Tridimensional Regression for Comparing and Mapping 3D Anatomical Structures

Shape analysis is useful for a wide variety of disciplines and has many applications. There are many approaches to shape analysis, one of which focuses on the analysis of shapes that are represented by the coordinates of predefined landmarks on the object. This paper discusses Tridimensional Regression, a technique that can be used for mapping images and shapes that are represented by sets of three-dimensional landmark coordinates, for comparing and mapping 3D anatomical structures. The degree of similarity between shapes can be quantified using the tridimensional coefficient of determination ( ). An experiment was conducted to evaluate the effectiveness of this technique to correctly match the image of a face with another image of the same face. These results were compared to the values obtained when only two dimensions are used and show that using three dimensions increases the ability to correctly match and discriminate between faces. 1. Introduction Tobler [1] proposed bidimensional regression as a tool for computing the degree of similarity between two planar configurations of points and to estimate mapping relations between two objects that are represented by a set of two-dimensional landmarks. Bidimensional regression is an extension of linear regression where both dependent and independent variables are represented by coordinate pairs, instead of scalar values. Specifically, Tobler [1] suggested that bidimensional regression may be useful for comparing signatures, geographical maps, or faces. The latter was done in the context of face recognition by Shi et al. [2] and Kare et al. [3]. Tobler’s [1] method has been extended to Tridimensional Regression for situations when both dependent and independent variables are represented by three-dimensional coordinates [4]. The purpose of this paper is to provide a summary of that extension, to illustrate the use of tridimensional regression for comparing and mapping anatomical structures, and to compare the effectiveness of the two-dimensional and three-dimensional methods. Widespread use of three-dimensional imaging devices in many areas of research makes this research timely. This technique is broadly applicable to any situation where spatial configurations of three-dimensional points are compared. Specific instances where tridimensional regression may be of use are three-dimensional mapping and comparison of objects or structures that are represented by their three-dimensional coordinates. The values derived from regression allow the degree of similarity between two objects to be quantified. 2. Methods