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

arXiv: https://arxiv.org/abs/2603.23199 · HTML: https://mlsanity.com/paper/2603.23199/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 38 references that are in the map.
- First posted on arXiv 24 Mar 2026.

## Builds on (most cited first, 25 of 25)

- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Swin UNETR: Swin Transformers for Semantic Segmentation of Brain Tumors in MRI Images](https://mlsanity.com/paper/2201.01266/) (2022)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [Self-Supervised Pre-Training of Swin Transformers for 3D Medical Image Analysis](https://mlsanity.com/paper/2111.14791/) (2021)
- [AMOS: A Large-Scale Abdominal Multi-Organ Benchmark for Versatile Medical Image Segmentation](https://mlsanity.com/paper/2206.08023/) (2022)
- [nnU-Net Revisited: A Call for Rigorous Validation in 3D Medical Image Segmentation](https://mlsanity.com/paper/2404.09556/) (2024)
- [The KiTS21 Challenge: Automatic segmentation of kidneys, renal tumors, and renal cysts in corticomedullary-phase CT](https://mlsanity.com/paper/2307.01984/) (2023)
- [Pre-training without Natural Images](https://mlsanity.com/paper/2101.08515/) (2021)
- [VoCo: A Simple-yet-Effective Volume Contrastive Learning Framework for 3D Medical Image Analysis](https://mlsanity.com/paper/2402.17300/) (2024)
- [Masked Image Modeling Advances 3D Medical Image Analysis](https://mlsanity.com/paper/2204.11716/) (2022)
- [UniMiSS: Universal Medical Self-Supervised Learning via Breaking Dimensionality Barrier](https://mlsanity.com/paper/2112.09356/) (2021)
- [Self-supervised 3D anatomy segmentation using self-distilled masked image transformer (SMIT)](https://mlsanity.com/paper/2205.10342/) (2022)
- [Revisiting MAE pre-training for 3D medical image segmentation](https://mlsanity.com/paper/2410.23132/) (2024)
- [A generalizable 3D framework and model for self-supervised learning in medical imaging](https://mlsanity.com/paper/2501.11755/) (2025)
- [Learning General-Purpose Biomedical Volume Representations using Randomized Synthesis](https://mlsanity.com/paper/2411.02372/) (2024)
- [MIS-FM: 3D Medical Image Segmentation using Foundation Models Pretrained on a Large-Scale Unannotated Dataset](https://mlsanity.com/paper/2306.16925/) (2023)
- [An OpenMind for 3D medical vision self-supervised learning](https://mlsanity.com/paper/2412.17041/) (2024)
- [Primus: Enforcing Attention Usage for 3D Medical Image Segmentation](https://mlsanity.com/paper/2503.01835/) (2025)
- [Pre-training Vision Transformers with Formula-driven Supervised Learning](https://mlsanity.com/paper/2206.09132/) (2022)
- [Advancing Volumetric Medical Image Segmentation via Global-Local Masked Autoencoder](https://mlsanity.com/paper/2306.08913/) (2023)
- [Primitive Geometry Segment Pre-training for 3D Medical Image Segmentation](https://mlsanity.com/paper/2401.03665/) (2024)

## Abstract (arXiv)

Deep learning-based 3D medical image segmentation methods relies on large-scale labeled datasets, yet acquiring such data is difficult due to privacy constraints and the high cost of expert annotation. Formula-Driven Supervised Learning (FDSL) offers an appealing alternative by generating training data and labels directly from mathematical formulas. However, existing voxel-based approaches are limited in geometric expressiveness and cannot synthesize realistic textures. We introduce Formula-Driven supervised learning with Implicit Functions (FDIF), a framework that enables scalable pre-training without using any real data and medical expert annotations. FDIF introduces an implicit-function representation based on signed distance functions (SDFs), enabling compact modeling of complex geometries while exploiting the surface representation of SDFs to support controllable synthesis of both geometric and intensity textures. Across three medical image segmentation benchmarks (AMOS, ACDC, and KiTS) and three architectures (SwinUNETR, nnUNet ResEnc-L, and nnUNet Primus-M), FDIF consistently improves over a formula-driven method, and achieves performance comparable to self-supervised approaches pre-trained on large-scale real datasets. We further show that FDIF pre-training also benefits 3D classification tasks, highlighting implicit-function-based formula supervision as a promising paradigm for data-free representation learning. Code is available at https://github.com/yamanoko/FDIF.