*4.2. Tests on CT Images*

Digital Imaging and Communications in Medicine (DICOM) is a standard protocol for the management and transmission of medical images, which is widely used in healthcare facilities. Here, we choose CT images as test datasets, which are stored in accordance with the DICOM standard. Each image contains 512 × 512 pixels, and each pixel is identified by its CT value. Figures 15 and 16 show the clustering and evaluation results of CVIBI for a group of typical CT images. The first column represents the original CT images. Columns 2–4 represent the partition results, which are clustered by DPC and C-means, respectively. Here, pseudo colors denote different clusters. The fifth column is the curves of CVIBI.

**Figure 15.** Tests of CVIBI on CT images with C-means.

Figures 15 and 16 show that we can obtain the optimal number of clusters when applying CVIBI with C-means and DPC to CT images, and the partitioned images show the shapes of various origins and tissues. Consequently, CVIBI with any clustering algorithms can take effect on automatic imaging segmentation in CT images, which can point out the categories of tissue in one particular CT layer.

Table 4 shows all evaluation results by using C-means and DPC with four indices, respectively, where *x* and *y* in sign image *x*/*y* refer to the investigated CT images and the correct number of clusters, respectively. The suggested numbers of clusters between the four validity indices are different; GS can identify the correct number of clusters no matter which clustering algorithm is applied; DC with C-means can identify the correct number of clusters, but DC with DPC fails for images 2 and 3; in terms of accuracy, CVIBI seems to be more efficient.

**Figure 16.** Tests of CVIBI on CT images with DPC.


<sup>√</sup> <sup>5</sup>

<sup>√</sup> <sup>5</sup>

<sup>√</sup> <sup>4</sup>

<sup>√</sup> <sup>3</sup>

<sup>√</sup> <sup>4</sup>

<sup>√</sup> <sup>5</sup>

<sup>√</sup> <sup>5</sup>

<sup>√</sup> <sup>3</sup>

<sup>√</sup> <sup>4</sup>

<sup>√</sup> <sup>5</sup>

<sup>√</sup> <sup>5</sup>

√

√

**Table 4.** Evaluation of clustering results by CVIBI, DB, DC and GS for 15 datasets.

### **5. Conclusions**

*Set* 4 25<sup>√</sup> <sup>3</sup>

*Set* 5 4<sup>√</sup> <sup>5</sup>

The clustering evaluation is an essential but difficult task in clustering analysis. Currently, the existing validity evaluation has to depend on a specific clustering algorithm, a specific cluster parameter (or several), and specific assumptions, and has very limited applicable range. In this paper, we proposed a novel validity index, which can evaluate the clustering results obtained either by a single clustering algorithm or by several clustering algorithms. Especially, it can be applied to select any clustering parameters besides the typical number of clusters. To our knowledge, the kind of necessary applications cannot be realized by existing validity indices. This novel index outperforms the existing validity indices on some benchmark datasets in terms of accuracy and generality. Experimental results validate this index. The boundary matching degree and connectivity degree are important notions in graph theory. Our future work is to combine these notions with graph theory to reduce time complexity.

**Author Contributions:** Conceptualization, Q.L. and S.Y.; methodology, Q.L. and S.Y.; software, Q.L. and Y.W.; validation, Q.L., M.D. and J.L.; formal analysis, Z.W.; investigation, J.L. and Z.W.; resources, Z.W.; data curation, Q.L. and M.D.; writing—original draft preparation, Q.L.; writing—review and editing, M.D. and J.L.; visualization, Y.W.; supervision, S.Y.; project administration, S.Y.; funding acquisition, S.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China, grant number 61573251 and 61973232.

**Conflicts of Interest:** The authors declare no conflict of interest.
