Shrinking: Another Method for Surface Reconstruction In-Kwon Lee (corresponding author) Dept. of Computer Science Yonsei University Seoul 120-749, South Korea iklee@yonsei.ac.kr Ku-Jin Kim The Graduate School of Information and Communication Ajou University Suwon 442-749, South Korea kujinkim@ajou.ac.kr Abstract We present a method to reconstruct a pipe or a canal sur- face from a point cloud (a set of unorganized points). A pipe surface is defined by a spine curve and a constant radius of a swept sphere, while a variable radius may be used to de- fine a canal surface. In this paper, by using the shrinking and moving least-squares methods, we reduce a point cloud to a thin curve-like point set which will be approximated to the spine curve of a pipe or canal surface. The distance be- tween a point in the thin point cloud and a corresponding point in the original point set represents the radius of the pipe or canal surface. 1. Introduction A pipe surface is defined as the envelope of a sphere with a constant radius moving through a spine curve C .A canal surface is a generalization of a pipe surface, where a variable radius function is used instead of . Pipe and canal surfaces are used in many practical applications such as surface blending and transition surfaces between pipes [1, 2]. Furthermore, there are many real or synthetic objects which can be represented by only pipe or canal surfaces (see Figure 1). In this paper, we consider the problem of reconstructing a pipe or canal surface from an unorganized point cloud having no ordering or structure of the point ele- ments. The input point cloud is usually scanned from a real pipe/canal surface by a 3D scanner. The motivation of this work comes from recent research in reverse engineering area, attempting to reconstruct sur- faces such as helical surfaces and surfaces of revolution [3], profile surfaces [4], developable surfaces [5], and pla- nar faces [6]. The details about these series of works can be found in [7] and [8]. The reconstruction of the sweep surfaces generated by translational sweeping [9] is also discussed by other researchers. Instead of reconstructing Figure 1. Some objects created using only canal surfaces spline surfaces or polygonal meshes, this research concen- trates on reconstructing a profile (cross-section) curve and a kinematic motion that defines a trajectory of the moving profile curve. This procedural description of a geometric model reduces the size of the object database and makes design/manipulation processes efficient and easy. Ramamoorthi and Arvo [10] suggested a system to re- construct various kinds of generative models, including pipe and canal surfaces, from point clouds with the aid of a pre- defined user-specified hierarchy of the various generative models. Unlike their work, we suggest an algorithm which avoids serious user interaction by narrowing down our inter- est to reconstruct pipe or canal surfaces only. Our solution uses only local linear optimization procedures that are much more efficient and robust than complex nonlinear optimiza- tion methods. In our previous papers [11, 12], we have roughly sketched out the idea behind reconstructing pipe surfaces (see Figure 2). First, an appropriate subregion of a given point cloud is taken (Figure 2(a)). Then, a torus is fitted to . Note that the local shape of a pipe surface fits a torus. The torus implies the radius of a swept sphere of the pipe surface. Each data point is translated by the radius towards a (target) spine curve along an estimated normal vector of the point (Figure 2(b)). After the translation, the point set