Comments on “Efficient skeletonization of volumetric objects”

Recently, a skeletonization algorithm for volumetric objects has been proposed. In this paper, we show that this algorithm cannot obtain correct results in the sense that both the connectivity and topological information of the original object cannot be maintained. Index Terms Skeleton, centerline, voxel-coding, distance transform Qingmao Hu and Wieslaw L. Nowinski are both with Kent Ridge Digital Labs 21 Heng Mui Kent Terrace Singapore 119613 E-mails are: [email protected], and [email protected] Recently, a skeletonization algorithm for volumetric objects has been proposed [1]. This algorithm was based on two kinds of voxel coding, i.e., the boundary-seeded (BS) voxel coding and single point seeded (SS) voxel coding. The BS coding adopted 3-4-5 metric to generate a traditional minimum distance field in order to achieve the ceneteredness of object skeleton, while the SS coding adopted a 1-2-3 metric to generate clusters to maitain object connectivity and topology information. Based on local maximum clusters, dividing clusters, and merging clusters, a cluster graph which reserved the topology information of the object was obtained. From the cluster graph, the skeleton was calculated through multiple centerline extraction, skeleton refinement, smoothing and connection. Even though the definition of skeleton is very difficult, it is widely accepted [2] that a skeleton should satisfy at least three conditions, i.e., connected, centered, and thin. The skeleton produced by [1] cannot guarantee the connectivity. Figure 1 is a 2D image that will have broken skeleton by [1]. In fact, Figure 1 is a quite simple binary image having one branch. The algorithm [1] is inherently weak in dealing with branches. In Figure 1, the point A was chosen as the RP, satisfying the authors’ assumption [1] to be a cusp point. Indicated is the BS code of the image strictly according to [1]. There were two local maximum clusters with cluster code 42 and 33. There was one dividing cluster with cluster code 18. Marked red crosses were those pixels on the Medpath. The pixel with cluster code 4 and circled blue (denoted as B) was supposed to be connected to the pixel with cluster code 15 and marked blue (denoted as C), where C was the medial point of the dividing cluster with cluster code 18. To deal with the large gap between B and C caused by branching, according to the authors [1], a new SS-field was produced taking C as a new RP and a shortest path from B to C was formed. All the non-end points of the path were replaced by the new local clusters’ medial points, which were marked blue crosses in Figure 1, and the connection relationship was indicated by arrows. The dotted arrow showed the new gap between two skeletal points that were supposed to be connected. The authors [1] stated “Each cluster can be approximately taken as a cross-section of the object, normal to the related centerline.” We show below that this is a wrong assumption and this wrong assumption will cause connectivity problem as well as topology problem. It’s obvious from Figure 2 that at least at dividing cluster, the cluster will contain two cross sections of the object. Thus at least the two medial points of the dividing cluster’s two predecessors were supposed to connect to the same point, i.e., the medial point of the dividing cluster. Suppose the two cross sections of the dividing cluster are denoted as CS1 and CS2, the two predecessors’ cross sections are denoted as CS11 and CS21. If the medial point of the dividing cluster corresponds to the local distance maximum in CS1, it is expected that the medial point of CS21 will have a large gap to the medial point of the dividing cluster since they are not supposed to connect. In addition, the authors in [1] made no effort to ensure connectedness between the medial points of neighboring clusters. Being medial points of neighboring clusters cannot guarantee connectivity. Besides the connectivity problem, we show that the wrong crossing problem below. Let’s denote the two end points of the dotted arrow as D and E respectively (see Figure 1). So according to the authors [1], E will be connected by D with quite a few pixels/voxels gaps; also assumed is that E will be a skeletal crossing (a skeletal crossing is a skeletal point that has more than two skeletal neighbors). But the true crossing is F (the pixel with distance code 19). We know crossing is an important topologic information of skeleton. Due to the fact that the skeleton produced by [1]will have wrong crossing position, the topologic information is not correctly reserved.So we conclude that the centerline or skeleton produced by [1] is not correct in thesense that both the connectivity and topology information of the object cannot bemaintained. References1. Y. Zhou, and A. W. Toga. “Efficient skeletonization of volumetric objects,” IEEETransactions on Visualization and Computer Graphics, vol. 5, no.3, pp.197-209,1999.2. C. W. Niblack, P. B. Gibbons, and D. W. Capson, “Generating skeletons andcenterlines from the distance transform,” CVGIP: Graphical Models and ImageProcessing, vol. 54, no. 5, pp. 420-437, 1992.