# Three-Dimensional Surface Reconstruction from Point Clouds Using [PERSON]'s Elastica Regularization
## 1 Introduction
Using [PERSON]'s elastica as a regularization term to address issues such as the staircase effect and contrast loss in low-order models is currently a popular topic [27,28]; however, due to the challenges associated with its application in point cloud reconstruction, there is currently a lack of research in this specific area. Through decades of research, this regularizer has been found to maintain some of the advantages of the TV regularization term and the curvature regularization term at the same time. Utilizing [PERSON]'s elastica regularization as the energy function effectively mitigates the stair-step effect and contrast loss problems inherent in the TV model. It can also address the problem of excessively smooth inflection points in curvature models, resulting in point cloud reconstructions that preserve more details. However, because of its complexity, this term comes with greater computational cost, so many new research works focused on designing a solution for [PERSON]'s elastica term, such as [PERSON] et al. 's ALM method [29] and [PERSON] et al. 's fast method [30]. With the help of these methods, the efficiency of [PERSON]'s elastica term applied to image processing problems is significantly improved. Given the aforementioned advantages, the decision is made to adopt [PERSON]'s elastica term as the regularization term. To resolve the issues previously identified with the traditional implicit reconstruction technique, we present a new method called Euler's elastica for creating precise surfaces from point cloud data. This approach transforms the surface reconstruction process into a task of implicit image segmentation. Our approach involves creating a boundary indicator function, a enclosed image, and an initial surface from the input point cloud data. The edge indicator function plays a crucial role in identifying the precise surface area that needs to be reconstructed. The bounded image acts as a marker, indicating if a specific grid point lies inside or outside the object. The level set method used in the proposed model is successfully initialized by the initial surface. Proposed surface reconstruction approach involves two stages. Stage 1: We use the fast sweeping method to compute a distance function from the point cloud data. This distance function is then utilized: (a) as an edge indicator function, (b) to generate an image (2D or 3D) surrounded by the point clouds, and (c) to determine an initial surface. Stage 2: The initial surface generated in Stage 1 is then advanced through the application of the proposed variational level set model, which effectively combines the edge indicator and the image from Stage 1. Optimizing this model involves utilizing the fast dual method, as detailed in [31], to simplify the computation of intricate curvature terms. The proposed method minimizes the variational model through gradient descent optimization to ultimately produce a precise surface from the point cloud data. Experiments indicate that the Euler elastica method that we propose surpasses other implicit methods in terms of accuracy. In this work, three main research contributions are presented. Firstly, we introduce [PERSON]'s elastica regularization for the first time in 3D surface reconstruction. By utilizing the Euler elastica function as the regularization term, it addresses the issues of the staircasing effect and contrast loss observed in the TV model. This is mainly because the Euler elastica energy function imposes constraints on the deformation field by considering the elastic properties of objects [32; 33], leading to more plausible and realistic deformations. It helps avoid unreasonable local deformations and ensures that the deformation field globally retains the shape and structural features of objects, thus mitigating the occurrence of the staircasing effect. Moreover, the TV model may encounter gradient vanishing problems in certain cases, resulting in the loss of reconstruction details.