Pacific Graphics 2011 Jan Kautz, Tong-Yee Lee, and Ming C. Lin (Guest Editors) Volume 30 (2011), Number 7 SSD: Smooth Signed Distance Surface Reconstruction F. Calakli and G. Taubin School of Engineering, Brown University, Providence, RI, USA Abstract We introduce a new variational formulation for the problem of reconstructing a watertight surface defined by an implicit equation, from a finite set of oriented points; a problem which has attracted a lot of attention for more than two decades. As in the Poisson Surface Reconstruction approach, discretizations of the continuous formula- tion reduce to the solution of sparse linear systems of equations. But rather than forcing the implicit function to approximate the indicator function of the volume bounded by the implicit surface, in our formulation the implicit function is forced to be a smooth approximation of the signed distance function to the surface. Since an indicator function is discontinuous, its gradient does not exist exactly where it needs to be compared with the normal vector data. The smooth signed distance has approximate unit slope in the neighborhood of the data points. As a result, the normal vector data can be incorporated directly into the energy function without implicit function smoothing. In addition, rather than first extending the oriented points to a vector field within the bounding volume, and then approximating the vector field by a gradient field in the least squares sense, here the vector field is constrained to be the gradient of the implicit function, and a single variational problem is solved directly in one step. The for- mulation allows for a number of different efficient discretizations, reduces to a finite least squares problem for all linearly parameterized families of functions, and does not require boundary conditions. The resulting algorithms are significantly simpler and easier to implement, and produce results of quality comparable with state-of-the-art algorithms. An efficient implementation based on a primal-graph octree-based hybrid finite element-finite dif- ference discretization, and the Dual Marching Cubes isosurface extraction algorithm, is shown to produce high quality crack-free adaptive manifold polygon meshes. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Ge- ometry and Object Modeling—Curve, surface, solid, and object representations G.1.2 [Numerical Analysis]: Approximation—Approximation of surfaces and contours 1. Introduction The problem of reconstructing watertight surfaces from ori- ented point clouds has received an immense amount of at- tention since the mid 80’s [BV91]. Oriented point clouds are nowadays usually obtained using optical measuring devices such as laser scanners and inexpensive structured lighting systems, by other computational means such as multi-view stereo reconstruction, and also result from large scale sim- ulations. Dense oriented point clouds have become a perva- sive surface representation in Computer Graphics [KB04]. The main challenges in this problem domain are: how to extrapolate to areas where the sampling is uneven, how to handle missing and noisy data, how to fill holes, and how to develop simple and efficient algorithms which gracefully scale up to very large data sets. As we show in section 9 the formulation introduced in this paper performs particu- larly well on unevenly sampled data sets. The primary con- tribution of this paper is a simple variational formulation of the problem developed from elementary geometric con- cepts. Various discretizations are proposed based on popu- lar surface representations. The particular hybrid FE/FD dis- cretization described in section 6 results in a simple algo- rithm which competes in quality and speed with the state-of- the-art methods. The prior art in surface reconstruction methods is exten- sive. We discuss only some of the existing methods, and refer the reader to [SS05] for a survey on recent developments. Despite the long history, the area is still very active. One family of algorithms produces interpolating polygon meshes c  2011 The Author(s) Journal compilation c 2011 The Eurographics Association and Blackwell Publishing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.