# TopoSinGAN: Learning a Topology-Aware Generative Model from a Single Image
## 1 Introduction
### The Need for Topological Accuracy  
Topological accuracy can have different meanings across fields such as graph theory and computer science. In image semantic segmentation models, it often refers to preserving the topological relationships between the input image and the predicted mask [17; 18; 19]. However, in the context of GANs, where there is no real image corresponding to each generated image for direct comparison, topological accuracy is generally defined by connectivity. This involves ensuring that objects expected to be connected, such as roads or agricultural field boundaries, do not display disconnections in the generated datasets [20; 21; 22]. Similarly, in this study, we define topological accuracy as minimizing open loops in the generated images. The case studies presented here, agricultural fields and dendrology data, involve scenarios where the generated masks should consist entirely of closed loops, as both agricultural fields and xylem cells in dendrology images are expected to appear as fully enclosed regions. Thus, our research focuses on reducing these discontinuities and enhancing the topological integrity of the generated masks. Ensuring topological accuracy is crucial in GAN applications across various domains. While GANs are highly effective at generating visually compelling outputs, they may overlook the intrinsic structure and connectivity of linear features that define the topology. This distinction from geometric representations is critical. In contexts such as road network mapping or medical imaging, maintaining topological accuracy in GAN outputs is essential to prevent the misrepresentation of crucial structures. For instance, an inaccurately depicted road junction in an autonomous driving map or an incorrect representation of a neuronal pathway in a medical image can have serious consequences [23]. Therefore, maintaining topological accuracy in GAN outputs not only preserves the interconnected nature of the data but also enhances the reliability and practical utility of the models in real-world applications. Attempts to improve the structural coherence of generated images began soon after the introduction of GANs. The work by [PERSON] et al. (2017), introducing Wasserstein GAN (WGAN), was an early example of this direction. WGAN aimed to improve training stability and address issues such as mode collapse by minimizing the Earth Mover (Wasserstein) distance, offering a more stable optimization process compared to traditional GANs [24]. In the domain of GANs that emphasize geometry, several models have been developed to capture and utilize geometric nuances from images. For example, geometricGAN [25] integrates the expansive margin principle from SVMs to guide both discriminator and generator training. Similarly, Localized GAN (LGAN) [26] utilizes local coordinates to focus on the specific geometry of the data manifold. GAGAN [27] is notable for its application to facial image generation, drawing on established facial geometries. Meanwhile, GeGAN [28] enforces geometry-consistency regulation to preserve the inherent semantics of images, limiting geometric modifications to simple rotations and flips. Recent advancements further highlight the integration of topological features in GAN models to enhance their application across different fields. The TR-GAN framework,for instance, improves retinal artery/vein (A/V) classification by employing a topology ranking discriminator and triplet loss, which significantly enhances vessel connectivity, outperforming conventional methods on the AV-DRIVE dataset [29]. TW-GAN further advances A/V classification by incorporating both topology and vessel width awareness, markedly improving performance on datasets such as AV-DRIVE, which is crucial for diagnosing conditions such as hypertension and diabetes [30]. Similarly, [PERSON] et al. (2019) employed a hierarchical architecture to extract and preserve both local and global graph features, leading to more accurate graph reconstruction [31].