# TopoSinGAN: Learning a Topology-Aware Generative Model from a Single Image
## 4 Discussion  
The research presented demonstrates the effectiveness of TopoSinGAN in enhancing topological accuracy through its integrated topology loss function. Unlike conventional losses, such as adversarial or reconstruction losses [13; 55], the topology loss optimizes for lower terminal counts using its specialized convolutional kernels. This approach allows TopoSinGAN to minimize topological anomalies that could otherwise go unnoticed, thereby addressing a critical need in applications requiring high topological integrity. While SinGAN retains visual similarities, structural discrepancies can arise without dedicated topological supervision, which TopoSinGAN successfully mitigates by penalizing these errors. A key feature of the topology loss function is its full differentiability, which facilitates end-to-end training via backpropagation. This seamless integration within the training cycle enables direct optimizations based on topological feedback, surpassing methods that rely solely on post-processing. The end-to-end nature of this approach ensures that topological feedback is continually refined throughout the training process, leading to more robust and accurate models. Furthermore, our study introduces a novel GAN evaluation metric focused purely on the topological attributes of the images, excluding geometric variations. This metric, termed the NTC score, provides a more realistic judgment of GAN models by quantifying topological integrity. Consistent improvements in NTC scores indicate the quantitative effectiveness of TopoSinGAN, as lower NTC values correspond to reduced loop openness and fewer topological anomalies. Importantly, these lower NTC scores denote higher topological accuracy, not merely structural similarity. Improved NTC scores reported in the comparative evaluation of TopoSinGAN with plain SinGAN, WGAN, and TopoSin further highlights TopoSinGAN's effectiveness in generating topologically  
Figure 8: Tuning of the \(\lambda_{3}\) parameter for the CREMI experiment, showing its impact on FID (blue line) and NTC (red line). The best balance between FID and NTC is achieved at \(\lambda_{3}=0.4\) for this experiment. enhanced images. According to Table 3, SinGAN and WGAN recorded similar NTC scores as they both use similar adversarial losses [13]. In Appendix C, we use the common graph metrics to demonstrate that our proposed topology loss has no significant negative impact on the learning flow of the generator compared to the original SinGAN (Table A2). The high similarity of common graph metrics for SinGAN and TopoSinGAN outputs suggests two observations. First, both models effectively replicate the general structure of real input images. This signifies that our topology loss has not significantly affected the realism of the generated image contexts which aligns with the result of modified FID evaluation. Second, it also reveals a limitation of common graph metrics to capture the topological inaccuracies within mask graphs. The NTC metric addresses this gap by focusing on a graph's topological structure, offering a more nuanced evaluation of topological accuracy. Unlike traditional metrics that compare graphs or graphlets for similarity [56; 57; 58; 59], the NTC metric emphasizes topological integrity, making it a specialized tool for assessing the precision of topological features in a graph. Additionally, the topological loss function in TopoSinGAN is uniquely characterized by its standalone application, assessing the model's topological loss based solely on the generated mask channel. This approach penalizes the model for topological inaccuracies without comparing the generated masks to real inputs, focusing on minimizing the occurrence of terminal nodes. Despite the modest additional computational cost, its benefits in preserving the integrity of topological features make it a valuable tool for various applications demanding precise structural fidelity. Nevertheless, TopoSinGAN has certain limitations that should be considered.