# TopoSinGAN: Learning a Topology-Aware Generative Model from a Single Image
## 5 Conclusions  
In this study, we introduced TopoSinGAN, an extension of SinGAN that incorporates a differentiable topology loss function to enhance topological accuracy in synthetic image generation. By minimizing terminal nodes, TopoSinGAN effectively addresses topological anomalies that conventional GAN losses often overlook, without significantly increasing computational complexity. Another contribution of this work is the proposal of the NodeTopology Clustering (NTC) metric, which focuses on evaluating topological integrity rather than just geometric similarity. Our experiments, involving agricultural and dendrological images, demonstrated that TopoSinGAN achieves significantly better topological fidelity than SinGAN, as evidenced by lower NTC indexes. This indicates fewer instances of loop openness and isolated line segments, which are crucial for applications requiring precise topological features. While TopoSinGAN excels in preserving topological integrity, it may not be ideal for contexts where open structures are expected. Nevertheless, it represents a crucial advancement in generating topologically accurate synthetic data, with potential applications in GIS, medical imaging, and other fields where topological accuracy is essential. Future work will focus on refining the topology loss function and extending its application to a broader range of tasks, such as image semantic segmentation. Additionally, we plan to adapt the proposed model to overcome the limitations outlined in Section 4. Moreover, the development of more sophisticated evaluation metrics for topological accuracy will remain a priority, aiming to enhance the robustness and applicability of generative models across diverse fields.  
Conceptualization, M.A. and E.S.; methodology, M.A.; software, M.A.; validation, M.A. and E.S.; formal analysis, M.A.; investigation, M.A. and E.S.; resources, E.S.; data curation, M.A.; writing--original draft preparation, M.A.; writing--review and editing, M.A. and E.S.; visualization, M.A.; supervision, E.S.; project administration, M.A. and E.S.; funding acquisition, E.S. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The code and data for this study are available in the project's GitHub repository [60].  
The authors would like to thank the Minnesota Supercomputing Institute (MSI) and the South Dakota State University Research Cyberinfrastructure (RCi) for providing the high-performance computing resources essential for our experimentation. We also extend our gratitude to everyone who contributed to the success of this project, including [PERSON], [PERSON], and [PERSON] for their valuable insights.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} GAN & Generative adversarial network \\ GT & Ground truth \\ TD & Terminal distance \\ NTC & Node topology clustering \\ PH & Persistence homology \\ STD & Standard deviation \\ GIS & Geographic information science \\ TDA & Topological data analysis \\ MSI & Minnesota supercomputing institute \\ FID & Fréchet inception distance \\ RGB & Red, green, blue \\ \end{tabular}  
Appendix A Demonstration of Independence of NTC Metric on the Placement of the Randomly Added Terminal Node  
To verify that the addition of a terminal node does not significantly impact the proposed topological evaluation metric, we randomly added a terminal node to both case study graphs 1000 times and calculated the percentage of LISA clusters. The results from both experiments, presented in Table 11, indicate that the percentage of clusters remained consistent, with a similar mean and a small standard deviation (STD).