Scattered Data Interpolation : Tests of Some Methods

This paper is concerned with the evaluation of methods for scattered data interpolation and some of the results of the tests when applied to a number of methods. The process involves evaluation of the methods in terms of timing, storage, accuracy, visual pleasantness of the surface, and ease of implementation. To indicate the flavor of the type of results obtained, we give a summary table and representative perspective plots of several surfaces. 1.0. Introduction. The basic problem which is being addressed here is evaluation of methods for obtaining a smooth (at least continuous first partial derivatives) bivariate function, F(x,_v), which takes on certain prescribed values, Fixk,yk) = fk, k = 1,.. ., N. The points {xk,yk) are not assumed to satisfy any particular conditions as to spacing or density, hence the term "scattered." It is usually convenient to think of the values fk as arising from some underlying (not necessarily known) function fix,y), so that/¿ = /(**,/*)> k = I, . . ., N. The problem of interpolation of scattered data in two or more independent variables has been addressed by numerous authors, as can be seen by the bibliography. Many of the basic ideas involved are discussed in two survey papers (both over a wider class of approximations than we consider here) due to Schumaker [52] and Barnhill [4]. A recent review of methods for contouring, which treats many of the same ideas from that point of view, is given by Sabin [51 ]. Many ideas put forth have not previously been explored computationally, or only to a limited extent. Thus, the capabilities of some plausible ideas were unexplored. In addition, most of the methods involve one or more ad hoc assumptions requiring a user to specify parameters (one or more). Generally only cursory attention has been paid to the appropriate choice of these parameters, and their overall effect on the interpolant has usually not been determined. Out of this situation arose a desire to attempt to answer a number of questions, basically all related: Which of these many methods deserve further study and development, and which should be discarded? Some methods require the user to specify an ad hoc parameter, and we have investigated the possibility of using a standard default value. The default value should give reasonably good results over a number of different sets of data, and preferably the interpolant should be rather stable with respect to changes in the parameter. Additionally, it is convenient for the user if the parameter is related to something about the data which can be easily Received July 21, 1980; revised May 6, 1981. 1980 Mathematics Subject Classification. Primary 65D05; Secondary 65D15. * Supported in part by the Foundation Research Program at the Naval Postgraduate School. © 1982 American Mathematical Society 0025-571 8/82/0000-0479/$06.00 181 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use