Date Published: October 4, 2012
Publisher: Hindawi Publishing Corporation
Author(s): Kendra K. Schmid, David B. Marx, Ashok Samal.
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 (R2). 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 R2 values obtained when only two dimensions are used and show that using three dimensions increases the ability to correctly match and discriminate between faces.
Tobler  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  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.  and Kare et al. .
In this section, a brief summary of bidimensional regression and its extension to three dimensions is provided. Details of the tridimensional regression models are provided.
An experiment was conducted to evaluate the effectiveness of tridimensional regression and its improvement over bidimensional regression. Three-dimensional landmark data obtained from human faces were used for this purpose. The landmarks were obtained by placing reflective markers on the faces of subjects and tracking the coordinates as the subjects moved through a series of poses using automated software. The landmarks were adapted from . They are shown in Figure 4 and described in Table 1.
Bidimensional regression  is a useful tool for comparing two geometric configurations that are each represented by a set of coordinate pairs. The scale, rotation, and translation relating the two configurations can be estimated by first estimating the parameters of the transformation model. As an application of the technique, [2, 3] used bidimensional regression analysis for relating faces in landmark-based face recognition.