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

Yukinori Yamamoto; Kazuya Nishimura; Tsukasa Fukusato; Hirokazu Nosato; Tetsuya Ogata; Hirokatsu Kataoka

cs.CVMar 24, 2026

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

Authors: Yukinori Yamamoto, Kazuya Nishimura, Tsukasa Fukusato, Hirokazu Nosato, Tetsuya Ogata, Hirokatsu Kataoka

Organizations: Waseda University · National Institute of Advanced Industrial Science and Technology · The University of Osaka · University of Oxford

Abstract

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.

Figures & tables

Figure 1: (a) Overview of FDIF (109 classes) and PrimGeoSeg (32 classes). (b) Comparison of FDIF and PrimGeoSeg. FDIF uses implicit functions and leverages it for texture generation. (c) A signed distance function (SDF) reflects distance to the nearest surface: positive outside, negative inside, and zero on the boundary.

Figure 2: Synthetic volume generation via primitive composition. Each primitive is assigned a base SDF from a diverse SDF library $\Phi$ , transformed with random spatial parameters, augmented with displacement $\Delta_{j}$ for geometric texture, and converted to intensity via mapper functions $g_{m}$ . Multiple primitives are merged to form the final labeled volume.

Architecture	Pre-training	Type	AMOS	ACDC	KiTS	Avg
SwinUNETR	Scratch	-	80.98	81.91	74.55	79.15
	PrimGeoSeg [ 23 ]	FDSL	81.89	85.02	75.80	80.90
	Ours (FDIF, Disp+Map)		82.28	83.82	77.20	81.10
	Ours (FDIF, Map only)		82.70	84.66	78.31	81.89
nnUNet ResEnc-L	Scratch	-	85.97	91.98	83.84	87.26
	S3D [ 27 , 28 ]	SSL	86.16	92.01	86.01	88.06

Table 1: Comparison of segmentation performance in terms of Dice score across different architectures and pre-training methods on three benchmark datasets. Best results are highlighted in bold and second-best results are underlined .

Figure 5: Qualitative comparison of segmentation results. Models pre-trained with FDIF demonstrate improved segmentation accuracy compared to baseline methods.

Pre-training	Avg	Spl	RKd	LKd	Gal	Eso	Liv	Sto	Aor	Pos	Pan	RAd	LAd	Duo	Bla	P/U
10 global-shape	87.95	96.84	96.00	95.58	83.32	82.59	97.65	91.17	94.91	89.87	87.15	75.90	77.27	81.09	86.90	82.95
109 global-shape	87.96	96.89	96.20	95.03	84.36	83.03	97.67	90.80	94.68	90.13	87.09	76.11	77.38	80.89	87.11	82.01
109 gs+10 Disp	88.02	96.77	95.96	95.49	84.66	82.79	97.68	91.10	94.87	90.24	87.34	75.50	77.62	81.51	86.77	82.00
109 gs+10 Map	88.04	96.86	96.14	95.47	84.19	83.00	97.69	90.93	94.87	90.21	87.49	76.07	77.47	81.53	86.24	82.45
109 gs+10 Disp+10 Map	87.66	96.78	96.06	95.55	84.11	82.53	97.69	90.58	94.82	90.24	87.46	76.06	77.41	81.59	81.15	82.89
PrimGeoSeg (32 gs)	87.57	96.80	95.76	95.44	83.20	81.96	97.56	90.37	94.78	89.96	87.04	75.85	76.89	80.36	85.62	82.01

Table 2: Ablation study on the AMOS dataset evaluating the effects of global-shape count, displacement, and mapper functions. Results are reported as Dice scores (%). Organ abbreviations: Spl=Spleen, RKd=Right Kidney, LKd=Left Kidney, Gal=Gallbladder, Eso=Esophagus, Liv=Liver, Sto=Stomach, Aor=Aorta, Pos=Postcava, Pan=Pancreas, RAd=Right Adrenal, LAd=Left Adrenal, Duo=Duodenum, Bla=Bladder, P/U=Prostate/Uterus.

Pre-training	Avg	Spl	RKd	LKd	Gal	Eso	Liv	Sto	Aor	Pos	Pan	RAd	LAd	Duo	Bla	P/U
10 gs (Extrusion only)	87.80	96.93	95.82	94.90	83.31	82.37	97.62	90.49	94.86	89.79	87.01	75.88	77.14	81.13	86.34	83.36
10 gs (Revolution & Hollowing only)	87.78	96.85	96.20	95.42	83.92	82.24	97.62	90.93	94.76	89.86	87.20	76.25	76.99	81.03	85.40	82.07
10 gs (Combined)	87.95	96.84	96.00	95.58	83.32	82.59	97.65	91.17	94.91	89.87	87.15	75.90	77.27	81.09	86.90	82.95

Table 3: Ablation study on global-shape construction methods: Comparison between extrusion-only, revolution/hollowing-only, and combined approaches on AMOS dataset.

Pre-training	MRNet	MedMNIST Organ	MedMNIST Nodule	MedMNIST Fracture	Avg
Ours (FDIF)	67.45	96.52	79.72	69.21	78.23
SwinUNETR (SSL) [ 25 ]	67.36	92.05	80.48	67.75	76.91
Scratch	62.19	85.32	85.12	56.22	72.21

Table 4: Transfer learning to 3D classification tasks. Balanced Accuracy (%) is reported for each dataset. Models are pre-trained with different strategies and fine-tuned on four classification benchmarks.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 1: SDF-based 3D shape construction methods. We generate 106 classes by applying extrusion, revolution, or hollowing to 2D base shapes. Each method is parameterised to produce multiple variants per base shape, yielding a rich and diverse shape library.

Figure 2: Displacement function families. We apply a variety of procedural displacement functions to the base shapes, generating diverse surface details that challenge the generalization capabilities of SDF regression models. Each family is parameterised to produce multiple variants, resulting in a rich dataset of 3D shapes with complex surface patterns.

Figure 3: Mapper function families. We apply a variety of mapper functions to convert signed distance values into intensity values, creating diverse surface patterns that challenge the regression capabilities of SDF models. Each family is parameterised to produce multiple variants, resulting in a rich dataset of 3D shapes with complex intensity distributions.

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Medical Image Segmentation Synthetic Data Pretraining 3D Medical Image Segmentation 3D ViTs

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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.

Abstract

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