IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 12, NO. 4, JULY/AUGUST 2006 581 3D Distance Fields: A Survey of Techniques and Applications Mark W. Jones, J. Andreas Bærentzen, and Milos Sramek Abstract—A distance field is a representation where, at each point within the field, we know the distance from that point to the closest point on any object within the domain. In addition to distance, other properties may be derived from the distance field, such as the direction to the surface, and when the distance field is signed, we may also determine if the point is internal or external to objects within the domain. The distance field has been found to be a useful construction within the areas of computer vision, physics, and computer graphics. This paper serves as an exposition of methods for the production of distance fields, and a review of alternative representations and applications of distance fields. In the course of this paper, we present various methods from all three of the above areas, and we answer pertinent questions such as How accurate are these methods compared to each other? How simple are they to implement?, and What is the complexity and runtime of such methods? Index Terms—Distance field, volume, voxel, fast marching method, level-set method, medial axis, cut locus, skeletonization, voxelization, volume data, visualization, distance transform. æ 1INTRODUCTION ERHAPS the earliest appearance of distance fields in the suitable antialiased representation of geometric objects for Pliterature is the 1966 image processing paper by the purposes of Volume Graphics. The term Volume Graphics Rosenfeld and Pfaltz [97], where they present the applica- was first introduced by Kaufman et al. in 1993 [64], where tion of a chamfer distance transform to an image, and also they presented the advantages of using volumetric models. create a skeleton which is a minimal representation of the Although they were working with binary representations original structure. Since then, many authors have improved which suffered from aliasing, many of the methods they the accuracy of chamfer distance transforms, and have proposed and discussed have adapted well to distance fields. introduced alternative algorithms such as vector distance Volume Graphics is now a subject area in its own right, transforms, fast marching methods, and level sets. Most of demonstrated by an annual “Volume Graphics” conference the earlier work concentrated on two-dimensional image series which started at Swansea in 1999 [24]. However, processing, but as three-dimensional data sets grew in distance fields have recently found many applications importance, recently much research has been targeted at unrelated to traditional volume graphics. For instance, they processing this and higher dimensional data. The literature can be used for collision detection, correcting the topology of seems broadly split between the computer vision commu- meshes, or to test whether a simplified mesh is within a given nity (for image processing), physics community (for distance threshold of the original. wavefront, Eikonal equation solving schemes), and compu- A distance field representation of an object can be ter graphics community (for object representation and particularly useful in situations where algorithms provide processing). This paper will draw together the literature for the fast processing of three-dimensional objects, and so from these communities and will, for the first time, this paper shall concentrate on the methods by which independently and thoroughly compare the various main distance fields are produced, and the applications that can algorithms and approaches. use these distance fields to accelerate modeling, manipula- For the purposes of this paper, we are most interested in tion and rendering techniques. Apart from a survey of the the application of algorithms using distance fields for the literature in those areas, the main contributions of this modeling, manipulation, and visualization of objects for paper are a summary of some of the very latest results in the computer graphics, and so we shall emphasize methods that production and use of distance fields, a new simplified enable such processes. Recently, it seems that there is general version of the Fast Marching Method (FMM) and the first widespread agreement that distance fields provide the most thorough comparison of FMM and its variants, Chamfer Distance Transforms (CDTs) and Vector Distance Trans- forms (VDTs) on both error and speed. M.W. Jones is with the Department of Computer Science, Swansea The remainder of this paper is organized as follows: In University, Singleton Park, Swansea, SA2 8PP, United Kingdom. E-mail: [email protected]. Section 2, we present some properties of the continuous . J.A. Bærentzen is with the Informatics and Mathematical Modelling distance field. Section 3 acquaints the reader with the main Department, Technical University of Denmark, Richard Petersens Plads approaches to calculating discrete distance fields. Aspects DTU, Building 321, DK-2800 Lyngby, Denmark. such as computing the distance field directly from the data, E-mail: [email protected]. and computing the sign for a signed distance field are . M. Sramek is with the OAW Visualisierung, Donau-City-Strasse 1, A- 1220 Vienna, Austria. E-mail: [email protected]. accounted in Section 3.1. Using a shell boundary condition as a basis it is possible to create distance fields using the Manuscript received 21 Mar. 2005; revised 6 Sept. 2005; accepted 7 Dec. 2005; published online 10 May 2006. vector and chamfer transform methods and the fast For information on obtaining reprints of this article, please send e-mail to: marching methods of Section 3.2. Section 3.6 provides the [email protected], and reference IEEECS Log Number TVCG-0039-0305. first in-depth comparison of all three approaches—vector, 1077-2626/06/$20.00 ß 2006 IEEE Published by the IEEE Computer Society 582 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 12, NO. 4, JULY/AUGUST 2006 1 chamfer, and fast marching methods on a variety of data M ¼ ðdxx þ dyy þ dzzÞ: ð5Þ sets. A thorough analysis of both time and error is given. 2 Section 4 examines alternative representation schemes for The Gaussian curvature is distance fields including Adaptive Distance Fields (ADFs), lossless compression schemes, wavelets, and the Complete dxx dxy dxx dxz dyy dyz G ¼ þ þ ð6Þ Distance Field Representation (CDFR). The main applica- dyx dyy dzx dzz dzy dzz tion areas using distance fields are briefly examined in and the principal curvatures are the two nonzero eigenva- Section 5. Finally, we conclude this paper with a discussion lues of H. The last eigenvalue is 0 reflecting the fact that d in Section 6. changes linearly in the gradient direction. Monga et al. give a good explanation of how the Hessian is related to 2CONTINUOUS DISTANCE FIELDS curvature [72]. 2.1 The Continuous Signed Distance Function 2.3 Continuity and Differentiability Assuming that we have a set Æ, we first define the unsigned It follows from the triangle inequality that the signed and distance function as the function that yields the distance from unsigned distance functions of a given surface are contin- a point p to the closest point in Æ: uous everywhere. However, neither is everywhere differ- entiable. This raises the question of where the signed distÆðpÞ¼inf kx À pk: ð1Þ distance function is differentiable which is a question a x2Æ number of authors have considered. Frequently, we are mainly interested in the signed distance In [67], it is demonstrated that for a Ck surface (k 1), function associated with a solid S. The signed distance the signed distance function is also Ck in some neighbour- function returns the distance to the boundary, @S, and the hood of the surface. In his technical report, Wolter [135] sign is used to denote whether we are inside or outside S. presents various theorems regarding the cut locus. The cut Here, we use the convention that the sign is negative inside. locus of a surface is the set of points equally distant from at This leads to the following formula for the signed distance least two points on the surface. Hence, the cut locus is the function corresponding to a solid S same as the union of the interior and exterior medial surfaces. Theorem 2 in [135] pertains directly to the dSðpÞ¼sgnðpÞ inf kx À pk; ð2Þ differentiability of the distance function. Specifically, it is x2@S shown that the unsigned distance function is differentiable, where and its gradient Lipschitz continuous at points which neither belong to the surface nor to its cut locus. However, À1ifp 2 S sgnðpÞ¼ the signed distance is differentiable also on the surface 1 otherwise: (except at points where the cut locus touches the surface), and the critical points1 of the signed distance coincide with When no ambiguity is possible, we drop the subscript S the cut locus. from d . S In the remainder of this paper, we are mostly concerned 2.2 Derivatives of the Distance Function with discrete distance fields. Such distance fields may be obtained by sampling a continuous distance function, and in An important property of the signed distance function d is that case it may be important to know whether two grid points that straddle the cut locus. If they do, it means that their distance krdk¼1 ð3Þ values correspond to surface points that may be very far apart. This can lead to poor reconstruction of the continuous almost everywhere, with the exception being points without field, its gradient, and other characteristics, since they are a unique closest point (e.g., the center of a sphere, see also typically estimated using finite reconstruction kernels which Section 2.3). In this case, the gradient is not defined. use a stencil of grid points.