FDIF: Formula-Driven supervised Learning with Implicit Functions for 3D Medical Image Segmentation

Vision Transformers (ViTs) excel in 3D medical segmentation but require massive annotated datasets. While Self-Supervised Learning (SSL) mitigates this using unlabeled data, it still faces strict privacy and logistical barriers. Formula-Driven Supervised Learning (FDSL) offers a privacy-preserving alternative by pre-training on synthetic mathematical primitives. However, a critical semantic gap limits its efficacy: generic shapes lack the morphological fidelity, fixed spatial layouts, and inter-organ relationships of real anatomy, preventing models from learning essential global structural priors. To bridge this gap, we propose an Anatomy-Informed Synthetic Supervised Pre-training framework unifying FDSL's infinite scalability with anatomical realism. We replace basic primitives with a lightweight shape bank with de-identified, label-only segmentation masks from 5 subjects. Furthermore, we introduce a structure-aware sequential placement strategy to govern the patch synthesis process. Instead of random placement, we enforce physiological plausibility using spatial anchors for correct localization and a topological graph to manage inter-organ interactions (e.g., preventing impossible overlaps). Extensive experiments on BTCV and MSD datasets demonstrate that our method significantly outperforms state-of-the-art FDSL baselines and SSL methods by 1.74% and up to 1.66%, while exhibiting a robust scaling effect where performance improves with increased synthetic data volume. This provides a data-efficient, privacy-compliant solution for medical segmentation. The code will be made publicly available upon acceptance.

Foundation models have transformed vision and language by learning general-purpose representations from large-scale unlabeled data, yet 3D medical imaging lacks analogous approaches. Existing self-supervised methods rely on low-level reconstruction or contrastive objectives that fail to capture the anatomical semantics critical for medical image analysis, limiting transfer to downstream tasks. We present MASS (MAsk-guided Self-Supervised learning), which treats in-context segmentation as the pretext task for learning general-purpose medical imaging representations. MASS's key insight is that automatically generated class-agnostic masks provide sufficient structural supervision for learning semantically rich representations. By training on thousands of diverse mask proposals spanning anatomical structures and pathological findings, MASS learns what semantically defines medical structures: the holistic combination of appearance, shape, spatial context, and anatomical relationships. We demonstrate effectiveness across data regimes: from small-scale pretraining on individual datasets (20-200 scans) to large-scale multi-modal pretraining on 5K CT, MRI, and PET volumes, all without annotations. MASS demonstrates: (i) few-shot segmentation on novel structures, (ii) matching full supervision with only 20-40% labeled data while outperforming self-supervised baselines by over 20 in Dice score in low-data regimes, and (iii) frozen-encoder classification on unseen pathologies that matches full supervised training with thousands of samples. Mask-guided self-supervised pretraining captures broadly generalizable knowledge, opening a path toward 3D medical imaging foundation models without expert annotations. Code is available: https://github.com/Stanford-AIMI/MASS.

Vision Transformers (ViTs) have revolutionized medical image analysis, yet their data-hungry nature clashes with the scarcity and privacy constraints of clinical archives. Formula-Driven Supervised Learning (FDSL) has emerged as a promising solution to this bottleneck, synthesizing infinite annotated samples from mathematical formulas without utilizing real patient data. However, existing FDSL paradigms rely on simple geometric shapes with homogeneous intensities, creating a substantial gap by neglecting tissue textures and noise patterns inherent in modalities like CT and MRI. In this paper, we identify a critical optimization conflict termed boundary aliasing: when high-frequency synthetic textures are naively added, they corrupt the image gradient signals necessary for learning structural boundaries, causing the model to fail in delineating real anatomical margins. To bridge this gap, we propose a novel Physics-inspired Spatially-Decoupled Synthesis framework. Our approach orthogonalizes the synthesis process: it first constructs a gradient-shielded buffer zone based on boundary distance to ensure stable shape learning, and subsequently injects physics-driven spectral textures into the object core. This design effectively reconciles robust shape representation learning with invariance to acquisition noise. Extensive experiments on the BTCV and MSD datasets demonstrate that our method significantly outperforms previous FDSL, as well as SSL methods trained on real-world medical datasets, by 1.43% on BTCV and up to 1.51% on MSD task, offering a scalable, annotation-free foundation for medical ViTs. The code will be made publicly available upon acceptance.