# Three-Dimensional Surface Reconstruction from Point Clouds Using [PERSON]'s Elastica Regularization
## 1 Introduction  
Three-dimensional reconstruction using point clouds has great academic and commercial significance, but the challenge lies in the fact that the point clouds are often disorganized and disconnected, making it difficult to accurately reconstruct surfaces. There are two main methods for the reconstruction of the surface, the data-based method and the model-based method. Data-based methods are typically deep learning methods, such as the Deep level set [1], DeepSDF [2], EdgeConv [3], Point-BERT [4], IterativePFN [5], etc. These methodcan achieve good results but require a large amount of training data. On the other hand, model-based methods can produce good results without training data [6,7]. Moreover, they can be used as a priori methods [8] to maintain the smoothness of the reconstructed surface, fill in missing surface points, and adjust misalignment [9]. For the above reasons, the classical model-based methods are still being actively researched and improved upon. There are two general approaches to describe a surface: explicit and implicit. Explicit representation [10, 11, 12] accurately represents the location of points on the reconstructed surface and is efficient and accurate, but it is less robust and not flexible enough to handle arbitrary and dynamic surface topologies. On the other hand, implicit representation [13, 14, 15, 16] starts with an initial surface and iteratively evolves it through a level set function [17] to achieve an accurate reconstruction. Unlike explicit methods, implicit methods have greater robustness and flexibility, making them better suited for reconstructing noisy and non-uniform point clouds [14,15,18]. The implicit surface reconstruction method proposed by [PERSON] et al. in [13,19] is a successful algorithm that uses the variational level set method and either the convection method or the energy-minimizing gradient flow method to reconstruct surfaces. This method is efficient in creating an initial surface, but it still has three main challenges. Firstly, the surface evolution process requires reinitialization, which ensures that the evolving level set is equivalent to a signed distance function, but this impacts the calculation efficiency and may lead to the final reconstructed surface shrinking [20]. Secondly, the model is non-convex, and the initial surface greatly affects the reconstruction results. Lastly, if the point cloud data encompasses delicate details or curved inward regions, the initial surface reconstruction may become trapped at a local extreme, despite the accuracy of the initial reconstruction. To better reconstruct the surface, a method is proposed for reconstructing point cloud surfaces using the L1-sparse method. This method effectively ignores the influence of Gaussian noise, and its global optimization property ensures that the reconstructed surface is not limited to the local extrema. Based on this work, [PERSON] [21,22] proposed a novel variational method using the TV method with a balloon force and a fitting term (TV\({}_{g}\)). The method can overcome the issue of getting stuck in local minima but still cause the surface to lose many details. However, when using the TV model for point cloud reconstruction, it can result in the stair-step effect and contrast loss, leading to clearly visible jagged regions in the reconstructed image and the loss of important texture details. To address these issues related to the stair-step effect and contrast loss, [PERSON] [23] proposed to adopt a curvature regularized term instead of the TV regularized term. Experiments show that the curvature regularized term can be used to better reconstruct the smooth sections of the surface but performs worse around the corners. It remains to be seen whether there is a method that can combine the advantages of the low-order model and the high-order model. On the other hand, [PERSON]'s elastica term has long been used in image processing, especially in image segmentation [24,25] and denoising [26].