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

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 (5)

• 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.
• Figure 3: Additive displacement on an SDF. Left: base sphere $\phi(\mathbf{x})$. Different displacement functions $\Delta(\mathbf{x})$ generate diverse textures while preserving a closed surface.
• Figure 4: Distance-to-intensity mapping on an SDF. Left: SDF $\phi(\mathbf{x})$. Mapper functions $g$ convert signed distance values into diverse intensity textures.
• Figure 5: Qualitative comparison of segmentation results. Models pre-trained with FDIF demonstrate improved segmentation accuracy compared to baseline methods.