MedSDF: SDF Representations in Medical Imaging

Updated 5 July 2026

MedSDF is a term describing continuous SDF-based representations that model protein surfaces, anatomical shapes, and segmentation masks.
In protein surface modeling, it employs smooth-min unions of atomic SDFs to approximate solvent-accessible boundaries while supporting backpropagation.
Its application in medical shape datasets and conditional segmentation models enables detailed anatomical reconstruction, uncertainty estimation, and efficient generative tasks.

Searching arXiv for the cited MedSDF-related papers to ground the article and verify identifiers. MedSDF is an overloaded term in recent arXiv literature rather than the name of a single unified framework. It has been used to denote a differentiable implicit representation of protein surfaces and interfaces built from unions of atom-centered signed distance functions (SDFs) (Scott et al., 31 Jul 2025), a large-scale multi-category medical shape dataset containing paired surface point clouds, SDF volumes, and meshes for anatomical generation (Zhang et al., 8 Mar 2026), and a conditional score-based generative segmentation model that predicts SDFs instead of binary masks (Bogensperger et al., 2023). Across these usages, the common technical substrate is the signed distance function: a continuous field whose zero level set defines a boundary and whose values can support implicit geometry, generative modeling, uncertainty estimation, and differentiable optimization.

1. Terminological scope

A recurrent source of confusion is terminological. In the cited literature, “MedSDF” refers to three distinct entities: a protein-surface representation, a medical shape dataset, and a segmentation model. The shared acronym reflects a common SDF-centric design pattern, not a single codebase or canonical benchmark (Scott et al., 31 Jul 2025, Zhang et al., 8 Mar 2026, Bogensperger et al., 2023).

Usage of “MedSDF”	Paper	Core object
Differentiable protein-surface representation	“Implicitly and Differentiably Representing Protein Surfaces and Interfaces” (Scott et al., 31 Jul 2025)	Smooth union of atom-centered sphere SDFs
Large-scale medical shape dataset	“High-Fidelity Medical Shape Generation via Skeletal Latent Diffusion” (Zhang et al., 8 Mar 2026)	Paired surface point clouds, SDF volumes, and meshes
Conditional score-based generative segmentation model	“Score-Based Generative Models for Medical Image Segmentation using Signed Distance Functions” (Bogensperger et al., 2023)	Conditional distribution over SDF masks

In all three cases, the motivating claim is that SDFs are a more geometrically faithful representation than explicit meshes or binary occupancy alone. What differs is the target domain: biomolecular geometry, anatomical shape generation, or medical image segmentation.

2. MedSDF as an implicit representation of protein surfaces and interfaces

In “Implicitly and Differentiably Representing Protein Surfaces and Interfaces,” MedSDF is a method for representing a protein surface, or the interface between two protein chains, as an implicit geometric object defined by signed distance functions rather than as an explicit mesh (Scott et al., 31 Jul 2025). Each atom is treated as a sphere with its van der Waals radius. For an atom centered at $\Vec{y}$ with radius $r$ , the analytic SDF is

$d(\Vec{x}) = ||\Vec{x} - \Vec{y}|| - r.$

The protein-level field is formed by combining atomic SDFs with the smooth-min operation

$\mathbf{d}(\Vec{x}) = -k \log \left( \sum_{i=1}^n e^{-\frac{1}{k} d_{i}(\Vec{x})} \right),$

where $k$ determines the smoothing radius. The resulting zero level set, $d(\mathbf{x}) = 0$ , approximates the protein’s solvent-accessible surface.

This construction is explicitly linked to constructive solid geometry and prior SDF blending work. The practical significance is that the representation remains implicit: it can be queried anywhere in space, and, because the implementation is built in PyTorch, those queries can participate in backpropagation. The paper emphasizes several ML-relevant properties of SDFs: unit gradient where defined, closed-form expressions for many primitives, and composability through simple boolean operations. In that sense, MedSDF is positioned as a bridge between classical molecular surface modeling and recent deep-learning work on neural and analytic SDFs.

For interfaces between two protein chains, the same SDF machinery is extended with boolean operators. The paper gives

$d_{d_1 \cap d_2}(\Vec{x}) = \max \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$

$d_{d_1 \cup d_2}(\Vec{x}) = \min \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$

and a rounding operation

$d^{(r)}(\Vec{x}) = d(\Vec{x}) - r$

to expand the zero level set outward by distance $r$ . The interface SDF is then defined so as to represent the parts of chain 1 within $r$ 0 of chain 2 and vice versa. In the experiments, the interaction radius is $r$ 1 Å, motivated by previous protein-protein interaction work.

The implementation is intentionally practical. It is built on top of the torch_sdf package, and Marching Cubes via SciKit-Image is used to convert implicit fields into meshes for visualization. The authors evaluate SDF queries over a $r$ 2 grid spanning the complex bounding box and compare bounding volume hierarchies (BVHs) with k-d trees for acceleration. Both reduce memory usage substantially relative to naively querying all atoms; the BVH is slightly faster, whereas the k-d tree suffers from CPU-GPU latency because there was, at the time of writing, no functional Torch+CUDA kd-tree implementation. The paper also notes a specific artifact: the BVH approach introduces rectilinear artifacts on the exterior because its bounding volumes are axis-aligned boxes.

Biologically, the framework targets solvent-accessible surfaces, solvent-excluded surfaces, and protein-protein interfaces. The demonstrations use bound conformations of protein complexes from Docking Benchmark 5 (DB5), computing $r$ 3 isosurfaces for whole proteins and producing interface meshes for all complexes in DB5. The paper suggests potential downstream use in protein-protein interaction prediction, protein surface analysis, ligand-binding site detection, optimization of amino-acid positions to fit a structural motif, and machine learning models that require shape-aware training signals.

A crucial limitation is stated explicitly. The smooth-min union of atomic spheres is not guaranteed to be a mathematically exact signed distance function everywhere. The zero isosurface is correct, but the interior distance values can be “broken” or only approximate; inside the protein, the field becomes a lower bound rather than the true distance to the surface. The paper therefore presents the method as a proof of concept and states that additional validation is needed to establish practical utility.

3. MedSDF as a multi-category medical shape dataset

In “High-Fidelity Medical Shape Generation via Skeletal Latent Diffusion,” MedSDF denotes a “large-scale multi-category medical shape dataset” introduced to train a skeletal latent diffusion framework and to provide a benchmark for high-fidelity anatomical shape reconstruction and generation (Zhang et al., 8 Mar 2026). Each sample includes a surface point cloud, an SDF volume of resolution $r$ 4, and a reconstructed surface mesh. The dataset contains 12,472 samples spanning 14 anatomical categories. It is constructed by processing over 10,000 samples from 12 anatomical categories in MedShapeNet and computing the corresponding SDF volumes, then augmenting the collection with vessel data from ImageCAS to increase morphological diversity.

The dataset is designed to support a two-stage pipeline. First, a shape auto-encoder maps an input surface point set and an online-computed skeleton into a latent representation; second, a diffusion model is trained over those latents. This requirement explains why the dataset pairs surface geometry with dense SDF supervision. The paper states that this paired supervision is necessary for point-cloud encoding and SDF decoding, and that the inclusion of vessel data is especially important because it extends the benchmark beyond organ-centric shapes to thin, elongated, topologically challenging geometries.

The auto-encoder is structured around differentiable skeletonization. Given a point cloud $r$ 5, the skeletonization module extracts skeleton points $r$ 6, where the four channels represent 3D position plus radius. The encoder forms a latent set

$r$ 7

and the decoder predicts signed distances for query points $r$ 8 through

$r$ 9

The skeleton itself is computed online by FPS initialization and iterative centroid refinement with KNN and DBSCAN clustering:

$d(\Vec{x}) = ||\Vec{x} - \Vec{y}|| - r.$0

with updates

$d(\Vec{x}) = ||\Vec{x} - \Vec{y}|| - r.$1

The paper uses $d(\Vec{x}) = ||\Vec{x} - \Vec{y}|| - r.$2 iterations and $d(\Vec{x}) = ||\Vec{x} - \Vec{y}|| - r.$3.

The data protocol is closely tied to training and inference. During training, each point cloud is subsampled to 2,560 points using farthest point sampling, and 2,560 voxel coordinates with ground-truth SDF values are used. Of these coordinates, 90% are sampled from the truncated SDF region with truncation value $d(\Vec{x}) = ||\Vec{x} - \Vec{y}|| - r.$4, and 10% are sampled from outside the truncation region to provide coarse supervision for distant space and improve scale generalization. At inference, the method selects the top 30% of voxels closest to the predicted skeleton and assigns the rest a constant value of $d(\Vec{x}) = ||\Vec{x} - \Vec{y}|| - r.$5.

The dataset supports conditional latent diffusion across 14 categories. The paper uses a probability flow ODE,

$d(\Vec{x}) = ||\Vec{x} - \Vec{y}|| - r.$6

and classifier-free guidance,

$d(\Vec{x}) = ||\Vec{x} - \Vec{y}|| - r.$7

The categories explicitly shown in qualitative figures include brain, colon, left coronary artery (LCA), duodenum, liver, and stomach, while experiments are reported across all 14 categories.

Quantitatively, the proposed method achieves the best average reconstruction scores over the 14 categories, with Chamfer Distance $d(\Vec{x}) = ||\Vec{x} - \Vec{y}|| - r.$8, EMD $d(\Vec{x}) = ||\Vec{x} - \Vec{y}|| - r.$9, Hausdorff Distance $\mathbf{d}(\Vec{x}) = -k \log \left( \sum_{i=1}^n e^{-\frac{1}{k} d_{i}(\Vec{x})} \right),$0, and F1 $\mathbf{d}(\Vec{x}) = -k \log \left( \sum_{i=1}^n e^{-\frac{1}{k} d_{i}(\Vec{x})} \right),$1, outperforming GeM3D at $\mathbf{d}(\Vec{x}) = -k \log \left( \sum_{i=1}^n e^{-\frac{1}{k} d_{i}(\Vec{x})} \right),$2, $\mathbf{d}(\Vec{x}) = -k \log \left( \sum_{i=1}^n e^{-\frac{1}{k} d_{i}(\Vec{x})} \right),$3, $\mathbf{d}(\Vec{x}) = -k \log \left( \sum_{i=1}^n e^{-\frac{1}{k} d_{i}(\Vec{x})} \right),$4, and $\mathbf{d}(\Vec{x}) = -k \log \left( \sum_{i=1}^n e^{-\frac{1}{k} d_{i}(\Vec{x})} \right),$5. On generation, it achieves the best FID and KID, with FID $\mathbf{d}(\Vec{x}) = -k \log \left( \sum_{i=1}^n e^{-\frac{1}{k} d_{i}(\Vec{x})} \right),$6 and KID $\mathbf{d}(\Vec{x}) = -k \log \left( \sum_{i=1}^n e^{-\frac{1}{k} d_{i}(\Vec{x})} \right),$7, while maintaining strong diversity metrics and competitive sampling speed. Ablation studies report that removing differentiable skeletonization, skeleton constraints, or latent attention consistently worsens reconstruction and generation. The paper therefore treats MedSDF not simply as a dataset release but as the backbone dataset for a compact skeletal latent representation.

4. MedSDF as a conditional score-based generative segmentation model

In “Score-Based Generative Models for Medical Image Segmentation using Signed Distance Functions,” MedSDF is a conditional score-based generative segmentation model that learns to generate signed distance functions of medical segmentation masks rather than directly generating binary masks (Bogensperger et al., 2023). The motivation is that binary masks are too discrete and brittle for diffusion-style corruption: Gaussian noise on a binary mask creates unnatural intermediate states, sharp transitions, and “hole-like” artifacts. By contrast, an SDF is continuous, smooth around boundaries, and more naturally aligned with object geometry.

The paper defines a truncated, normalized SDF over a mask domain $\mathbf{d}(\Vec{x}) = -k \log \left( \sum_{i=1}^n e^{-\frac{1}{k} d_{i}(\Vec{x})} \right),$8, with object region $\mathbf{d}(\Vec{x}) = -k \log \left( \sum_{i=1}^n e^{-\frac{1}{k} d_{i}(\Vec{x})} \right),$9 and boundary $k$ 0. The construction uses Euclidean distance to the nearest boundary point, truncation threshold $k$ 1, and a factor $k$ 2 for inside pixels. A sampled SDF map $k$ 3 is converted back to a binary mask by thresholding at $k$ 4, although the paper also notes a practical threshold of $k$ 5 because residual noise remains at the smallest diffusion scale.

The training set is

$k$ 6

where $k$ 7 is the input medical image and $k$ 8 is the corresponding SDF map. The goal is to learn the conditional distribution $k$ 9. The method adopts a variance-exploding SDE, learns the conditional score $d(\mathbf{x}) = 0$ 0 by denoising score matching, schedules noise as

$d(\mathbf{x}) = 0$ 1

and generates samples with a predictor-corrector sampler. The predictor follows the reverse-time dynamics, while the corrector uses Langevin MCMC refinement. This yields an implicit distribution over masks rather than a single deterministic segmentation.

A central consequence is uncertainty estimation. Multiple samples from $d(\mathbf{x}) = 0$ 2 can be thresholded and aggregated into mean predictions and variance maps. The paper states that high uncertainty tends to concentrate near boundaries, ambiguous structures, and wrongly detected or over-segmented regions. This is presented as useful for downstream analysis, quality control, and robustness assessment.

The experiments are on MoNuSeg and GlaS. For MoNuSeg, the data comprise 30 training images and 14 test images of size $d(\mathbf{x}) = 0$ 3, with over 21,000 annotated nuclei. For GlaS, the data comprise 85 training and 80 test H&E images of colorectal cancer tissue glands. Preprocessing includes structure-preserving color normalization, grayscale conversion for MoNuSeg, resizing, overlapping $d(\mathbf{x}) = 0$ 4 crops for MoNuSeg, and random horizontal and vertical flips. Training uses Adam with learning rate $d(\mathbf{x}) = 0$ 5, $d(\mathbf{x}) = 0$ 6, and $d(\mathbf{x}) = 0$ 7.

Quantitatively, the method is best on GlaS and competitive on MoNuSeg. On MoNuSeg, it reports F1 $d(\mathbf{x}) = 0$ 8 and mIoU $d(\mathbf{x}) = 0$ 9, with MMSE averaging over 128 sampling runs improving this to F1 $d_{d_1 \cap d_2}(\Vec{x}) = \max \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$0 and mIoU $d_{d_1 \cap d_2}(\Vec{x}) = \max \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$1. On GlaS, it reports F1 $d_{d_1 \cap d_2}(\Vec{x}) = \max \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$2 and mIoU $d_{d_1 \cap d_2}(\Vec{x}) = \max \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$3, with MMSE averaging improving this to F1 $d_{d_1 \cap d_2}(\Vec{x}) = \max \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$4 and mIoU $d_{d_1 \cap d_2}(\Vec{x}) = \max \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$5. An ablation that removes the SDF representation and trains the same score model directly on binary masks lowers performance to mean IoU / mean F1 of $d_{d_1 \cap d_2}(\Vec{x}) = \max \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$6 on MoNuSeg and $d_{d_1 \cap d_2}(\Vec{x}) = \max \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$7 on GlaS. The paper therefore attributes a substantial part of its performance to the SDF representation itself.

Several adjacent works clarify what is specific to each MedSDF usage and what belongs to a broader SDF-based methodological family. “VesselSDF: Distance Field Priors for Vascular Network Reconstruction” is explicitly described as a medical signed distance field approach tailored to sparse CT reconstruction of thin, branching vessels (Esposito et al., 19 Jun 2025). It replaces binary voxel classification with continuous SDF regression, using a two-stage 3D U-Net-based pipeline: an occupancy predictor followed by an SDF refinement network with detached gradients. The paper adds an anisotropy-aware eikonal regularizer with $d_{d_1 \cap d_2}(\Vec{x}) = \max \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$8, an adaptive Gaussian regularizer to suppress floating artifacts, and a surface regularization term. On the Hepatic Vessels dataset, VesselSDF is best on all reported metrics, with Dice $d_{d_1 \cap d_2}(\Vec{x}) = \max \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$9, IoU $d_{d_1 \cup d_2}(\Vec{x}) = \min \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$0, JD $d_{d_1 \cup d_2}(\Vec{x}) = \min \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$1, CD $d_{d_1 \cup d_2}(\Vec{x}) = \min \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$2, and HD $d_{d_1 \cup d_2}(\Vec{x}) = \min \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$3; on IRCADb, Dice and IoU match nnU-Net at $d_{d_1 \cup d_2}(\Vec{x}) = \min \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$4 and $d_{d_1 \cup d_2}(\Vec{x}) = \min \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$5, while CD and HD improve to $d_{d_1 \cup d_2}(\Vec{x}) = \min \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$6 and $d_{d_1 \cup d_2}(\Vec{x}) = \min \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$7.

“FlowSDF: Flow Matching for Medical Image Segmentation Using Distance Transforms” replaces score-based diffusion with image-guided conditional flow matching in SDF space (Bogensperger et al., 2024). It represents segmentation masks as truncated distance transforms, learns a conditional vector field over a prescribed interpolation path

$d_{d_1 \cup d_2}(\Vec{x}) = \min \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$8

with $d_{d_1 \cup d_2}(\Vec{x}) = \min \left( d_1(\Vec{x}), d_2(\Vec{x}) \right),$9, and integrates the learned ODE with torchdiffeq. The network uses additive fusion of image and noisy-mask features, NFE is set to 4, MMSE evaluation typically uses $d^{(r)}(\Vec{x}) = d(\Vec{x}) - r$0 samples, and optional noise injection uses $d^{(r)}(\Vec{x}) = d(\Vec{x}) - r$1. On MoNuSeg, FlowSDF reaches F1 $d^{(r)}(\Vec{x}) = d(\Vec{x}) - r$2 and mIoU $d^{(r)}(\Vec{x}) = d(\Vec{x}) - r$3 for $d^{(r)}(\Vec{x}) = d(\Vec{x}) - r$4, and F1 $d^{(r)}(\Vec{x}) = d(\Vec{x}) - r$5 and mIoU $d^{(r)}(\Vec{x}) = d(\Vec{x}) - r$6 for MMSE with $d^{(r)}(\Vec{x}) = d(\Vec{x}) - r$7. On GlaS, it reaches F1 $d^{(r)}(\Vec{x}) = d(\Vec{x}) - r$8 and mIoU $d^{(r)}(\Vec{x}) = d(\Vec{x}) - r$9 for $r$ 0, and F1 $r$ 1 and mIoU $r$ 2 for MMSE with $r$ 3. The paper positions this as the first application of conditional flow matching to image segmentation.

“Quasi-Medial Distance Field (Q-MDF): A Robust Method for Approximating and Discretizing Neural Medial Axis” is not named MedSDF, but it is described as being in the same conceptual family as MedSDF-style neural medial surface and distance field methods (Kong et al., 2024). Its core quantity is the quasi-medial distance field

$r$ 4

which is treated as a surrogate unsigned distance field to the medial axis. The medial axis is extracted as the zero level set of this field, then converted into a compact medial membrane via a modified double covering method. The paper reports that, with marching cubes depth 8, $r$ 5 works well in most cases, and evaluates on ABC, Thingi10K, and SRB using Chamfer distance, Hausdorff distance, Euler characteristic, and a curvature-based boundary smoothness metric. It emphasizes robustness to point clouds with missing regions, non-uniform sampling, noise, thin structures, open boundaries, triangle soups, and CAD models with sharp features.

Taken together, these works show that the technical content associated with “MedSDF” extends far beyond one segmentation paper. This suggests a broader SDF-centered research idiom spanning segmentation, anatomical generation, vascular reconstruction, medial-axis extraction, and protein geometry.

6. Limitations, ambiguities, and research directions

The primary limitation of the term is semantic rather than mathematical: “MedSDF” does not identify a single model family, evaluation protocol, or application domain. A reader encountering the term must first determine whether it refers to a protein-surface representation, a medical shape dataset, or a score-based segmentation model. A plausible implication is that direct cross-paper comparisons under the name alone are potentially misleading.

Method-specific limitations are also explicit. The protein-surface formulation states that its smooth-min union is not guaranteed to be a mathematically exact signed distance function everywhere; interior values can be “broken,” and the current work is framed as a proof of concept that still requires further validation for practical biological utility (Scott et al., 31 Jul 2025). The 2023 segmentation model reports competitive but not uniformly state-of-the-art performance, focuses on binary segmentation rather than multi-class settings, and requires repeated sampling for uncertainty estimation, which is computationally expensive; it also does not provide a full theoretical or empirical analysis of uncertainty-error calibration (Bogensperger et al., 2023). The dataset-centered usage of MedSDF is tightly coupled to a skeletal latent representation, differentiable skeletonization, and skeleton-guided coordinate sampling, so its reported gains should be read in the context of that full pipeline rather than as a dataset-only effect (Zhang et al., 8 Mar 2026).

The cited papers also outline several forward directions. The protein paper proposes optimization of protein shape, conformation, and position using SDF-based losses; combining explicit atomic SDF construction with neural implicit models such as DeepSDF, possibly conditioned on pretrained protein transformer embeddings; incorporating atomic features into generated surfaces; and using protein-specific acceleration structures such as ChainTree (Scott et al., 31 Jul 2025). FlowSDF proposes extending SDF-based conditional generation beyond binary segmentation and beyond the two histopathology benchmarks studied, and suggests integration with latent diffusion models or foundation models (Bogensperger et al., 2024). The anatomical-generation paper presents MedSDF as a sufficiently large benchmark to support latent diffusion across diverse anatomies, including vessel structures, which in turn supports more general shape priors and multi-category evaluation (Zhang et al., 8 Mar 2026).

In the narrow sense, MedSDF names three separate contributions. In the broader technical sense, it marks a convergence around SDFs as a representation in which geometry remains continuous, boundaries are implicit zero sets, and learning can exploit shape-aware supervision rather than discrete occupancy alone.