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3D Shape Similarity Metric based on the Correspondence of Points

Recently, due to the rapid progress of computers, it has come to be easy to display 3D objects. Displaying the 3D objects is necessary in many fields such as virtual reality. Usually, 3D objects are represented as 3D shape models.

As many 3D shape models become available, it is necessary to classify and retrieve these models. For this purpose, a similarity measure between two shape models is needed. In classification, models are classified by clustering similar ones according to the similarity measure. In retrieval, a model is retrieved by searching the most similar one to the query shape model given by a user. The similarity measure is also useful for object recognition.

So far, several similarity measures using parametric representations of shape have been proposed. Their target shapes are limited to those which can be described with parametric representation. They are suitable for the rough comparison between shapes, but not for the detailed comparison between similar shapes.

In this paper, we propose a new similarity measure between 3D shape models based on the correspondence of the points on the models. It is applied to any similar shapes, if enough number of points are set on the model to represent the shape accurately. It is suitable for detailed comparison between similar shapes, because the shape is represented with the set of points on the model. It is, however, not suitable for the comparison between different shapes, because it is difficult to obtain the correspondence of the points.

In our method, we set the same number of points uniformly on both models, and obtain the correspondence of the points on the two models. If the two shape models are similar, the distance between two points on one model is close to that between the two correspondent points on the other. When the average of the distance difference is small, the two models are considered to be similar. Therefore, we define the dissimilarity as the minimum value of the average of the distance difference over all combinations of the correspondences of the points. Here, we minimize the average of the distance difference by the heuristic search over the correspondences of the points. To make the amount of computation small and to avoid falling into local minima, we start the minimization from the initial correspondence of the points obtained by the method of Gunsel and Tekalp.

The desirable properties of a shape similarity measure are as follows: (1)If the shapes themselves do not change, their similarity should not change. The similarity measure should be invariant for rigid transformation(translation, rotation and reflection) and scaling. (2)The similarity measure should satisfy the metric properties(positivity, identity, symmetry and triangle inequality). If it satisfies the metric properties, each shape is represented in the feature space, and many advantages of the metric are obtained in classification and recognition. (3)The similarity measure is used by human. It should agree with human feeling.

Our similarity measure is invariant for rigid transformation, because it is defined with the distances between the points on the models. It also satisfies the metric properties.

We conducted the experiment of applying our method to ten human body models to obtain the similarity. As a result, the correspondence of the points on the two models obtained by our method are natural. We also measured the similarity of human feeling by questionnaire for the same ten human models. We projected the models to the points on the 2-dimensional space so that the distance between the points reflected the dissimilarity between the models obtained by our method and that obtained by the questionnaire, respectively, using the fourth method of quantification, and we compared those distributions of the points. As a result, our similarity metric is close to the similarity of human feeling.

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