During fossilization, crania are often fractured and not all the fragments are recovered. To restore the antemortem appearance of a fossil skull, different steps are required. The first process is assembling the fragments, the second process is eliminating distortions, and third process is compensating for missing parts. In this article, we consider the third step: interpolation of the missing areas. Conventionally, such interpolation is created manually with plaster as filler, and based on the knowledge and experience of skilled anthropologists. However, in order to interpolate more precisely missing surfaces in fossil crania, recently researchers have attempted to digitally interpolate missing parts in a virtual space using a composite technique involving X-ray computed tomography (CT) and computer-assisted morphology. Two methods have been proposed for such interpolation: geometric interpolation using a spline function and statistical interpolation using multivariate regression (Gunz et al., 2009). Geometric interpolation interpolates a missing part based on data mapped from a complete reference specimen using a thin-plate spline function (TPS). Specifically, the existing portion of the specimen to be interpolated is used to define a mapping function, and the corresponding portion of the complete specimen is mapped to the partial specimen in order to reconstruct the missing parts. This method yields anatomically natural and morphologically consistent interpolations of the missing parts, but the results may vary depending on the reference specimen used. On the other hand, statistical interpolation based on multivariate regression is a method to estimate missing coordinates (positions of landmarks) based on a sample of complete specimens as reference dataset. Multivariate regressions are calculated with the missing coordinates as dependent variables and the other coordinates as independent variables. These equations are then applied to predict the missing coordinates. This method can estimate the position of missing landmarks more precisely than geometric interpolation if we have enough number of reference dataset. Therefore, the number of reference dataset is the limit for this method. To compute a multivariate regression analysis, the number of the samples in the reference dataset has to exceed the number of landmark coordinates used as independent variables. However, the number of the reference specimens that can be used for analysis is often limited, although more landmark coordinates are necessary to precisely capture morphological characteristics of each cranial specimen, hindering the use of the multivariate regression analysis for the interpolation of missing parts in fossil crania. Both methods have been applied in paleoanthropology, and they succeeded to get good results (Gunz et al., 2009 Weaver and Hublin, 2009), despite some limitations. The first limitation is that the result of interpolation is always influenced by reference data. For example, if we try to interpolate missing parts of Neanderthals by using modern humans as reference sample, the interpolation will be influenced by modern human characters . At present, it is difficult to avoid this problem. A second limitation is that both methods are based on landmarks. Locating landmarks is a useful way to quantify the morphology of cranium. However, it is difficult to locate enough number of landmark on the whole skull to capture every morphological characteristics. Some methods to distribute landmark and semilandmark have been proposed (ex: sliding methods, equal spaced points), but the debate is still open. In conclusion, the existing interpolation methods are very useful and effective, but there are still some limitations which merit attention. We have to keep on developing more and more methods and techniques … I wrote a brief article about statistical interpolating methods in this book.

*Hideki Amano*