This paper presents a new approach to construct surface models from extremely large point sets using piecewise quadratic functions that capture the local shape of the surface, and then blending them together using multi-level partitions of unity as weighting functions. This is the first method to use partitions of the point set into regions to fit implicit functions to, as opposed to function fitting at every point, in previous methods. The paper presents a surface representation using multi-level partition of unity (MPU) implicit surfaces. Presents an approach with the 3 key concepts: 1. Piecewise quadratic functions used as local shape estimates for the surface. 2. Weighing functions (partition of unity) to blend the local shape functions. 3. Octree subdivision that adapts based on shape complexity. This approach is adaptive based on the desired accuracy. The use of implicit functions provide data repairing capabilities for scattered data and an easy way to perform implicit modeling operations. This paper proposes a method to implicitly define a surface as the zero level of a distance function. The space is partitioned in octree cells and a shape function is fitted to each cell. In contrast to previous approaches (such as Hoppe 92), the distance to the surface (i.e., the implicit function evaluation) is not computed respect to a single element of the space partition (in this case, an octree cell) but is a weighted sum. Due to the partition of unity approach and the use of smooth weighting function, the distance function changes smoothly along the 3D space. The paper introduces the partition of unity approach for approximation of the signed distance function of a set of scattered points with normals. The method breaks up the point set using octree subdivision (can handle complex topology). It then uses piecewise quadratic function as local approximations of the surface. These local approximations are blended together using smooth, local weights (can represent sharp features). This paper proposes an efficient, octree-based, algorithm for reconstructing surfaces from oriented point samples. The approach falls somewhere between global implicit methods and MLS approaches, as the approach partitions space, computes a local implicit surface around each partition, and then glues these together using a partition of unity. In addition, the authors also describe how to handle sharp features by clustering points based on their normal orientations, fitting implicits to each of the clusters independently,
Ohtake et al. REF introduce the multi-level partition of unity implicits.