🤖 AI Summary
This work proposes a novel method for jointly estimating an implicit surface and its noise distribution from noisy measurements. The surface is modeled as the zero level set of a function in a reproducing kernel Hilbert space, while a parametric noise model is introduced to capture measurement uncertainty. The resulting optimization framework requires no hyperparameter tuning or preprocessing and scales linearly with data size, offering theoretical convergence guarantees. Implicit surface representations are constructed using polynomial and trigonometric basis functions, enabling applicability to high-dimensional data. Extensive experiments on 2D and 3D benchmark datasets demonstrate that the proposed approach consistently outperforms state-of-the-art methods such as Poisson reconstruction and Encoder-X in both reconstruction accuracy and computational efficiency.
📝 Abstract
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.