While conventional computer vision emphasizes pixel-level and feature-based objectives, medical image analysis of intricate biological structures necessitates explicit representation of their complex topological properties. Despite their successes, deep learning models often struggle to accurately capture the connectivity and continuity of fine, sometimes pixel-thin, yet critical structures due to their reliance on implicit learning from data. To address this challenge, we introduce Conformable Convolution, a novel convolutional layer designed to explicitly impose topological consistency. Conformable Convolution learns adaptive kernel offsets that focus on regions of high topological significance within an image. This prioritization is guided by our proposed Topological Posterior Generator (TPG) module, which leverages persistent homology. The TPG module identifies key topological features and guides the convolutional layers by applying persistent homology to feature maps transformed into cubical complexes. Unlike existing approaches that are merely aware of topology, our method explicitly constrains the learning process to ensure topological correctness. The proposed modules are architecture-agnostic, enabling them to be integrated seamlessly into various architectures. We showcase the effectiveness of our framework in the segmentation task, where preserving the interconnectedness of structures is critical. The results on three diverse datasets demonstrate that our framework effectively preserves the topology both quantitatively and qualitatively.

Scaling by training on large datasets has been shown to enhance the quality and fidelity of image generation and manipulation with diffusion models; however, such large datasets are not always accessible in medical imaging due to cost and privacy issues, which contradicts one of the main applications of such models to produce synthetic samples where real data is scarce. Also, fine-tuning on pre-trained general models has been a challenge due to the distribution shift between the medical domain and the pre-trained models. Here, we propose Latent Drift (LD) for diffusion models that can be adopted for any fine-tuning method to mitigate the issues faced by the distribution shift or employed in inference time as a condition. Latent Drifting enables diffusion models to be conditioned for medical images fitted for the complex task of counterfactual image generation, which is crucial to investigate how parameters such as gender, age, and adding or removing diseases in a patient would alter the medical images. We evaluate our method on three public longitudinal benchmark datasets of brain MRI and chest X-rays for counterfactual image generation. Our results demonstrate significant performance gains in various scenarios when combined with different fine-tuning schemes.