approximation with maximal regularity fora given spatial support.

Acknowledgments

The research that led to the writing of this paper was financially supported by the Nether-lands Organization for Scientific Research (NWO) and was carried out at the BiomedicalImaging Group (BIG) of the Swiss Federal Institute of Technology in Lausanne (EPFL),Switzerland. The author is grateful to Dr. Thierry Blu (BIG) for his helpful expla-nations concerning the theoretical aspects of interpolation and approximation, and toProf. Dr. Michael Unser (BIG) for his support and comments on the manuscript. He alsowishes to express his gratitude to Dr. Erik Jan Dubbink (Josephine Nefkens Institute, Eras-mus University Rotterdam, the Netherlands) and Mr. Steven Gheyselinck (BibliothequeCentrale, EPFL, Switzerland) for providing him with copies of references [162] and [130],respectively, as well as to the anonymous reviewers, who brought to his attention somevery interesting additional references.

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