Software Support for Advanced Cephalometric Analysis in Orthodontics

Geometric Morphometrics

Geometric morphometrics uses mathematical and statistical tools to quantify and study shape (Bookstein, 1991; Dryden & Mardia, 1998; Slice, 2005). In the domain of GM, shape is defined as the geometric properties of an object that are invariant to location, orientation and scale (Dryden & Mardia, 1998). Thus, the concept of shape is restricted to the geometric properties of an object, without regard to other characteristics such as, for example, material or colour. Relating this definition to cephalometrics, one could consider the conventional cephalometric measurements of angles, distances and ratios as shape variables. Angles and ratios have the advantage that they are location- and scale-invariant, whereas distances, although not scale-invariant, can be adjusted to a common size. Unfortunately, such variables pose significant limitations, a major one being that they need to be of sufficient number and carefully chosen in order to describe the shape of the object in a comprehensive, unambiguous manner. Consider, for example, a typical cephalometric analysis, which may consist of 15 angles, defined between some 20 landmarks. It is obvious that the position of the landmarks cannot be recreated from the 15 measurements, even if these have been carefully selected. The information inherent in these shape variables is limited and biased; multiple landmark configurations exist that give the same set of measurements. A solution to this problem (not without its own difficulties) is to use the Cartesian (x, y) coordinates of the landmarks as the shape variables. Notice that these coordinates are also distance data (the distance of each landmark to a set of reference axes), so they include location and orientation information, in addition to shape. However, the removal of this ‘nuisance’ information is now more easily accomplished, using what is known as Procrustes superimposition.