This paper proposes a method for estimating a surface that contains a given set of points from noisy measurements. More precisely, by assuming that the surface is described by the zero set of a function in the span of a given set of features and a parametric description of the distribution of the noise, a computationally efficient method is described that estimates both the surface and the noise distribution parameters. In the provided examples, polynomial and sinusoidal basis functions were used. However, any chosen basis that satisfies the outlined conditions mentioned in the paper can be approximated as a combination of trigonometric, exponential, and/or polynomial terms, making the presented approach highly generalizable. The proposed algorithm exhibits linear computational complexity in the number of samples. Our approach requires no hyperparameter tuning or data preprocessing and effectively handles data in dimensions beyond 2D and 3D. The theoretical results demonstrating the convergence of the proposed algorithm have been provided. To highlight the performance of the proposed method, comprehensive numerical results are conducted, evaluating our method against state-of-the-art algorithms, including Poisson Reconstruction and the Neural Network-based Encoder-X, on 2D and 3D shapes. The results demonstrate the superiority of our method under the same conditions.
Figures & tables
Fig. 1 : Point cloud of the Clebsch cubic surface corrupted by noise at a 20% level.
Fig. 2 : Convergence and parameter estimation for the Clebsch cubic. (a) Absolute cosine similarity ∣cosθ∣ between the estimated and optimal coefficient vectors, demonstrating asymptotic recovery across varying noise levels. (b) Grid search over the noise parameter; the global minimum singular value of M^DL accurately identifies the true injected noise level ( 20% ).
Fig. 3 : Zero-level set reconstruction of the Clebsch cubic utilizing the proposed RAIN-FIT algorithm on the 20% corrupted dataset. The global topology is accurately recovered without relying on local normal estimates.
Fig. 4 : Failure of standard Poisson surface reconstruction on the 20% corrupted Clebsch dataset. The stochastic disruption of the oriented normal vector field results in severe topological distortions.
Fig. 5 : Fitting results employing RAIN-FIT (diamond blue) and Encoder-X (circle red). The black line depicts the original shape to be used as a reference for the quality of the fit. The columns (from left to right) correspond to the results for noise levels of 0%, 10%, 20%, and 30%.