from ancient astronomy to modern signal and image processing

More documents

Recommendations

Info

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
that had been shown earlier [349] to yield better results thanlinear interpolation only. Here, we could also mention severalstudies in the field of remote sensing [193], [350], [351],which also showed the superiority of cubic convolution overlinear and nearest-neighbor interpolation.Grevera and Udupa [197] compared interpolation methodsfor the very specific task of doubling the number of slices of3-D medical data sets. Their study included not only convolution-basedmethods, but also several of the shape-based interpolationmethods [334], [352] mentioned earlier. The experimentsconsisted in subsampling a number of magneticresonance (MR) and computed tomography (CT) data sets,followed by interpolation to restore the original resolutions.Based on the results they concluded that there is evidencethat shape-based interpolation is the most accurate methodfor this task. Note, however, that concerning the convolutionbasedmethods, the study was limited to nearest-neighbor,linear, and two forms of cubic convolution interpolation (althoughthey referred to the latter as cubic spline interpolation).In a later more task-specific study [353], which ledto the same conclusion, shape-based interpolation was comparedonly to linear interpolation.A number of independent large-scale evaluations ofconvolution-based interpolation methods for the purpose ofgeometrical transformation of medical image data have recentlybeen carried out. Lehmann et al. [354], e.g., compareda total of 31 kernels, including the nearest-neighbor, linear,and several quadratic [258] and cubic convolution kernels[187], [188], [192], [355], as well as the cubic B-splineinterpolator, various Lagrange- [255] and Gaussian-based[356] interpolators and truncated and Blackman-Harris[357] windowed-sinc kernels of different spatial support.From the results of computational-cost analyses and forward-backwardtransformation experiments carried outon CCD-photographs, MRI sections, and X-ray images, itfollowed that, overall, cubic B-spline interpolation providesthe best cost-performance tradeoff.An even more elaborate study was presented by the presentauthor [358], who carried out a cost-performance analysis ofa total of 126 kernels with spatial support ranging from two toten grid intervals. Apart from most of the kernels studied byLehmann et al., this study also included higher degree generalizationsof the cubic convolution kernel [260], cardinalspline, and Lagrange central interpolation kernels up to ninthdegree, as well as windowed-sinc kernels using over a dozendifferent window functions well known from the literatureon harmonic analysis of signals [357]. The experiments involvedthe rotation and subpixel translation of medical datasets from many different modalities, including CT, three differenttypes of MRI, PET, SPECT, as well as 3-D rotationaland X-ray angiography. The results revealed that of all mentionedtypes of interpolation, spline interpolation generallyperforms statistically significantly better.Finally, we mention the studies by Thévenaz et al. [276],[277], who carried out theoretical as well as experimentalcomparisons of many different convolution-based interpolationschemes. Concerning the former, they discussedthe approximation-theoretical aspects of such schemes andpointed at the importance of having a high approximationorder, rather than a high regularity and a small value for theasymptotic constant, as discussed in the previous subsection.Indeed, the results of their experiments, which included allof the aforementioned piecewise polynomial kernels, as wellas the quartic convolution kernel by German [259], severaltypes of windowed-sinc kernels and the O-MOMS [278],confirmed the theoretical predictions and clearly showedthe superiority of kernels with optimized properties in theseterms.V. SUMMARY AND CONCLUSIONThe goal in this paper was to give an overview of thedevelopments in interpolation theory of all ages and toput the important techniques currently used in signal andimage processing into historical perspective. We pointedat relatively recent research into the history of science, inparticular of mathematical astronomy, which has revealedthat rudimentary solutions to the interpolation problem dateback to early antiquity. We gave examples of interpolationtechniques originally conceived by ancient Babylonian aswell as early-medieval Chinese, Indian, and Arabic astronomersand mathematicians and we briefly discussed thelinks with the classical interpolation techniques developedin Western countries from the 17th until the 19th century.The available historical material has not yet given reasonto suspect that the earliest known contributors to classicalinterpolation theory were influenced in any way by mentionedancient and medieval Eastern works. Among theseearly contributors were Harriot and Briggs who, in the firsthalf of the 17th century, developed higher order interpolationschemes for the purpose of subtabulation. A generalizationof their rules for equidistant data was given independentlyby Gregory and Newton. We saw, however, that it isNewton who deserves the credit for having put classical interpolationtheory on a firm foundation. He invented the conceptof divided differences, allowing for a general interpolationformula applicable to data at arbitrary intervals and gaveseveral special formulae that follow from it. In the courseof the 18th and 19th century, these formulae were furtherstudied by many others, including Stirling, Gauss, Waring,Euler, Lagrange, Bessel, Laplace, and Everett, whose namesare nowadays inextricably bound up with formulae that caneasily be derived from Newton’s regula generalis.Whereas the developments until the end of the 19th centuryhad been impressive, the developments in the past centuryhave been explosive. We briefly discussed early resultsin approximation theory, which revealed the limitations ofinterpolation by algebraic polynomials. We then discussedtwo major extensions of classical interpolation theory introducedin the first half of the 20th century: first, the conceptof the cardinal function, mainly due to E. T. Whittaker, butalso studied before him by Borel and others and eventuallyleading to the sampling theorem for bandlimited functions asfound in the works of J. M. Whittaker, Kotel’nikov, Shannon,and several others and second, the concept of osculatory interpolation,researched by many and eventually resulting inMEIJERING: A CHRONOLOGY OF INTERPOLATION 335
Page 1: A Chronology of Interpolation: From
Page 4 and 5: nomial of given degree , i.e.,. It
Page 6 and 7: D. Studies on More General Interpol
Page 8 and 9: literature parallel to Shannon. In
Page 10 and 11: discussed the design of alternative
Page 12 and 13: spline interpolation, this approach
Page 14 and 15: 1) The kernel has th-order zeroes i
Page 18 and 19: Schoenberg’s theory of mathematic
Page 20 and 21: [82] M. Gasca and T. Sauer, “Poly
Page 22 and 23: [204] L. R. Watkins and J. Le Bihan
Page 24: [316] N. S.-N. Lam, “Spatial inte

from ancient astronomy to modern signal and image processing

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?