A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing ERIK MEIJERING, MEMBER, IEEE (Encouraged Paper) This paper presents a chronological overview of the develop- communication of information, it is not difficult to find ex- ments in interpolation theory, from the earliest times to the present amples of applications where this problem occurs. The rel- date. It brings out the connections between the results obtained atively easiest and in many applications often most desired in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary approach to solve the problem is interpolation, where an ap- of the insights and recommendations that follow from relatively re- proximating function is constructed in such a way as to agree cent theoretical as well as experimental studies concludes the pre- perfectly with the usually unknown original function at the sentation. given measurement points.2 In view of its increasing rele- Keywords—Approximation, convolution-based interpolation, vance, it is only natural that the subject of interpolation is history, image processing, polynomial interpolation, signal pro- receiving more and more attention these days.3 However, cessing, splines. in times where all efforts are directed toward the future, the past may easily be forgotten. It is no sinecure, scanning the “It is an extremely useful thing to have knowledge of the literature, to get a clear picture of the development of the true origins of memorable discoveries, especially those that subject through the ages. This is quite unfortunate, since it have been found not by accident but by dint of meditation. implies a risk of researchers going over grounds covered ear- It is not so much that thereby history may attribute to each lier by others. History has shown many examples of this and man his own discoveries and others should be encouraged to several new examples will be revealed here. The goal of the earn like commendation, as that the art of making discoveries present paper is to provide a systematic overview of the de- should be extended by considering noteworthy examples of velopments in interpolation theory, from the earliest times to it.”1 the present date and to put the most well-known techniques currently used in signal and image processing applications into historical perspective. The paper is intended to serve as I. INTRODUCTION a tutorial and a useful source of links to the appropriate liter- The problem of constructing a continuously defined func- ature for anyone interested in interpolation, whether it be its tion from given discrete data is unavoidable whenever one history, theory, or applications. wishes to manipulate the data in a way that requires informa- As already suggested by the title, the organization of tion not included explicitly in the data. In this age of ever-in- the paper is largely chronological. Section II presents an creasing digitization in the storage, processing, analysis, and 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 of information. Manuscript received July 19, 2001; revised November 17, 2001. This It seems that the word was introduced in the English literature for the first work was supported by the Netherlands Organization for Scientific time around 1612 and was then used in the sense of “to alter or enlarge Research. [texts] by insertion of new matter” [3]. The original Latin word appears [4] The author is with the Biomedical Imaging Group, Swiss Federal In- to have been used first in a mathematical sense by Wallis in his 1655 book stitute of Technology Lausanne, CH-1015 Lausanne, Switzerland (e-mail: on infinitesimal arithmetic [5]. [email protected]). 3A search in the multidisciplinary databases of bibliographic information Publisher Item Identifier S 0018-9219(02)02899-2. collected by the Institute for Scientific Information in the Web of Science will reveal that the number of publications containing the word “interpola- 1Leibniz, in the opening paragraph of his Historia et Origo Calculi Dif- tion” in the title, list of keywords, or the abstract has dramatically increased ferentialis [1]. The translation given here was taken from a paper by Child over the past decade, even when taking into account the intrinsic (and like- [2]. wise dramatic) increase in the number of publications as a function of time. 0018-9219/02$17.00 © 2002 IEEE PROCEEDINGS OF THE IEEE, VOL. 90, NO. 3, MARCH 2002 319 overview of the earliest known uses of interpolation in so-called “chord function” (related to the sine function) antiquity and describes the more sophisticated interpolation for the purpose of computing the positions of celestial methods developed in different parts of the world during bodies. Later examples are found in the Almagest (“The the Middle Ages. Next, Section III discusses the origins of Mathematical Compilation,” ca. 140 AD) of Claudius the most important techniques developed in Europe during Ptolemy, the Egypt-born Greek astronomer-mathematician the period of Scientific Revolution, which in the present who propounded the geocentric view of the universe which context lasted from the early 17th until the late 19th century. prevailed until the 16th century. Apart from theory, this A discussion of the developments in what could be called influential work also contains numerical tables of a wide the Information and Communication Era, covering roughly variety of trigonometric functions defined for astronomical the past century, is provided in Section IV. Here, the focus purposes. To avoid the tedious calculations involved in the of attention is on the results that have had the largest impact construction of tables of functions of more than one variable, on the advancement of the subject in signal and image Ptolemy used an approach that amounts to tabulating the processing, in particular on the development of techniques function only for the variable for which the function varies for the manipulation of intensity data defined on uniform most, given two bounding values of the other variable and grids. Although recently developed alternative methods for to provide a table of coefficients to be used in an “adaptive” specific interpolation tasks in this area will also be men- linear interpolation scheme for computation of the function tioned briefly, the discussion in this part of the paper will for intermediate values of this latter variable [10]. be restricted mainly to convolution-based methods, which is justified by the fact that these are the most frequently used B. Interpolation in Early-Medieval China and India interpolation methods, probably because of their versatility Analysis of the computational techniques on which early- and relatively low complexity. Finally, summarizing and medieval Chinese ephemerides are based often reveals the concluding remarks are made in Section V. use of higher order interpolation formulae.4 The first person to use second-order interpolation for computing the positions II. ANCIENT TIMES AND THE MIDDLE AGES of the sun and the moon in constructing a calendar is said to be the astronomer Liù Zhuó. Around 600 AD, he used this In his 1909 book on interpolation [6], Thiele character- technique in producing the so-called Huáng jí lì or “Impe- ized the subject as “the art of reading between the lines in a rial Standard Calendar.” According to Yan˘ and Shírán [12], [numerical] table.” Examples of fields in which this problem the formula involved in his computations reads in modern arises naturally and inevitably are astronomy and, related to notation5 this, calendar computation. Because man has been interested in these since day one, it should not surprise us that it is in these fields that the first interpolation methods were con- ceived. This section discusses the earliest known contribu- (1) tions to interpolation theory. with , , and A. Interpolation in Ancient Babylon and Greece and with and the observed results at times In antiquity, astronomy was all about time keeping and and , respectively. This formula is closely related to making predictions concerning astronomical events. This later Western interpolation formulae, to be discussed in the served important practical needs: farmers, e.g., would base next section. Methods for second-order interpolation of un- their planting strategies on these predictions. To this end, equal-interval observations were later used by the astronomer it was of great importance to keep up lists—so-called Monk Yì Xíng in producing the so-called “Dà Yan˘ Calendar” ephemerides—of the positions of the sun, moon, and (727 AD) and by XúÁng in producing the “Xuan¯ Míng Cal- the known planets for regular time intervals. Obviously, endar” (822 AD). The latter also used a second-order formula these lists would contain gaps, due to either atmospherical for interpolation of equal-interval observations equivalent to conditions hampering observation or the fact that celestial the formula used by Liù Zhuó. bodies may not be visible during certain periods. From Accurate computation of the motion of celestial bodies, his study of ephemerides found on ancient astronomical however, requires more sophisticated interpolation tech- cuneiform tablets originating from Uruk and Babylon niques than just second order. More complex techniques in the Seleucid period (the last three centuries BC), the were later developed by GuoSh¯ oujìng¯ and others. In historian-mathematician Neugebauer [7], [8] concluded that interpolation was used in order to fill these gaps. Apart from 4The paragraphs on Chinese contributions to interpolation theory are linear interpolation, the tablets also revealed the use of more based on the information provided in the books by Martzloff [11] and Yan˘ and Shírán [12]. For a more elaborate treatment of the techniques and complex interpolation methods. Precise formulations of the formulae mentioned here, see the latter. This reference was brought to the latter methods have not survived, however.