sociated with the surface point, called shape diameter function (SDF) [73]. The SDF is an approximation of medial axis transform (MAT) [11] that uses local shape diameter to approximate the radius of the maximal inscribed ball. Both MAT and SDF have the same problem that the volume information at surface points is quite local. For example, the volume information associated with a point refers only to the maximal inscribed ball asso-ciated with that point in the MAT approach. For the slab of a non-cylindrical part, the union of several balls is required for evaluating the entire volume. The SDF has the same limi-tation since it is conceptually derived from MAT. There are other shape properties defined in intermediate-scale. Liu et al. measured the possibility of surface region to be within the same object part by the visibility of surface region inside the object [53].
2.2 Mesh Segmentation
In the past decade, many mesh segmentation methods have been proposed. Based on the objective, mesh segmentation methods generally fall into two categories: patch-type and part-type [5, 71, 10]. Patch-type segmentation usually decomposes the mesh into several patches by analyzing the surface properties such as dihedral angles [46, 75], curvature [56, 60, 75], geodesic distance [88], and planarity [21]. Part-type segmentation tends to segment a complex object into several meaningful components, usually based on the concepts from cognition theory [28, 29]. For example, the minima rule states that human perception tends to break an object into parts along the region of minimum negative curvature [28].
Moreover, the salience of parts determined by relative volume, boundary strength, and degree of protrusion is important for human perception [29]. Our method is a part-type segmentation, and we will mainly review this type of segmentation and refer readers to the excellent survey paper by Shamir [71] for the patch-type segmentation.
Locating the boundary between parts can be done by either boundary-based or region-based approach. Mangan and Whitaker used the watershed to segment the object into several parts according to the curvature on the surface [56]. Lee et al. [45] cut the object into parts by first finding the loop along the minimum negative curvature, and then test the salience of the divided parts based on the part salience theory [29]. However, the surface curvature is too local for describing the shape of object and locating cut boundaries based on the curvature cannot always result in a meaningful part segmentation. Moreover, the iterative segmentation process proposed leads to a binary hierarchical segmentation on
2.2 Mesh Segmentation 11
which each level does not provide an intuitive meaning for object parts.
The geodesic distance is another attribute widely used in mesh segmentation [77, 36, 51, 52]. The averaged geodesic distance (AGD) derived as the average of geodesic distance from a surface point to all other points can be used to represent the degree of protrusion of a part. However, such attributes are useful only for models that have an obvious core part and feature parts.
The Medial Axis Transform (MAT) is a global shape descriptor of the object [11].
MAT or skeleton can be used as a guideline for segmentation. For example, Li et al.
segmented the object by moving a sweep plane along the skeleton of object [48]. Since the size of the cutting section of the sweeping plane can be regarded as local volumetric information of the object, the cut boundaries are usually at the regions where the size of cutting section varies rapidly. Oscar et al. segmented the skeletal mesh by measuring thickness corresponding to the skeleton nodes derived from the skeletonization process [6].
The most concave region for the cutting is searched for each skeleton branch by comparing the thickness of the skeleton nodes with their neighbors [6]. Reniers and Telea observed that the junction between two skeleton paths has high potential to be a good place for separating two parts [66, 67]. Shapira et al. proposed a hierarchical segmentation method by fitting k Gaussian functions to the histogram of SDF values, and clustering the mesh faces according to their corresponding Gaussian functions [73]. However, the fitting of the global histogram can not reflect the difference between the object parts and some small parts with no salient feature may be segmented. The segmentations generated by using different number of Gaussian functions do not have consistent part correspondence and the part boundaries may not always lie on meaningful regions.
An iterative approach for decomposing the object into several parts is based on the k-means clustering [54]. Shlafman et al. used k-means clustering to segment the object into a user-specified number of components [77]. This work was later refined to achieve hierarchical segmentation [36]. However, the geodesic distance used in [36] describes only the protrusion of object parts and hence the resultant hierarchical segmentation tends to cut the object along the longest parts at the higher level of the hierarchy, which may not meet the concept of perception. Moreover, the method is usually suitable only for objects having obvious core-salient features. The challenge to the k-means clustering methods is that the value of k needs to be given a priori. Liu and Zhang overcame such problems by applying
2.2 Mesh Segmentation 12
the spectral analysis on the affinity matrix constructed using the mesh faces [51].
Another segmentation approach was based on the fitting of primitives. Attene et al.
extended the hierarchical face clustering [21] and replaced the clustering metric by other similarity measures for the predefined primitives, such as spheres, cylinders and planes [4].
Mortara et al. detected the parts with tubular shape from the whole object [59]. They extracted the core part by excluding all the tubular parts from the object to complete the segmentation.
Pose-invariant mesh segmentation has attracted more attention in recent years. Such works focused on finding the consistent segmentation over different poses of a model.
Katz et al. transformed the original model into a pose-invariant representation using multi-dimensional scaling and then used spherical mirroring to extract the core of the object and feature points to segment the objects [35]. In character animation, the pose-invariant segmentation can be achieved by finding the rigid components during the animation [41, 44, 43]. However, such methods are usually suitable only for articulated models and the requirement of animation sequences also poses some restrictions on the usability.
Consistent mesh segmentation aims to produce consistent segmentations for a set of meshes [78, 72, 84]. Golovinskiy and Funkhouser [23] employed rigid alignment [9] in a hierarchical clustering approach for consistent segmentation. Both the geometric features of individual meshes and the correspondence information between the set of meshes are considered. However, rigid alignment may not be able to correctly align the meshes in some cases, as reported in [34]. Kalogerakis et al. [34] proposed a scheme to compute segmentations and to assign labels for a set of meshes. The assignment of labels was formulated as an optimization problem and the objective function measured the consistency of primitives (i.e. triangles) with labels. The objective function was computed via a training process and then applied to the other meshes for computing consistent segmentation. Their approach handled various segmentations for a wide range of meshes. However, the training process is time consuming and usually takes hours of computing.
Golovinskiy and Funkhouser [22] defined a surface function called partition function that indicates how likely each edge is to lie on the boundary of a random segmentation drawn from a set of existing segmentations. Based on the partition function, a cut is asso-ciated with a consistency measure as the length-weighted average of the partition function values of its edges. The most consistent cuts defined as the set of cuts with highest