Topological correctness plays a critical role in many image segmentation tasks, yet most networks are trained using pixel-wise loss functions, such as Dice, neglecting topological accuracy. Existing topology-aware methods often lack robust topological guarantees, are limited to specific use cases, or impose high computational costs. In this work, we propose a novel, graph-based framework for topologically accurate image segmentation that is both computationally efficient and generally applicable. Our method constructs a component graph that fully encodes the topological information of both the prediction and ground truth, allowing us to efficiently identify topologically critical regions and aggregate a loss based on local neighborhood information. Furthermore, we introduce a strict topological metric capturing the homotopy equivalence between the union and intersection of prediction-label pairs. We formally prove the topological guarantees of our approach and empirically validate its effectiveness on binary and multi-class datasets, demonstrating state-of-the-art performance with up to fivefold faster loss computation compared to persistent homology methods.

Direct image-to-graph transformation is a challenging task that involves solving object detection and relationship prediction in a single model. Due to this task's complexity, large training datasets are rare in many domains, making the training of deep-learning methods challenging. This data sparsity necessitates transfer learning strategies akin to the state-of-the-art in general computer vision. In this work, we introduce a set of methods enabling cross-domain and cross-dimension learning for image-to-graph trans-formers. We propose (1) a regularized edge sampling loss to effectively learn object relations in multiple domains with different numbers of edges, (2) a domain adaptation frame-work for image-to-graph transformers aligning image- and graph-level features from different domains, and (3) a projection function that allows using 2D data for training 3D transformers. We demonstrate our method's utility in cross-domain and cross-dimension experiments, where we utilize labeled data from 2D road networks for simultaneous learning in vastly different target domains. Our method consistently outperforms standard transfer learning and self-supervised pretraining on challenging benchmarks, such as retinal or whole-brain vessel graph extraction.11Code: github.com/AlexanderHBerger/cross-dim_i2g

Biophysical modeling, particularly involving partial differential equations (PDEs), offers significant potential for tailoring disease treatment protocols to individual patients. However, the inverse problem-solving aspect of these models presents a substantial challenge, either due to the high computational requirements of model-based approaches or the limited robustness of deep learning (DL) methods. We propose a novel framework that leverages the unique strengths of both approaches in a synergistic manner. Our method incorporates a DL ensemble for initial parameter estimation, facilitating efficient downstream evolutionary sampling initialized with this DL-based prior. We showcase the effectiveness of integrating a rapid deep-learning algorithm with a high-precision evolution strategy in estimating brain tumor cell concentrations from magnetic resonance images. The DL-Prior plays a pivotal role, significantly constraining the effective sampling-parameter space. This reduction results in a fivefold convergence acceleration and a Dice-score of 95%.