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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 (

Tobler [

Tobler’s [

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.

Nakaya [

where

where

where

Tobler [

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 [

Example of a bidimensional affine transformation.

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

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 [

The equations for the projective transformation can be rewritten using homogeneous coordinates and put in matrix notation as shown in (

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 [

The similarity of the two objects is assessed using the bidimensional correlation coefficient [

The bidimensional regression models proposed by Tobler [

In this paper, the linear transformations discussed by Tobler [

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

The format of the rotation matrix depends on the axis of rotation. The formats for each of the three rotations are shown below, where

As in the two-dimensional case, the transformation can be linearized by reparameterization, where the new transformation matrix (

For rotation about the

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

For rotation about

The extension of the affine transformation from two dimensions into three dimensions includes additional parameters for translation, scaling, rotation, and shear. Figure

Example of a tridimensional affine transformation [

The extension of the projective transformation from two to three dimensions involves the conversion to homogeneous coordinates (

Let

where

As described in [

Nonlinear regression is an extension of linear regression where the expected responses are nonlinear functions of the parameters [

For all transformations, parameter estimates can be found in the usual manner,

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 [

Description of landmarks used for evaluation (adapted from [

tr | The point on the hairline in the midline of the forehead. |

go | The most lateral point on the mandibular angle close to the bony gonion. |

gn | The lowest median landmark on the lower border of the mandible. |

en | The point at the inner commissure of the eye fissure. |

ex | The point at the outer commissure of the eye fissure. |

sci | The highest point on the upper boarder in the midportion of each eyebrow. |

The midpoint of both the nasal root and the nasofrontal structure. | |

prn | The most protrudent point of the apex nasi. |

ac | The most lateral point in the curved baseline of each ala. |

ls | The midpoint of the upper vermillion line. |

li | The midpoint of the lower vermillion line. |

ch | The point located at each labial commissure. |

sa | The highest point of the free margin of the auricle. |

sba | The lowest point of the free margin of the ear lobe. |

pa | The most posterior point on the free margin of the ear. |

jm | The most protrudent point of the muscle when the jaw is clenched. |

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

For each transformation, both in two and three dimensions, the distributions of

Error rates for each transformation.

Bidimensional regression | Tridimensional regression | ||||

False positive | False negative | False positive | False negative | ||

Euclidean | Observed | 59.7% | 36.0% | 19.3% | 12.0% |

Expected | 51.9% | 33.6% | 16.7% | 9.0% | |

Affine | Observed | 57.5% | 38.7% | 17.0% | 3.3% |

Expected | 49.0% | 37.2% | 14.9% | 7.5% | |

Projective | Observed | 56.8% | 35.3% | 23.5% | 7.3% |

Expected | 51.2% | 34.5% | 18.4% | 16.2% |

Within and between person

Within and between person

Within and between person

Table

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

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

Bidimensional regression [

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 [

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.