CrossSDF: 3D Reconstruction of Thin Structures From Cross-Sections¶

Conference: CVPR 2025
arXiv: 2412.04120
Code: Project Page
Institution: University of Edinburgh / University of British Columbia
Area: Medical Imaging / 3D Reconstruction
Keywords: SDF, cross-section reconstruction, thin structures, neural implicit fields, hash encoding, medical imaging

TL;DR¶

CrossSDF is proposed to reconstruct a 3D SDF from 2D cross-sectional Signed Distance Fields. By combining hybrid encoding (hash grid + random Fourier features) and symmetric difference loss, it achieves accurate reconstruction of thin tubular structures (such as blood vessels) for the first time.

Background & Motivation¶

Background¶

Background: Reconstructing 3D structures from planar cross-sectional data is a classic problem in medical imaging, manufacturing, and topography. CT/MRI scans generate sparse 2D cross-sectional data, from which complete 3D shapes need to be reconstructed.

Limitations of Prior Work:

Limitations of Prior Work¶

Limitations of Prior Work: Point cloud reconstruction methods tend to fail due to sparse data between cross-sections.

Key Challenge¶

Key Challenge: Traditional interpolation methods (such as OReX) suffer from severe "laddering artifacts," leading to staircase-like structures between parallel cross-sections.

Mechanism¶

Mechanism: Indicator function methods suffer from normal and smoothness artifacts due to discontinuities, and degrade into an unstable binary classification problem.

Supplementary Notes¶

Supplementary Notes: All existing methods perform poorly when reconstructing thin structures (such as blood vessels and nerve tissue), suffering from either over-smoothing or fragmentation.

Key Challenge: In general, 2D SDF and the desired 3D SDF are inconsistent—directly fitting 2D SDF forces 3D surface normals to be parallel to the cross-sections, resulting in "laddering effects".

Key Insight: Leveraging only the zero-set information of the 2D SDF (inside/outside classification) guides the 3D SDF learning through a symmetric difference loss, freeing the model from the constraint of "must be a 2D SDF".

Core Idea: Symmetric difference loss (releasing in-plane constraints) + hybrid encoding (hash + RFF, eliminating grid artifacts) + adaptive sampling = accurate reconstruction of thin structures.

Method¶

Inputs & Outputs¶

Input: $n$ arbitrary-oriented 2D planes, with closed contours (intersection lines between cross-sections and target geometry) on each plane.
Output: Neural 3D SDF $f(\mathbf{x}; \boldsymbol{\theta})$, where the zero-set defines the reconstructed geometry.

Key Designs¶

Adaptive Sampling
- Function: Ensures that the interior of thin structures is sufficiently sampled.
- Problem: Uniform sampling under-samples structures with small cross-sectional areas; OReX's fixed-radius sampling is also unsuitable for thin structures.
- Solution: Samples directly inside each contour up to a threshold count, leveraging the inside/outside labels of 2D cross-sections for precise partitioning.
- Effect: Under-represented contours with small cross-sectional areas are not ignored.
Hybrid Encoding
- Function: Combines hash grid encoding $\gamma_{\text{hash}}$ and Random Fourier Feature (RFF) encoding $\gamma_{\text{RF}}$.
- Hash Grid: Captures fine details but introduces interpolation artifacts at grid boundaries.
- Random Fourier Features: Encourages smooth representation and reduces grid boundary artifacts.
- Fusion Method: $\mathbf{z}_{\text{comb}} = \mathbf{z}_{\text{hash}} + \alpha \cdot \mathbf{z}_{\text{RF}}$, where $\alpha$ is a balancing factor.
- Final Input: $\mathbf{z}_{\text{final}} = [\mathbf{z}_{\text{comb}} | \mathbf{x}]$, concatenating the original 3D coordinates.
Symmetric Difference Loss
- Function: Drives optimization only in regions where the predicted and target inside/outside classifications are inconsistent.
- Core Insight: The unique information regarding 3D geometry contained in the 2D SDF on a cross-section is the inside/outside classification.
- Two Sets: $\Omega_{\text{on}} = \{\mathbf{x} | f_{2D}(\mathbf{x}) = 0\}$ (zero-set sampling points) and the symmetric difference region.
- Advantage: The loss is zero when the predicted and target contours completely coincide, freeing the network to autonomously learn the 3D SDF.
- Effect: Eliminates laddering artifacts caused by directly fitting 2D SDFs.
Eikonal Regularization
- Constraints $|\nabla f| = 1$ to guarantee the SDF property.
- Particularly effective for sparse inputs.

Overall Architecture¶

Two channels: a hash grid channel and an RFF channel, each equipped with a single-layer MLP.
After fusion, the features are fed into an SDF MLP to output the final distance value.

Key Experimental Results¶

Quantitative Comparison with Existing Methods (Synthetic Data)¶

Method	Applicable Scenarios	Thin Structure Preservation	Topological Continuity	Surface Smoothness
OReX	Arbitrary Cross-Sections	Poor	Poor	Poor (Laddering Artifacts)
Points2Surf	Point Clouds	Moderate	Moderate	Moderate
POCO	Point Clouds	Moderate	Moderate	Good
DeepSDF	Latent Space	Poor	Good	Good (Over-smoothed)
CrossSDF	Arbitrary Cross-Sections	Good	Good	Good

Ablation Study¶

Configuration	Thin Structure Quality	Surface Smoothness	Laddering Artifacts
Hash encoding only	Good	Poor (Grid artifacts)	Few
RFF encoding only	Poor (Over-smoothed)	Good	Few
Directly fitting 2D SDF	Good	Poor	Severe
Without adaptive sampling	Poor (Missing small structures)	—	—
Full Method	Good	Good	None

Key Findings¶

The symmetric difference loss is critical for eliminating laddering artifacts.
Hybrid encoding achieves a better balance between detail preservation and smoothness than a single encoding.
Adaptive sampling is essential for structures with small cross-sectional areas.

Additional Technical Details¶

Network Parameters¶

Hash grid resolution: Multi-scale, ranging from $16^3$ to $2048^3$.
RFF encoding dimension $d$: Set to match the output dimension of the hash encoding.
Scaling factor $\alpha$: Configured to make the amplitudes of both encodings approximately equal during initialization.
SDF MLP: Standard 8-layer fully-connected network with skip connections.

Eikonal Regularization¶

$$\mathcal{L}_{eik} = \mathbb{E}_{\mathbf{x}}[(|\nabla f(\mathbf{x})| - 1)^2]$$ Applied on randomly sampled points in the 3D space between cross-sections to provide a smoothness prior.

Highlights & Insights¶

The design of the symmetric difference loss is highly elegant: it relaxes the geometric constraint from "learning 2D SDF" to "only learning matching inside/outside classifications." This retains adequate supervision signals while liberating the degrees of freedom.
The hybrid encoding solution is concise: one captures detail while the other ensures smoothness, offering strong complementarity.
Thin vessel reconstruction in medical imaging represents a critical real-world demand, giving high practical value to this method.
Supports non-parallel cross-sections of arbitrary orientation, demonstrating greater versatility compared to methods restricted to parallel plans.
The core of the method lies in the design of the loss function; the network architecture requires no special modifications, allowing easy transfer to other implicit field representations.