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Enforcing 3D Topological Constraints in Composite Objects via Implicit Functions

Medical applications often require accurate 3D representations of complex organs with multiple parts, such as the heart and spine. Their individual parts must adhere to specific topological constraints to ensure proper functionality. Yet, there are very few mechanisms in the deep learning literature to achieve this goal. This paper introduces a novel approach to enforce topological constraints in 3D object reconstruction using deep implicit signed distance functions. Our method focuses on heart and spine reconstruction but is generalizable to other applications. We propose a sampling-based technique that effectively checks and enforces topological constraints between 3D shapes by evaluating signed distances at randomly sampled points throughout the volume. We demonstrate it by refining 3D segmentations obtained from the nn-UNet architecture.

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