Tridimensional Regression for Comparing and Mapping 3D Anatomical Structures

Abstract

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.

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 𝑅² values derived from regression allow the degree of similarity between two objects to be quantified.

2. Methods

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.

2.1. Bidimensional Regression

Nakaya [5] defines a bidimensional regression model as𝑢_𝑖𝑣_𝑖=𝑔𝑥_𝑖,𝑦_𝑖ℎ𝑥_𝑖,𝑦_𝑖+𝜀_𝑖𝜂_𝑖, (1)

where (𝑢_𝑖,𝑣_𝑖) is the dependent variable, (𝑥_𝑖,𝑦_𝑖) represent the corresponding coordinates of the independent variable, 𝑔 and ℎ are transformation functions used to estimate mapping relations between independent and dependent variables, and (𝜀_𝑖,𝜂_𝑖) is an error vector that is assumed to be normally and independently distributed. Both Tobler [1] and Nakaya [5] discuss obtaining estimates for parameters in 𝑔 and ℎ using the method of least-squares so that ^𝑛_𝑖=1𝑢_𝑖−𝑔𝑥_𝑖,𝑦_𝑖²+𝑣_𝑖−ℎ𝑥_𝑖,𝑦_𝑖², (2)

where 𝑔 and ℎ are the transformation functions evaluated at the parameter estimates, is minimized. Here 𝑛 is the number of landmark points used in the analysis. The normal equations are obtained in the usual manner [1], and by solving 𝜷_𝑗 in𝐗^𝑇_𝑗𝐗_𝑗𝜷_𝑗=𝐗^𝑇_𝑗𝐘, (3)

where 𝐗_𝑗 is the design matrix of transformation 𝑗 and 𝐘=[^𝐮𝐯]^𝑇 is a (2𝑛×1) vector for the dependent variable partitioned by the coordinates, will yield the least-squares parameter estimates [6]. The design matrix (𝐗_𝑗) will depend on the transformation used and the number of parameters to estimate; hence, the dimension of 𝜷_𝑗 will also be determined by the type of transformation.

Tobler [1] proposes four bidimensional regression models, three of which are intrinsically linear and one is curvilinear. Friedman and Kohler [7] argue that the curvilinear model may be too general for practical use and describe the linear transformations in more detail. Each of the other three transformations is linearized by reparameterization prior to solving the parameter estimates.

The three linear transformations yield the Euclidean, affine, and projective models where in each model the original coordinates are scaled, rotated, and translated. These transformations form a hierarchy with the Euclidean being the simplest (fewest parameters) and the projective the most complex (most parameters) of the models.

Details of bidimensional regression models can be found in [1, 5, 7, 8]. Briefly, the Euclidean model is a similarity transformation in that the overall shape remains unchanged. The coordinates are translated, rotated, and isotropically scaled [8], thus preserving the original shape and angles. The affine model allows for 𝑋 and 𝑌 coordinates to be scaled independently, and the configuration could exhibit shear (𝛾) (e.g., a square may become a parallelogram; Figure 1). The projective transformation, which is the most complex, allows the size, shape, and orientation to change as a function of viewpoint [7]. An example of a projective transformation is shown in Figure 2.

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Figure 1 

Example of a bidimensional affine transformation.

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Figure 2 

Example of a bidimensional projective transformation (ABCD→FGHE) [9].

In the Euclidean and affine transformations, the models are linearized by reparameterization, and then the normal equations can be derived in the usual manner. Once the parameters have been estimated [7], provide equations for calculating the scale and rotation values for the Euclidean transformation and the scale, shear, and rotation values for the affine transformation.

The equations for the projective transformation can be rewritten using homogeneous coordinates and put in matrix notation as shown in (16). Homogeneous coordinates can be used with any of the models to provide a uniform framework for all transformations. For rotation, scaling, and shear, the transformed coordinates can be expressed as the product of a transformation matrix and the original coordinates. For translation, however, the coordinates are derived by addition of the translation vector to the original coordinates. Use of homogeneous coordinates makes all the transformations multiplicative. This is accomplished by adding an additional coordinate (𝑡), called the homogeneous coordinate.

The homogeneous coordinate is added purely for mathematical simplification and has no effect on the transformation of coordinates. For example, it is convenient to represent a sequence of transformations as the product of the corresponding transformation matrices. Thus, in the Euclidean and affine models, the translation parameters become multiplicative and one matrix could be used for all of the transformation parameters [10]. With the projective model, the conversion is used to linearize the model, and once the object is mapped using homogeneous coordinates, the original coordinates are restored by dividing by the homogeneous coordinate, 𝑡. However, when this is done, the restriction placed on 𝑡 results in parameter estimates of 𝛽₃₁=0, 𝛽₃₂=0,and𝛽₃₃=1. Consequently, the projective transformation is reduced to the affine transformation and the results are identical. The conversion to homogeneous coordinates is adequate for determining the location of transformed points, but not for obtaining transformation parameter estimates. If left in terms of the original equations, the parameters of the projective transformation can be estimated using nonlinear regression. When extended to three dimensions, a similar approach is used.

The similarity of the two objects is assessed using the bidimensional correlation coefficient [1],𝑅_2𝐷=∑1−^𝑖𝑢_𝑖−𝑢_𝑖²+𝑣_𝑖−𝑣_𝑖²∑_𝑖𝑢_𝑖−𝑢²+𝑣_𝑖−𝑣². (4)

2.2. Tridimensional Regression

The bidimensional regression models proposed by Tobler [1] can be extended to instances where three-dimensional data are used for comparison. A specific instance may include anatomical structures that are represented by three-dimensional landmark coordinates, but tridimensional regression can be useful for determining the degree of similarity between any two objects that are represented by three-dimensional coordinates.

In this paper, the linear transformations discussed by Tobler [1] will be extended to three dimensions. Extensions to the Euclidean, affine, and projective transformations are described in detail where the dependent and independent variables are represented by their three-dimensional coordinates,𝑢_𝑖𝑣_𝑖𝑤_𝑖,𝑥_𝑖𝑦_𝑖𝑧_𝑖,respectively. (5)

2.2.1. Euclidean Transformation

The three-dimensional Euclidean transformation is similar to the two-dimensional case in that coordinates are simply translated, rotated, and isotropically scaled. The overall shape and the angles of the original object are preserved, and parallel lines in the original object are mapped to parallel lines in the transformed space. There is an additional translation parameter, and the rotation matrix differs depending on which axis(es) are used for the rotation. In general, the number of rotation parameters is 𝑘(𝑘−1)/2, where 𝑘 is the number of dimensions. Therefore, there are three rotation parameters for the general three-dimensional Euclidean transformation. However, for instances when it is known that all three rotations are not necessary, the transformation can be reduced to one or two rotations. These special cases are discussed in detail in [4].

Three-Dimensional Euclidean Transformation with One Angle of Rotation
The format of the rotation matrix depends on the axis of rotation. The formats for each of the three rotations are shown below, where 𝛾 is the angle of rotation about the 𝑥-axis, 𝜃 is the angle of rotation about the 𝑦-axis, and 𝜙 is the angle of rotation about the 𝑧-axis: 𝐑_𝐗=,𝐑1000cos𝛾−sin𝛾0sin𝛾cos𝛾_𝐘=,𝐑cos𝜃0sin𝜃010−sin𝜃0cos𝜃_𝐙=.cos𝜙−sin𝜙0sin𝜙cos𝜙0001 (6)   The general form of the three-dimensional Euclidean transformation is 𝑢_𝑖𝑣_𝑖𝑤_𝑖=𝛼₁𝛼₂𝛼₃𝑥+𝑠𝐑𝑦^𝑖_𝑖𝑧_𝑖, (7) where 𝐑 is one of the rotation matrices.
As in the two-dimensional case, the transformation can be linearized by reparameterization, where the new transformation matrix (𝐑^′) is a combination of the scale and rotation parameters. The reparameterized transformations and their normal equations follow.
For rotation about the 𝑥-axis, 𝐑^′_𝐗=1000𝛽₁−𝛽₂0𝛽₂𝛽₁,𝑢_𝑖=𝛼₁+𝑥_𝑖,𝑣_𝑖=𝛼₂+𝛽₁𝑦_𝑖−𝛽₂𝑧_𝑖,𝑤_𝑖=𝛼₃+𝛽₂𝑦_𝑖+𝛽₁𝑧_𝑖, (8) and deriving the normal equations in the usual manner yields N00000N0𝑦_𝑖−𝑧_𝑖00N𝑧_𝑖𝑦_𝑖0𝑦_𝑖𝑧_𝑖𝑦²_𝑖+𝑧²_𝑖00−𝑧_𝑖𝑦_𝑖0𝑦²_𝑖+𝑧²_𝑖𝛼₁𝛼₂𝛼₃𝛽₁𝛽₂=𝑢_𝑖−𝑥_𝑖𝑣_𝑖𝑤_𝑖𝑣_𝑖𝑦_𝑖+𝑤_𝑖𝑧_𝑖𝑤_𝑖𝑦_𝑖−𝑣_𝑖𝑧_𝑖. (9) Similar details for rotation about the 𝑦 and 𝑧 axes can be found in [4].

Three-Dimensional Euclidean Transformation with Multiple Angles of Rotation
When more than one rotation is used, the reparameterization to linearize the model is not obvious; therefore, the rotation matrices remain in terms of the rotation parameters and nonlinear regression is used. The advantage of using nonlinear regression is that the rotation and scale parameters are directly estimated instead of being solved in terms of 𝛽_𝑖; the disadvantage in using nonlinear regression is convergence may not be reached and starting values must be specified. The similarity of the two objects is assessed using the Pseudo- 𝑅² as defined by [11]. The Pseudo-𝑅² is calculated in the same manner as 𝑅², but, in general, is not guaranteed to be greater than zero. Again, the rotation matrix differs depending upon the axes of rotation. An example of a two-rotation Euclidean transformation is shown below.
For rotation about 𝑥 and 𝑦 axes, 𝑢_𝑖𝑣_𝑖𝑤_𝑖=𝛼₁𝛼₂𝛼₃𝑥+𝑠cos𝜃0sin𝜃sin𝛾sin𝜃cos𝛾−sin𝛾cos𝜃−cos𝛾sin𝜃sin𝛾cos𝛾cos𝜃^𝑖𝑦_𝑖𝑧_𝑖. (10)  In the general form of the three-dimensional Euclidean transformation, the order in which the transformations are applied will result in different parameter estimates. Permuting this order will result in different estimates of the rotation parameters, but the measure of similarity will remain the same regardless of the order of transformations. The following system of equations shows the rotations in the order of 𝑥-axis, 𝑦-axis, and then 𝑧-axis:𝑢_𝑖𝑣_𝑖𝑤_𝑖=𝛼₁𝛼₂𝛼₃𝑥+𝑠cos𝜃cos𝜙−cos𝜃sin𝜙sin𝜃sin𝛾sin𝜃cos𝜙+cos𝛾sin𝜙−sin𝛾sin𝜃sin𝜙+cos𝛾cos𝜙−sin𝛾cos𝜃−cos𝛾sin𝜃cos𝜙+sin𝛾sin𝜙cos𝛾sin𝜃sin𝜙+sin𝛾cos𝜙cos𝛾cos𝜃^𝑖𝑦_𝑖𝑧_𝑖. (11)

2.2.2. Affine Transformation

The extension of the affine transformation from two dimensions into three dimensions includes additional parameters for translation, scaling, rotation, and shear. Figure 3 shows an example of a three-dimensional affine transformation. The transformed coordinates in affine transformations are given by𝑢_𝑖=𝛼₁+𝛽₁₁𝑥_𝑖+𝛽₁₂𝑦_𝑖+𝛽₁₃𝑧_𝑖,𝑣_𝑖=𝛼₂+𝛽₂₁𝑥_𝑖+𝛽₂₂𝑦_𝑖+𝛽₂₃𝑧_𝑖,𝑤_𝑖=𝛼₃+𝛽₃₁𝑥_𝑖+𝛽₃₂𝑦_𝑖+𝛽₃₃𝑧_𝑖,𝑢_𝑖𝑣_𝑖𝑤_𝑖=𝛼₁𝛼₂𝛼₃+𝛽₁₁𝛽₁₂𝛽₁₃𝛽₂₁𝛽₂₂𝛽₂₃𝛽₃₁𝛽₃₂𝛽₃₃𝑥_𝑖𝑦_𝑖𝑧_𝑖. (12) Deriving the normal equations in the usual manner yields 𝐈₃⊗N𝐈₃⊗𝑥_𝑖𝑦_𝑖𝑧_𝑖𝐈₃⊗𝑥_𝑖𝑦_𝑖𝑧_𝑖𝐈₃⊗𝑥²_𝑖𝑥_𝑖𝑦_𝑖𝑥_𝑖𝑧_𝑖𝑥_𝑖𝑦_𝑖𝑦²_𝑖𝑦_𝑖𝑧_𝑖𝑥_𝑖𝑧_𝑖𝑦_𝑖𝑧_𝑖𝑧²_𝑖𝛼₁𝛼₂𝛼₃𝛽₁₁𝛽₁₂𝛽₁₃𝛽₂₁𝛽₂₂𝛽₂₃𝛽₃₁𝛽₃₂𝛽₃₃=𝑢_𝑖𝑣_𝑖𝑤_𝑖𝑢_𝑖𝑥_𝑖𝑢_𝑖𝑦_𝑖𝑢_𝑖𝑧_𝑖𝑣_𝑖𝑥_𝑖𝑣_𝑖𝑦_𝑖𝑣_𝑖𝑧_𝑖𝑤_𝑖𝑥_𝑖𝑤_𝑖𝑦_𝑖𝑤_𝑖𝑧_𝑖, (13) where 𝐈₃ is a 3 × 3 identity matrix and ⊗ is the direct product of the two matrices.

Figure 3 

Example of a tridimensional affine transformation [12].

2.2.3. Projective Transformation

The extension of the projective transformation from two to three dimensions involves the conversion to homogeneous coordinates (16). Additional parameters are added corresponding to the coordinate of the third dimension. In a projective transformation, the size, shape, and orientation can all change as a function of viewpoint. While this is a nonlinear transformation, by using homogeneous coordinates, the model can be linearized in order to obtain the normal equations and estimate the parameters. The equations to obtain the transformed coordinates are𝑢_𝑖=𝛽₁₁𝑥_𝑖+𝛽₁₂𝑦_𝑖+𝛽₁₃𝑧_𝑖+𝛽₁₄𝛽₄₁𝑥_𝑖+𝛽₄₂𝑦_𝑖+𝛽₄₃𝑧_𝑖+𝛽₄₄,𝑣_𝑖=𝛽₂₁𝑥_𝑖+𝛽₂₂𝑦_𝑖+𝛽₂₃𝑧_𝑖+𝛽₂₄𝛽₄₁𝑥_𝑖+𝛽₄₂𝑦_𝑖+𝛽₄₃𝑧_𝑖+𝛽₄₄,𝑤_𝑖=𝛽₃₁𝑥_𝑖+𝛽₃₂𝑦_𝑖+𝛽₃₃𝑧_𝑖+𝛽₃₄𝛽₄₁𝑥_𝑖+𝛽₄₂𝑦_𝑖+𝛽₄₃𝑧_𝑖+𝛽₄₄,𝑢_𝑖𝑡𝑣_𝑖𝑡𝑤_𝑖𝑡𝑡=𝛽₁₁𝛽₁₂𝛽₁₃𝛽₁₄𝛽₂₁𝛽₂₂𝛽₂₃𝛽₂₄𝛽₃₁𝛽₃₂𝛽₃₃𝛽₃₄𝛽₄₁𝛽₄₂𝛽₄₃𝛽₄₄𝑥_𝑖𝑦_𝑖𝑧_𝑖1. (14)

Let 𝑢^′_𝑖=𝑢_𝑖𝑣𝑡,_′𝑖=𝑣_𝑖𝑤𝑡,_′𝑖=𝑤_𝑖𝑡, (15) then 𝑢^′_𝑖𝑣^′_𝑖𝑤^′_𝑖𝑡=𝛽₁₁𝛽₁₂𝛽₁₃𝛽₁₄𝛽₂₁𝛽₂₂𝛽₂₃𝛽₂₄𝛽₃₁𝛽₃₂𝛽₃₃𝛽₃₄𝛽₄₁𝛽₄₂𝛽₄₃𝛽₄₄𝑥_𝑖𝑦_𝑖𝑧_𝑖1, (16) and deriving the normal equations in the usual manner yields𝐈₄⊗𝑥²𝑥𝑦𝑥𝑧𝑥𝑥𝑦𝑦²𝑦𝑧𝑦𝑥𝑧𝑦𝑧𝑧²𝛽𝑧𝑥𝑦𝑧𝑁¹¹𝛽₁₂𝛽₁₃𝛽₁₄𝛽₂₁𝛽₂₂𝛽₂₃𝛽₂₄𝛽₃₁𝛽₃₂𝛽₃₃𝛽₃₄𝛽₄₁𝛽₄₂𝛽₄₃𝛽₄₄=𝑢𝑥𝑢𝑦𝑢𝑧𝑢𝑣𝑥𝑣𝑦𝑣𝑧𝑣𝑤𝑥𝑤𝑦𝑤𝑧𝑤𝑡𝑥𝑡𝑦𝑡𝑧𝑡, (17)

where 𝐈₄ is a 4 × 4 identity matrix and ⊗ is the direct product of the two matrices.

As described in [4], this linearization results in parameter estimates that reduce the transformation to affine. The linearization is adequate to determine the transformed points, but not for the optimization to determine the transformation parameters or for measuring the degree of similarity between the two objects. Therefore, the transformation is left in terms of the original equations and nonlinear regression is used to obtain parameter estimates.

Nonlinear regression is an extension of linear regression where the expected responses are nonlinear functions of the parameters [13]. Finding least-squares estimates for linear models is straightforward as they have a closed-form solution. For nonlinear models, the least-squares estimates must be found using an iterative procedure. In this paper, the Gauss-Newton algorithm is used. This iterative procedure utilizes a Taylor series expansion to find the least-squares estimates [13].

For all transformations, parameter estimates can be found in the usual manner, 𝜷_𝑗=(𝐗^𝑇_𝑗𝐗_𝑗)⁻¹𝐗^𝑇_𝑗𝐘, and subsequently used to solve for rotation, scale, and sheer parameters. The similarity of the two objects can be assessed using the tridimensional correlation coefficient, 𝑅_3𝐷, given by𝑅_3𝐷=∑1−^𝑖𝑢_𝑖−𝑢_𝑖²+𝑣_𝑖−𝑣_𝑖²+𝑤_𝑖−𝑤_𝑖²∑_𝑖𝑢_𝑖−𝑢²+𝑣_𝑖−𝑣²+𝑤_𝑖−𝑤², (18) which is an extension to the bidimensional correlation coefficient [1].

3. Results and Discussion

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 [14]. They are shown in Figure 4 and described in Table 1.

Table 1 

Description of landmarks used for evaluation (adapted from [14]).

Figure 4 

Landmarks used for evaluating tridimensional regression.

The landmarks were obtained for three subjects at two different sittings and five poses per sitting. The objective was to compare 𝑅² values within a subject to the 𝑅² values between subjects using both tridimensional regression and bidimensional regression. One would expect the degree of similarity to be higher, thus a higher 𝑅² value, for two samples from the same person than for samples from two different people. All pairwise 𝑅² values were calculated for bidimensional and tridimensional regressions. Poses of the same individual within a sitting were not compared since the markers were not removed between poses and using these poses would result in inflated 𝑅² values.

For each transformation, both in two and three dimensions, the distributions of 𝑅² values for within and between subjects were obtained by fitting a theoretical distribution over the histograms of observed values. Overlaying these theoretical distributions allowed for the estimation of a threshold value (𝜏) as a cutoff for determining if two images were from the same subject. 𝑅² values greater than 𝜏 lead to the decision that the two images are of the same subject (match) while 𝑅² values less than 𝜏 indicate that the images are of two different subjects (nonmatch). The threshold value was determined to be where the two distributions cross, as to simultaneously minimize the false-positive and false-negative error rates. A false positive is when images of two different subjects are incorrectly determined to be from the same subject (an 𝑅² value greater than 𝜏 for different subjects); a false-negative occurs when two images from the same subject are incorrectly determined to be from different subjects (an 𝑅² value less than 𝜏 for the same subject). In addition to calculating the observed error rates, the expected error rates were found by evaluating the cumulative distribution functions of the 𝑅² values at 𝜏. Table 2 summarizes the observed and expected error rates, and Figures 5, 6, and 7 show the within-subject (dotted line) and between subject (solid line) distributions for each transformation.

Table 2 

Error rates for each transformation.

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Figure 5 

Within and between person 𝑅2 for bidimensional (𝑙) and tridimensional (𝑟) Euclidean transformation.

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Figure 6 

Within and between person 𝑅2 for bidimensional (𝑙) and tridimensional (𝑟) affine transformation.

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Figure 7 

Within and between person 𝑅2 for bidimensional (𝑙) and tridimensional (𝑟) projective transformation.

Table 2 shows that both the observed and expected error rates for tridimensional regression are much smaller than those for bidimensional regression using any of the three transformations. Bidimensional regression resulted in both error rates being very high, false-positives often over fifty percent. Tridimensional regression shows a substantial decrease in both false-positive and false-negative error rates which indicates that the three-dimensional method is better at correctly matching a subject to him or herself.

In this application, the Euclidean and affine transformations were comparable to one another with the affine performing slightly better. The projective transformation had the largest observed false-positive rate. This result is not surprising as the flexibility of the projective transformation allows it to map objects into many other shapes. This flexibility results in the ability to match even two very dissimilar objects quite well with certain transformation parameters. Consequently, the 𝑅² values are very high for all matches. This shifts the between-person distribution closer to the within person-distribution which results in a larger false-positive error rate.

Additionally, a sixth pose was taken on each of the subjects in each setting. This pose was not used to build the within- and between-subject distributions, or to determine the threshold. These six sets of points (two for each subject) were compared to all other poses not taken in the same setting of the same subject (30 comparisons per pose, 6 possible correct matches). The highest 𝑅² for all six was a correct match. In addition, a minimum of the top 4 matches were correct matches, illustrating that tridimensional regression can be very good at identifying correct matches and discriminating between different objects.

4. Conclusion

Bidimensional regression [1] 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.

In this paper, the bidimensional technique has been extended to three dimensions. Such an extension may prove useful in the analysis of three-dimensional landmark data. The underlying foundations for tridimensional regression have been developed with different transformations: Euclidean, affine, and projective. Its use is demonstrated through an application to compare human faces using three-dimensional landmarks. Results show that tridimensional regression improves the ability to correctly match objects that are represented by landmark data. Both the Euclidean and affine transformations work well to reduce the error rates. The projective transformation also shows improved error rates, but its flexibility may make it too general for some practical applications. Choice of transformation should be given careful consideration given the goals of the application. While there is improvement over Bidimensional regression, the observed and expected error rates are likely higher in this experiment due to the small number of subjects involved and comparing several poses of the same subject. A larger-scale study is needed to better estimate the expected error rates.

This work can be extended in several different directions. The focus here was in developing the theory of tridimensional regression and conducting an initial investigation for shape matching with a feasibility experiment. An investigation with a larger amount of three-dimensional landmark data is needed to more fully understand its effectiveness. In addition to a larger-scale study, it is also of interest to develop weighted tridimensional regression techniques which would allow some landmarks to be weighted more or less heavily than others. Weighting landmarks allows for less weight to be placed on landmarks that are highly variable. Some landmarks could be more variable because they are less reliably extracted or simply due to more natural variability. Weighting has been shown to improve the matching ability in bidimensional regression [15], specifically in a face matching application [16], and is expected to improve the matching ability and precision in mapping for tridimensional regression as well.

Acknowledgment

The authors wish to thank Dr. Jordan Green of the University of Nebraska-Lincoln for his help in obtaining the three-dimensional landmark data used for this research.