from ancient astronomy to modern signal and image processing

consider sampled data as realizations of a random processat given spatial locations and to make inferences on the unobservedvalues of the process by means of a statisticallyoptimal predictor. Here, “optimal” refers to minimizing themean-squared prediction error, which requires a model ofthe covariance between the data points. This approach tointerpolation is generally known as kriging—a term coinedby the statistician Matheron [310] in honor of D. G. Krige[311], a South-African mining engineer who developed empiricalmethods for determining true ore-grade distributionsfrom distributions based on sampled ore grades [312], [313].Kriging has been studied in the context of geostatistics [312],[314], [315], cartography [316], and meteorology [317] andis closely related to interpolation by thin-plate splines or radialbasis functions [318], [319]. In medical imaging, thetechnique seems to have been applied first by Stytz and Parrot[320], [321]. More recent studies related to kriging in signaland image processing include those of Kerwin and Prince[322] and Leung et al. [323].A special type of interpolation problem arises whendealing with binary data, such as, e.g., segmented images.It is not difficult to see that in that case, the use of anyof the aforementioned convolution-based interpolationmethods followed by requantization practically boils downto nearest-neighbor interpolation. In order to cope withthis problem, it is necessary to consider the shape of theobjects—that is to say, their contours, rather than their“grey-level” distributions. For the interpolation of slicesin a three-dimensional (3-D) data set, e.g., this may beaccomplished by extracting the contours of interest andto apply elastic matching to estimate intermediate contours[324], [325]. An alternative is to apply any of thedescribed convolution-based interpolation methods to thedistance transform of the binary data. This latter approachto shape-based interpolation was originally proposed in thenumerical analysis literature [326] and was first adapted tomedical image processing by Raya and Udupa [327]. Laterauthors have proposed variants of their method by usingalternative distance transforms [328] and morphologicaloperators [329]–[332]. Extensions to the interpolation oftree-like image structures [333] and even ordinary grey-levelimages [334], [335] have also been made.In a way, kriging and shape-based interpolation may beconsidered the precursors of more recent image interpolationtechniques, which attempt to incorporate knowledge aboutthe image content. The idea with most of these techniques isto adapt or choose between existing interpolation techniques,depending on the outcome of an initial image analysis phase.Since edges are often the more prevalent image features,most researchers have focused on gradient-based schemesfor the analysis phase, although region-based approacheshave also been reported. Examples of specific applicationswhere the potential advantages of adaptive interpolationapproaches have been demonstrated are image resolutionenhancement or zooming [205], [336]–[342], spatial andtemporal coding or compression of images and imagesequences [343], [344], texture mapping [250], and volumerendering [345]. Clearly, the results of such adaptivemethods depend on the performance of the employedanalysis scheme as much as on that of the eventual (oftenconvolution-based) interpolators.M. Evaluation Studies and Their ConclusionsTo return to our main topic: apart from the study by Parkeret al. [199] discussed in Section IV-H, many more comparativeevaluation studies of interpolation methods have beenpublished over the years. Most of these appeared in the medical-imaging-relatedliterature. Perhaps this can be explainedfrom the fact that especially in medical applications, the issuesof accuracy, quality, and also speed can be of vital importance.The loss of information and the introduction of distortionsand artifacts caused by any manipulation of imagedata should be minimized in order to minimize their influenceon the clinicians’ judgements [276].Schreiner et al. [346] studied the performance ofnearest-neighbor, linear and cubic convolution interpolationin generating maximum intensity projections (MIPs) of3-D magnetic-resonance angiography (MRA) data forthe purpose of detection and quantification of vascularanomalies. From the results of experiments involving botha computer-generated vessel model and clinical MRA data,they concluded that the choice for an interpolation methodcan have a dramatic effect on the information contained inMIPs. However, whereas the improvement of linear overnearest-neighbor interpolation was considerable, the furtherimprovement of cubic convolution interpolation was foundto be negligible in this application. Similar observations hadbeen made earlier by Herman et al. [191] in the context ofimage reconstruction from projections.Ostuni et al. [347] analyzed the effects of linearand cubic spline interpolation, as well as truncated andHann-windowed sinc interpolation on the reslicing offunctional MRI (fMRI) data. From the results of forward–backwardgeometric transformation experiments onclinical fMRI data, they concluded that the interpolationerrors caused by cubic spline interpolation are much smallerthan those due to linear and truncated-sinc interpolation.In fact, the errors produced by the latter two types ofinterpolation were found to be similar in magnitude, evenwith a spatial support of eight sample intervals for thetruncated-sinc kernel compared to only two for the linearinterpolation kernel. The Hann-windowed sinc kernel with aspatial support extending over six to eight sample intervalsperformed comparably to cubic spline interpolation in theirexperiments, but it required much more computation time.Using similar reorientation experiments, Haddad andPorenta [348] found that the choice for an interpolationtechnique significantly affects the outcome of quantitativemeasurements in myocardial perfusion imaging based onsingle photon emission computed tomography (SPECT).They concluded that cubic convolution is superior to severalalternative methods, such as local averaging, linearinterpolation and what they called “hybrid” interpolation,which combines in-plane 2-D linear interpolation withthrough-plane cubic Lagrange interpolation—an approach334 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 3, MARCH 2002