SIMPC: Learning Self-Induced Mirror-Point Consistency for Unsupervised Point Cloud Denoising

Technical Summary: SIMPC: Learning Self-Induced Mirror-Point Consistency for Unsupervised Point Cloud Denoising

Problem Statement
Point cloud denoising is critical for downstream tasks like surface reconstruction and semantic understanding, yet it faces unique challenges compared to 2D image denoising. In images, pixel-wise indexing allows for fixed spatial correspondences across noisy observations, enabling unsupervised methods to treat noisy values at the same location as random realizations of a clean signal. In contrast, point clouds lack fixed spatial indexing; noise perturbs the coordinates themselves, which encode both spatial location and geometry. This breaks correspondence consistency across noisy observations.

Existing unsupervised point cloud denoising methods attempt to learn statistical mappings across noisy variants but suffer from "correspondence ambiguity."

• Noise-based methods (e.g., Noise4Denoise, Noise2Score3D) generate noisier variants by injecting random noise. The induced correspondences are driven purely by stochastic noise and do not explicitly point to the target surface.
• EMD-based methods (e.g., NoiseMap, U-CAN) use Earth Mover's Distance to impose optimal transport between denoised outputs and noisy variants. While this yields more structured correspondences than noise-based approaches, the transport operates at the point-set distribution level. It does not explicitly encourage point-level consistency with the target surface, leading to ambiguous mappings where points may not represent different observations of the same underlying surface.

Methodology: Self-Induced Mirror-Point Consistency (SIMPC)
The authors propose SIMPC, a framework that learns deterministic one-to-one correspondences between points and the underlying surface without paired clean data. The core insight is to establish correspondences self-induced from a single noisy point rather than matching independently sampled noisy observations.

The framework operates on two noisy variants ($X_a, X_b$) processed independently through an iterative denoising pipeline consisting of an Encoder, Denoiser Blocks, and a Mirror-Point Generation Module (MPGM).

1. Encoder: Utilizes a T-layer DGCNN with skip connections to extract point-wise features from the input noisy point cloud, constructing dynamic neighborhoods in feature space.
2. Denoiser Block: Employs Point Self-Attention (PSA) to refine features and enhance local geometric perception, followed by a decoder that predicts a point-wise denoising vector ($d_i$). The point coordinates are updated iteratively: $x_i^l = x_i^{l-1} + d_i^l$.
3. Mirror-Point Generation Module (MPGM): This is the novel component designed to create deterministic correspondences. For a noisy seed point $x_i$ and its denoising vector $d_i$:
   • A standard denoised seed point is generated: $\hat{x}_i = x_i + w_1 d_i$ (where $w_1=1$).
   • A mirror-point is generated on the opposite side of the underlying surface by extending the denoising vector with a larger scaling factor: $\tilde{x}_i = x_i + w_2 d_i$ (where $w_2=2$).
   • The local neighborhood for this mirror-point is recomputed in the original noisy cloud (excluding the original point).
   • The same PSA and decoder operations are applied to the mirror-point's neighborhood to predict a mirror denoising vector ($\tilde{d}_i$) and obtain a denoised mirror-point: $\bar{x}_i = \tilde{x}_i + \tilde{d}_i$.
4. Mirror-Point Consistency Loss (MPCL): The method enforces consistency between the denoised seed point ($\hat{x}_i$) and the denoised mirror-point ($\bar{x}_i$), which are deterministically associated with the same underlying surface location but exhibit different neighboring noise states. The loss is defined as:
   $$L_{MPC} = \sum_{i=1}^{N} \|\hat{x}_i - \bar{x}_i\|_2^2$$
   This encourages both points to converge to a common target surface, effectively localizing the surface position.
5. Overall Objective: The total loss combines the MPCL with a Chamfer Distance-based similarity regularization ($L_{SR}$) between the two independent noisy variants to provide coarse similarity constraints.

Key Contributions

• Novel Framework: Introduction of SIMPC, an unsupervised framework that establishes deterministic one-to-one correspondences between noisy points associated with the same target surface, addressing the correspondence ambiguity inherent in previous methods.
• MPGM and MPCL: The design of the Mirror-Point Generation Module to extract geometric priors and generate self-induced mirror counterparts, coupled with the Mirror-Point Consistency Loss to enforce consistent denoising targets.
• Performance: Extensive experiments demonstrate that SIMPC achieves state-of-the-art (SOTA) performance among unsupervised methods on both synthetic (PUNet, PCNet) and real-world (Paris-Rue-Madame, Kinect) datasets. Notably, it surpasses several strong supervised counterparts.

Experimental Results
The authors evaluated SIMPC on synthetic datasets with Gaussian and non-Gaussian (Laplacian, Discrete, Anisotropic, Uniform) noise, as well as real-world scans.

• Quantitative Metrics: SIMPC consistently outperformed existing unsupervised methods (e.g., Noise2Score3D, U-CAN, Score-U) and optimization-based methods (e.g., GLR, MRPCA) in terms of Chamfer Distance (CD) and Point-to-Mesh (P2M) distance. In many cases, it achieved lower error rates than supervised methods like PD-Refiner and StraightPCF.
• Qualitative Results: Visualizations showed that SIMPC effectively suppressed residual noise and produced smoother surfaces with better alignment to the underlying geometry compared to methods that suffered from clustering artifacts or linear structure convergence.
• Ablation Studies: Removing the MPCL and relying solely on Chamfer Distance or EMD-based constraints resulted in significantly degraded performance, validating the necessity of deterministic correspondences. The geometric symmetry design ($w_2=2$) was found to be optimal compared to non-symmetric settings.

Significance and Claims
The paper claims that SIMPC addresses the fundamental limitation of correspondence ambiguity in unsupervised point cloud denoising. By leveraging geometric priors extracted during the denoising process to generate self-induced mirror points, the method learns deterministic correspondences that allow for accurate localization of the underlying surface. The authors assert that this approach not only closes the gap with supervised methods but, in several instances, surpasses them, demonstrating that high-quality denoising can be achieved without the need for paired noisy-clean training data. The work is supported by the National Key Research and Development Program of China.