Statistical methods for the analysis of tooth shape
The study of tooth shape has traditionally involved analysing distances or angles between established points of correspondence, known as landmarks. Digital imaging has aided this process, yet improved statistical techniques, which offer advantages by retaining information on the geometry of objects throughout the analysis, have so far received little attention. Since methods must be suitable for use on un-extracted teeth, a key difficulty is that unwanted variation in recorded shape results from differences in the position of patients' gingival (gum) tissue. Here we present new methodology for addressing this problem and for use in more general applications, where objects are analysed as configurations of landmarks and one would wish to account for lack of precise correspondence between certain points, in a better way than is possible using existing techniques. After introducing the ideas of Procrustes analysis to this field, we use newly proposed methods of reliability assessment to show how, in addition to failing to allow for gum variation, implementation of this technique in its standard form is problematic, due to the poor reproducibility of particular landmarks. Use of Bookstein's (1996a, d, e) semi- landmark method, which aims to overcome lack of precise correspondence along certain directions by allowing landmarks to move iteratively along chords during Procrustes registration, is investigated but found to produce unrealistic results in certain situations. Novel modifications of this method are then proposed and evaluated in terms of addressing the issues noted above. Alternatives to minimising the `bending energy' of a pair of splines mapping from the mean shape, in order to determine new semi- landmark positions are explored and two new methods, using a `nearest point' or `full Procrustes' criterion, identified as most promising. Further investigation, by application to tooth shape problems (including a simulation study of gingival tissue variation) and use on distorted configurations generated from simple geometric shapes, show that these methods offer improvements over existing techniques in terms of filtering out unwanted variation.