README.md# Dual-Pull : NURBS-Based Fitting of Dual Point Clouds for Fast Surface Reconstruction Using Thick-slice Scans

A. Supplementary Material

Here, we provide supplementary material for the proposed NURBS fitting of dual point clouds and the pull-back operation to guide implicit neural representations of shapes. In Section A.1, we discuss the NURBS theory, the NURBS fitting process. In Section A.2, we discuss the dual point clouds pullback operation. In Section A.3, we provide more experimental results, including 3D reconstruction visualisation of different organs and different modal data. Finally, we provide a Ablation Study.

A.1. NURBS theory and the NURBS fitting process

A.1.1 NURBS theory

NURBS (Non-Uniform Rational B-Splines) is a powerful method for representing curves and surfaces, widely used in computer graphics, CAD (Computer-Aided Design), and medical imaging reconstruction. Its features include:

• Non-Uniform: Allows precise control over local shapes.
• Rational: Uses weights to control the shape of the curve, enabling precise fitting.
• B-Splines: Ensures smooth connections and avoids abrupt changes.

NURBS Curve Mathematical Expression

The mathematical expression of NURBS curves is as follows:

$$ C(u) = \frac{\sum_{i=0}^{n} N_{i,p}(u) w_i P_i}{\sum_{i=0}^{n} N_{i,p}(u) w_i} $$

Where:

• $C(u)$ is a point on the curve;
• $P_i$ are the control points;
• $N_{i,p}(u)$ are the rational B-spline basis functions;
• $p$ is the degree of the curve;
• $u$ is the parameter that controls the shape of the curve.

A.1.2 the NURBS fitting process

Algorithm 1 is the organ contour processing procedure, while Algorithm 2 is the dual point cloud fitting process.

Algorithm 1: NURBS Fitting for Initial Contours

Input:

• P = Extracted contour points from MRI slices

Output:

• NURBS-fitted contour curves

Step	Description
1	Sort contour points in a counterclockwise order using sort_contour_point(P).
2	Initialize NURBS curve N:
2.1	Set degree p = 3.
2.2	Define control points.
2.3	Append the first p + 1 control points to P' for closure.
2.4	Set the knot vector T as a uniform distribution.
3	Compute NURBS curve points C using NURBS interpolation.
4	Reconstruct 3D points:
4.1	For each point, assign z from the original contour layer.
4.2	Store the final 3D contour.
5	Return the fitted NURBS curve.

Algorithm 2: Dual Contour Construction

Input:

• NURBS-fitted contours from different MRI slices

Output:

• Dual contour curves for smooth 3D reconstruction

Step	Description
1	For each index i in ([1, m]) (each corresponding contour point along the layers):
1.1	Extract corresponding points from all slices.
2	Construct a new NURBS curve N_d:
2.1	Set degree p = 3.
2.2	Define control points.
2.3	Generate a non-uniform knot vector.
3	Compute the dual contour curve.
4	Store the result in the dual contour set.
5	Return the dual contour curves.

A.2.pullback operation

The figure 1 represents the pull-back process of the dual point cloud to the initial point cloud.

Figure 1: pullback operation

In the absence of a skeletonization algorithm, the extracted point cloud forms a ring-like structure, potentially compromising the accuracy of the subsequent 3D reconstruction, as illustrated in the figure below.

fig1：no Skeleton	fig2：with Skeleton

### A.3. Additional experiments and detail

A.3.1 Evaluation metrics

In real-world scenarios, obtaining ground-truth 3D models of organs is often impractical, making it difficult to directly compare reconstruction results with absolute ground truth. Following the evaluation protocol proposed in IGR , we uniformly sample 1 million points from the reconstructed mesh and compare them with the original scanned point cloud.

To measure the similarity between the reconstructed mesh and the initial (scanned) mesh, we adopt the following commonly used evaluation metrics:

• Chamfer Distance (CD)
• Hausdorff Distance (HD)
• Earth Mover’s Distance (EMD)
• Average Symmetric Surface Distance (ASSD)

These metrics together provide a comprehensive evaluation of geometric accuracy, point-wise correspondence, and boundary consistency of the reconstructed shapes.

A.3.2 Experimental setup and timing performance

The proposed Dual-Pull model is implemented based on the deep learning framework PyTorch~¹. All experiments were conducted on a single-GPU server running Ubuntu. The GPU used was NVIDIA GeForce RTX 4090 with 24GB of memory, and the CUDA version was 12.1. The Dual-IGR module was developed using PyTorch 2.4.1 and programmed in Python 3.10. A summary of the experimental environment is provided in Table 2.

Table 2. Experimental Environment Configuration

Environment	Version / Specification
Operating System	Ubuntu 18.04
CPU Model	Intel(R) Xeon(R) Gold 6133
GPU Model	NVIDIA GeForce RTX 4090
System Memory	32GB
GPU Memory	24GB
PyTorch Version	2.4.1
Python Version	3.10
CUDA Version	12.1

For all experiments, the Adam optimizer~² was used. The specific training hyperparameters are listed in Table 3.

Table 3. Training Hyperparameter Settings

Hyperparameter	Value
Batch Size	1
Optimizer	Adam
Initial Learning Rate	0.001
Learning Rate Milestone	100 epochs
Learning Rate Decay Factor	0.99

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A.3.3 Quantitative Evaluation on ACDC Dataset(MRI)

To further validate the effectiveness and accuracy of the proposed Dual-Pull framework in 3D reconstruction of cardiac anatomy, we conducted experiments on the public ACDC dataset, focusing on three key anatomical structures: left ventricle (LV), right ventricle (RV), and myocardium (Myo). We performed a comprehensive quantitative comparison with existing state-of-the-art methods, including IGR, DiGS, PINC, IPSR, and SPSR.

Table 1 presents the evaluation results in terms of four metrics: Chamfer Distance (CD), Hausdorff Distance (HD), Average Symmetric Surface Distance (ASSD), and Earth Mover’s Distance (EMD).

From the overall results, the Dual-Pull method consistently achieves excellent performance on CD and ASSD, particularly in the reconstruction of the RV and LV, where CD scores are 2.58e-03 and 1.86e-02, respectively. These values are significantly better than traditional methods, indicating superior reconstruction of overall geometry. Moreover, in terms of ASSD, Dual-Pull also achieves optimal or sub-optimal results for the RV and Myo, showcasing strong capabilities in boundary smoothness and spatial consistency.

Regarding the Hausdorff Distance, Dual-Pull outperforms all other methods on LV and RV with minimum HD values of 2.48e-01 and 1.47e-01, respectively, demonstrating robust handling of outlier point errors. Although IPSR performs slightly better on HD in the Myo reconstruction, Dual-Pull remains competitive and stable.

For the EMD metric, differences among all methods are minimal (within the 1e-4 range), suggesting convergence in terms of point cloud density distribution. Thus, EMD more so reflects the uniformity of sampling strategies than the reconstruction quality itself.

From the organ-wise analysis, Dual-Pull excels particularly in structures with regular morphology and smooth surfaces (e.g., LV and RV), which can be attributed to its NURBS-based contour-dual point cloud construction. This strategy enhances geometric continuity across slices and significantly improves anatomical shape recovery.

However, for more structurally complex regions such as the myocardium, while Dual-Pull remains competitive in CD and ASSD, its HD value suggests room for improvement. Given the intricate surface details and irregular boundaries of the myocardium, future work may explore more expressive generative models—such as Diffusion Models—as alternatives to NURBS fitting, aiming to further boost reconstruction precision for complex anatomical structures.

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Table 1. Quantitative Comparison on the ACDC Dataset (Lower is Better ↓)

Organ	Method	CD ↓	HD ↓	ASSD ↓	EMD ↓
LV	Dual-Pull (Ours)	1.86e-02	2.48e-01	8.62e-02	5.89e-04
	IGR	1.09e-02	5.75e-01	5.84e-02	5.89e-04
	DiGS	4.92e-03	5.65e-01	3.44e-02	5.89e-04
	PINC	2.24e-02	5.89e-01	9.04e-02	5.89e-04
	IPSR	1.23e-02	5.49e-01	4.62e-02	5.89e-04
	SPSR	5.52e-02	5.49e-01	1.40e-01	5.89e-04

Organ	Method	CD ↓	HD ↓	ASSD ↓	EMD ↓
RV	Dual-Pull (Ours)	2.58e-03	1.47e-01	2.80e-02	5.48e-04
	IGR	6.19e-03	4.02e-01	4.03e-02	5.48e-04
	DiGS	2.92e-03	2.74e-01	2.77e-02	5.48e-04
	PINC	6.26e-02	5.22e-01	1.55e-01	5.48e-04
	IPSR	8.20e-03	3.74e-01	3.83e-02	5.48e-04
	SPSR	1.97e-02	3.74e-01	7.56e-02	5.48e-04

Organ	Method	CD ↓	HD ↓	ASSD ↓	EMD ↓
Myo	Dual-Pull (Ours)	1.55e-02	4.95e-01	7.69e-02	2.89e-04
	IGR	3.42e-02	2.70e-01	1.17e-01	2.89e-04
	DiGS	3.31e-03	4.77e-01	3.12e-02	2.89e-04
	PINC	3.71e-02	5.75e-01	1.11e-01	2.89e-04
	IPSR	4.25e-03	2.10e-01	3.53e-02	2.89e-04
	SPSR	4.32e-02	2.10e-01	1.27e-01	2.89e-04

Figure 2: ACDC Dataset Surface Reconstruction

Figure 3: 3D-Slicer compare with ours on ACDC Dataset

A.3.3 Surface reconstruction on CHAOS(CT)

The proposed Dual-Pull 3D reconstruction framework was originally designed for thick-slice MRI data. However, thanks to its generalized implicit field representation and robust geometric constraint mechanisms, the method also demonstrates strong cross-modality generalization capability when applied to CT slice data. Experimental results show that even when the input modality shifts from MRI to CT, the Dual-Pull model consistently achieves accurate reconstruction of organ surfaces, effectively addressing challenges such as sparsity and topological defects caused by large inter-slice spacing. These findings further validate the versatility and generalizability of the proposed method in the field of medical imaging.

Fig1 and fig2 shows the 3D reconstruction results of the liver from CT images in the CHAOS dataset. The proposed Dual-Pull 3D reconstruction framework was originally designed for thick-slice MRI data. However, due to its generalized implicit field representation and robust geometric constraint mechanisms, the method also demonstrates strong cross-modality generalization capability when applied to CT slice data. Experimental results indicate that even when the input modality shifts from MRI to CT, the Dual-Pull model can still achieve stable and accurate reconstruction of organ surfaces. It effectively overcomes challenges such as sparsity and topological defects caused by large inter-slice spacing, reasonably extrapolates the missing parts of the point cloud, and completes the reconstruction of the missing regions. These results confirm the versatility and generalizability of the proposed method in the field of medical imaging.

To further validate the effectiveness of Dual-Pull on CT datasets, we also conducted experiments on the MM-WHS dataset. As shown in Fig3 and fig4, our method successfully removes erroneous outlier noise and achieves high-quality reconstruction results.

fig1：liver point clouds	fig2：ours

fig3：Point Cloud of the Ascending Aorta	fig4：3D Reconstruction Result of the Ascending Aorta

## 🔍 Ablation Study

To validate the contribution of each proposed component in improving 3D reconstruction performance, we conduct a detailed ablation study on the right ventricle masks of the ACDC dataset. The study evaluates the effectiveness of each core module in our proposed Dual-Pull framework, including:

• Dual-contour point cloud generation
• Pull-back loss
• Hessian smoothness regularization
• Minimal surface (area) loss

The baseline is set to IGR, and the modules are incrementally added to analyze performance improvements. Results are summarized in the table below.

Key Observations:

1. 📌 Dual-contour point cloud module (Base model) shows significant gains:
   Compared to the IGR baseline, adding the dual-contour point cloud module (Base) reduces Chamfer Distance (CD), Hausdorff Distance (HD), and Average Symmetric Surface Distance (ASSD) by 42.49%, 35.95%, and 23.26% respectively. This shows the module effectively reduces reconstruction errors caused by sparse slice intervals and enhances continuity across anatomical layers.

2. 🔧 Individual loss functions contribute differently:
   Adding Pull-back, Hessian, and Area losses on top of the base model each improve different aspects:

   • Pull-back loss leads to the most substantial improvements, with HD reduced by 51.94%.
   • Hessian regularization improves local surface smoothness, reflected in ASSD (↑23.56%).
   • Minimal area loss improves global consistency, with HD improved by 41.51%.

3. 🔗 Combined loss functions exhibit synergy:
   When two losses are combined:

   • Area + Hessian performs best overall, achieving the highest improvement across all three metrics (CD: ↑59.54%, HD: ↑64.66%, ASSD: ↑30.27%).
   • Pull + Area ranks second with comparable gains, indicating synergistic effects across loss types.

📊 Quantitative Results

Method	Chamfer Distance ↓	Hausdorff Distance ↓	ASSD ↓
IGR (Baseline)	0.006187 (0.00%)	0.401772 (0.00%)	0.040283 (0.00%)
+ Dual cloud points (Base)	0.003558 (↑42.49%)	0.257340 (↑35.95%)	0.030915 (↑23.26%)
+ Base + Pull-back	0.002773 (↑55.18%)	0.193097 (↑51.94%)	0.028979 (↑28.06%)
+ Base + Hessian	0.003198 (↑48.31%)	0.257882 (↑35.81%)	0.030793 (↑23.56%)
+ Base + Area	0.003516 (↑43.17%)	0.235006 (↑41.51%)	0.031351 (↑22.17%)
+ Base + Area + Hessian	0.002503 (↑59.54%)	0.142005 (↑64.66%)	0.028089 (↑30.27%)
+ Base + Pull-back + Hessian	0.003317 (↑46.39%)	0.272452 (↑32.19%)	0.029886 (↑25.81%)
+ Base + Pull-back + Area	0.002578 (↑58.33%)	0.146873 (↑63.44%)	0.027947 (↑30.62%)

📌 All metrics are "the lower the better". Improvement percentages are relative to IGR.

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🖼 Visual Comparison

The figure below shows visual reconstruction comparisons of the right ventricle on the ACDC dataset. It is evident that the minimal surface loss provides more natural extrapolation in regions with missing slices.

Figure: Visual comparison of right ventricle 3D reconstruction on ACDC dataset
Left: Ours without Area loss Right: Ours with Area loss

Footnotes

1. Imambi, S. M. et al. PyTorch: An imperative style, high-performance deep learning library. arXiv preprint arXiv:1912.01703 (2021). ↩

2. Kinga, D., & Ba, J. (2015). Adam: A method for stochastic optimization. International Conference on Learning Representations (ICLR). ↩