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Noise2Score3D: Tweedie's Approach for Unsupervised Point Cloud Denoising

Conference: ICCV 2025 arXiv: 2503.09283
Code: GitHub
Area: 3D Vision Keywords: Point cloud denoising, unsupervised learning, Tweedie's formula, score function, total variation

TL;DR

This paper proposes Noise2Score3D, a fully unsupervised point cloud denoising framework based on Tweedie's formula. It learns the score function directly from noisy data and achieves single-step denoising, while introducing point cloud total variation to estimate unknown noise parameters.

Background & Motivation

3D point clouds are frequently corrupted by noise due to sensor errors and environmental factors. Existing deep learning-based denoising methods face two fundamental challenges:

Supervised methods depend on paired data — obtaining clean data in real-world scenarios is difficult or impossible.

Existing unsupervised methods suffer from significant limitations: - TotalDn relies on spatial priors and involves slow iterative processes. - The "score" in ScoreDenoise is based on displacement rather than likelihood, precluding the use of Bayesian statistical tools. - Poor generalization (retraining required across datasets and noise levels). - Low inference efficiency (iterative denoising required).

Noise2Score has demonstrated the power of Tweedie's formula in 2D image denoising: by reformulating denoising as a score function estimation problem, it offers a theoretically elegant and flexible framework. This paper extends this idea to 3D point clouds.

Method

Theoretical Foundation — Tweedie's Formula

For Gaussian noise: \(\mathbf{y} = \mathbf{x} + \sigma\epsilon\)

Tweedie's formula yields an explicit expression for the posterior expectation: $\(E_{p(\mathbf{x}|\mathbf{y})}(\mathbf{x}) = \mathbf{y} + \sigma^2 \nabla_\mathbf{y} \log p(\mathbf{y})\)$

where \(\nabla_\mathbf{y} \log p(\mathbf{y})\) is the score function of the noisy data distribution. Key advantage: given the score function and noise parameters, denoising is accomplished in a single step.

Score Estimation Network

Based on the KPConv architecture: - 5 encoding stages (14 blocks): 3→256→512→1024→2048 channels - 4 decoding blocks: 3072→128 - Final fully connected layer outputting a 3D score vector - 24.3M parameters

The Amortized Residual DAE (AR-DAE) loss is employed: $\(\mathcal{L}_{AR-DAE} = \frac{1}{N}\sum_{i=1}^N \|\sigma_t \cdot S(y'_i) + u\|^2\)$

where \(y'_i = y_i + u \cdot \sigma_t\) is the perturbed version and \(u \sim \mathcal{N}(0, I)\).

Unknown Noise Parameter Estimation — Point Cloud Total Variation

\[TV_{PC} = \sum_{i=1}^N \sum_{j \in \text{neighbors}(i)} w_{i,j} \cdot \sqrt{\|\mathbf{p}_i - \mathbf{p}_j\|^2 + \epsilon^2}\]

The optimal noise parameter is estimated by minimizing the total variation of the denoised result: $\(\sigma^* = \arg\min_\sigma TV_{PC}(\hat{x}(\sigma))\)$

Denoising Pipeline

  1. Training: Learn the score function \(S(y)\) from noisy point clouds.
  2. Denoising: \(\hat{x} = y + \sigma^2 S(y)\) (completed in a single step).
  3. Unknown noise: Search for the optimal \(\sigma\) over a set of candidates.

Key Experimental Results

Denoising Performance on ModelNet-40 (CD×\(10^4\))

Method 10K pts 1% noise CD 10K pts 3% noise CD 50K pts 1% noise CD
Bilateral 5.865 31.034 3.711
GLR 6.592 12.890 1.860
TotalDn 8.079 29.617 5.044
Score-U 5.514 18.239 2.696
Noise2Score3D 4.891 12.456 1.654

Generalization Evaluation

Training Set Test Set CD↓ P2M↓
ModelNet-40 ModelNet-40 Best Best
ModelNet-40 PU-Net Still superior Still superior

Key Findings

  1. Achieves state-of-the-art performance among unsupervised methods, approaching supervised methods in certain settings.
  2. Strong generalization — a single pretrained model transfers across datasets and noise levels without retraining.
  3. Efficient inference — single-step denoising when noise parameters are known, outperforming iterative approaches.
  4. Point cloud total variation serves as an effective no-reference quality metric for automatic noise parameter selection.

Highlights & Insights

  1. Integration of Bayesian statistical tools — Tweedie's formula decouples denoising from score estimation, offering theoretical elegance.
  2. Single-step denoising — distinguishes this work from all existing unsupervised methods that rely on iterative procedures.
  3. Noise model agnosticism — the same loss function and pretrained weights apply across different noise models.
  4. Point cloud total variation — introduces a no-reference quality metric to point cloud denoising for the first time.

Limitations & Future Work

  • Validation is primarily conducted under Gaussian noise; performance on other noise distributions remains to be verified.
  • The KPConv-based network has a relatively large parameter count (24.3M).
  • When noise parameters are unknown, a search over a candidate range is required.
  • Traditional methods: Bilateral filtering, low-rank approximation, graph Laplacian regularization.
  • Supervised methods: PointCleanNet, ScoreDenoise, IterativePFN.
  • Unsupervised methods: TotalDn, DMR-U, Score-U.
  • Score matching: Denoising Score Matching, AR-DAE.

Rating

  • Novelty: ⭐⭐⭐⭐ (Extension of Tweedie's formula to 3D point clouds)
  • Technical Depth: ⭐⭐⭐⭐⭐ (Solid theoretical foundation, complete Bayesian framework)
  • Experimental Thoroughness: ⭐⭐⭐⭐ (Multiple noise levels, cross-dataset generalization)
  • Value: ⭐⭐⭐⭐⭐ (No clean data required, single-step denoising, automatic noise estimation)