This article proposes a 3D object classification approach based on volumetric parts, where Superquadricbased Geon (SBG) description is implemented for representing the volumetric constituents of 3D object. In the approach, 3D object classification is decomposed into the constrained search on interpretation tree and the similarity measure computation. First, a set of integrated features and corresponding constraints are presented, which are used for defining efficient interpretation tree search rules and evaluating the model similarity. Then a similarity measure computation algorithm is developed to evaluate the shape similarity of unknown object data and the stored models. By this classification approach, both whole and partial matching results with model shape similarity ranks can be obtained; especially, focus match can be achieved, in which different key parts can be labeled and all the matched models with corresponding key parts can be obtained. Some experiments are carried out to demonstrate the validity and efficiency of the approach for 3D object classification.
TopProblem Description
The specific problem we consider in this article is to classify 3D object into one of the known object classes and obtain models with shape similarity measures. We do not discuss how 3D data is segmented and fitted by superquadrics, which have performed well in much research work (Gupta & Bajcsy, 1993; Jaklic, Leonardis, & Solina, 2000; Liu & Yuan, 2001).
We are given a set of object data parts represented by SBGs, and the prestored models with object class labels. The classification of 3D objects can be regarded as the match between 3D object data and models, which is structured as an Interpretation Tree (Grimson, 1990). In Figure 1, let {m1, m2, ..., mi, ..., mp}be a set of model parts and {d1, d2, ..., dj, ..., dq}, be the object data parts, starting at a root node, we construct a tree in a depth-first fashion, assigning one model part to different data parts at each level of the tree.
Figure 1. Complete interpretation tree structure
In order to deal with the possible nonexistence of a feasible match between the current model part and the data parts, the “wild card” (*) is introduced to match a null data part with a model part in the tree for improving the robustness. A path through this tree represents a set of feasible correspondences or matching pairs, that is, a consistent interpretation.
Volumetric Representation: Superquadric-Based Geon (SBG)
RBC Theory and Geons
Biederman’s RBC theory (Biederman, 1987) provides a promising computational framework for the object recognition and postulates a finite set of distinct volumetric shapes called “geons” as the basis for representation of objects. RBC theory maintains that the set of geons apparently have sufficient representational power to express humans’ capacity for basic visual recognition, and the visual system readily decomposes the objects into such components and represents them in terms of their geons and the invariant relationships among them. Geons are classified due to four qualitative geometrical properties: axis shape, cross-section edge shape, cross-section size sweeping function, and cross-section symmetry.
These attributes provide distinct shape characteristics useful for symbolic object recognition (Bergevin & Levine, 1993). Psychological experimentation has provided supports for the descriptive power of geon-based description (Biederman & Gerhardstein, 1993) and geon models have been proposed as a basis for numbers of object recognition systems (Bergevin & Levine, 1993; Dickinson, 1997).
Superquadric Models
Superquadrics as a family of parametric shapes can describe a wide variety of relatively complex and realistic 3D primitive shapes effectively with compact parameters (Barr, 1981). A basic superquadric surface may be defined by an implicit equation:
(1) where
ax,
ay,
az are defined for the size along
X,
Y,
Z axis, ε
1, ε
2 are square parameters and control the shape of superquadric model.
The modeling power of superquadrics is augmented by applying various deformation operations which include bending, tapering, and so forth (Barr, 1981; Solina & Bajcsy, 1990) to the basic superquadrics.
Superquadric-Based Geons
Because the information offered by geons is only qualitative in nature (Dickinson, 1997), using geon-based description for 3D object recognition directly would not be very efficient in usual cases. In the article, superquadrics as a unified quantitative parametric shape models are implemented for representing geons because of superquadric powerful modeling capability and extensive implementation in computer vision, which brings a compositive volumetric representation called Superquadric-based Geons (SBG) in this article.
Geons are classified by a labeling A-B-C, where A∈{s=straight, b=bent}, axis shape; B∈{s=straight, c=curved}, cross-section edge shape; C∈{co=constant, id=increasing-and-decreasing, t=tapered}, cross-section size sweeping function.
Obviously, SBG combines superquadric quantitative parametric description and the geon’s qualitative geometrical attributes, which have powerful representative and discriminative capability. In this article, we use 12 geon types, shown in Figure 2.