An Algorithm for Clustering Objects According to Their Shapes

The main goal in image analysis is to detect and recognize objects of interest present in the observed images. Imaged objects can be characterized in many ways: according to their colors, textures, shapes, movements, and locations. The past decade has seen large efforts in modeling and analysis of pixel values or textures in images to attain these goals albeit with limited success. An emerging opinion in the scientific community is that complicated features such as shapes be taken into account. Characterization of complex objects using their shapes is fast becoming a major tool in computer vision and image understanding. Analysis of shapes, especially those of complex objects, is a challenging task and requires sophisticated mathematical tools. Applications of shape analysis include biomedical image analysis, homeland security, biometrics, military target recognition and general computer vision. Image analysis using full shape characteristics is a distant future goal. Towards that goal, one step is to develop an algorithm to categorize or cluster objects according to the shapes of their boundaries. There are several applications that can benefit from such clustering: (i) It can be used in efficient, hierarchical organization of databases (of shapes). Such organization can significantly improve database searches [2] and systems with shape-based queries. While retrieving shapes, the test shape can be compared with a representative shape of each cluster instead of the whole shape database and then searched for the closest shape in that cluster. (ii) Clustering can provide efficient encoding of shapes. For example, one can describe a database of shapes using means, variances, and other statistics, of each cluster. (iii) It can contribute in robust algorithms for computer vision by incorporating shape-based analysis. Past research in shape analysis has been mostly restricted to landmark-based analysis. Researchers [4, 6] have studied shapes using a finite number of landmarks (points in Euclidean space) establishing equivalences with respect to shape preserving transformations, i.e. rigid rotation and translation, and non-rigid uniform scaling. Shape clustering based upon the resulting Procrustes mean shape is discussed in [1]. We consider the shapes of continuous, closed curves in R, without the need for defining landmarks. Using the geometric representations that were introduced in [3], we develop an algorithm for clustering shapes. This clustering algorithm can be used to organize large databases of shapes into categories of similar shapes, with similarity specified using the geodesic length metric on the shape space. In Section 2, we summarize the geometric representation of planar shapes, and describe the use of geodesic length to impose a metric on that shape space. Section 3 develops an algorithm for clustering shapes and illustrates it with an example.