Hierarchical Spline Approximation of the Signed Distance Function Xinghua Song, Bert Juettler, Adrien Poteaux To cite this version: Xinghua Song, Bert Juettler, Adrien Poteaux. Hierarchical Spline Approximation of the Signed Dis- tance Function. SMI 2010, Jun 2010, Aix-en-provence, France. pp.241-245, 10.1109/SMI.2010.18. inria-00506000 HAL Id: inria-00506000 https://hal.inria.fr/inria-00506000 Submitted on 26 Jul 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. IEEE INTERNATIONAL CONFERENCE ON SHAPE MODELING AND APPLICATIONS (SMI) 2010 1 Hierarchical Spline Approximation of the Signed Distance Function Xinghua Song1, Bert J¨uttler2, Adrien Poteaux2 1Projet GALAAD, INRIA Sophia Antipolis, France, 2Institute of Applied Geometry, JKU Linz, Austria Abstract—We present a method to approximate Pekerman et al. [13]. Seong et al. [15] used distance the signed distance function of a smooth curve or maps for trimming local and global self-intersections in surface by using polynomial splines over hierarchical offset curves/surfaces, see also [10]. T-meshes (PHT splines). In particular, we focus on The remainder if this paper is organized as follows. closed parametric curves in the plane and implicitly After a short introduction to polynomial splines over hi- defined surfaces in space. erarchical T-meshes and a summary of properties of the signed distance function, we describe the computation Keywords—signed distance function, hierarchical of the approximate signed distance function, and provide T-spline, trimmed offsets some examples for trimmed offset computation. 1. INTRODUCTION 2. PHT-SPLINES The signed distance function of a closed curve or Spline functions over T-meshes have recently been surface Γ in the plane or in three-dimensional space analyzed in [6]. For our application, we are mainly assigns to any point x the shortest distance between x interested in the special case of C1-smooth bi- and and Γ, with a positive sign if x is inside Γ and a negative tricubic polynomial splines over hierarchical T-meshes, one otherwise. It is a highly useful representation in a which is summarized in this section. We will call these large number of problems in geometric computing. For spline functions PHT-splines for short. instance, several approaches to surface reconstruction A two-dimensional T-mesh is a partition of an axis- use the signed distance function, see for example [3], aligned box (e.g., the unit square) into smaller axis– [4], [17]. aligned boxes. The edges of the boxes form a rectangular The signed distance function is also closely related to grid that may possess T-junctions. If a grid point of the concept of level set methods, which were introduced a two-dimensional T-mesh is a crossing vertex (i.e., it in [11]. For instance, the distance function preservation possesses valency 4), or belongs to the boundary of the has several advantages from the geometric and numeri- domain, then we call it a base vertex. cal point of view [7], [12], [18]. Kimmel and Bruckstein A three-dimensional T-mesh is the extension of this [9] used level sets for shape offsetting in the plane, in concept to the three-dimensional space. It is a partition particular for computing trimmed offsets. of an axis-aligned box such that each cell is another In the present paper, we propose a new method to axis-aligned box. A vertex of a 3D T-mesh is called a compute a hierarchical approximation of the signed base vertex if it either distance function. While algorithms developed in the • belongs to a boundary edge, context of level set methods use a computational grid, • is a crossing vertex on a boundary facet, or we use spline approximation, which enable us to get • possesses valency 6. smooth approximations. More precisely, we interpolate In this paper, we consider hierarchical meshes. More the signed distance function of the boundary of the precisely, we assume that the T-mesh has been obtained given shape by using a spline function defined over a by repeatedly applying a splitting step, where an axis- hierarchical T-mesh [6], [14] level by level. aligned box is subdivided into two smaller axis-aligned A simple but potentially useful application of our boxes, to the original axis-aligned box representing the method is to generate the trimmed offsets of the bound- entire domain of the spline functions and to the boxes ary of a given shape. Since the signed distance function obtained by subdividing it. of the given boundary curve/surface can be approxi- For a given hierarchical T-mesh T , a C1-smooth bi- mated with any desired precision, we can obtain an or tricubic PHT-spline f(x) is a function which is a implicit representation for any offset of the given bound- bicubic (2D) or tricubic (3D) polynomial within each ary. We illustrate this observation by several examples cell of T , such that the collection of these polynomial in Section 5. segments forms a globally C1-smooth function. The The computation of trimmed offset surfaces is space of these functions admits a simple local construc- still a challenging geometrical problem. The issue of tion, as follows. self-intersection detection and elimination in general At each base vertex, we specify the value, the first freeform curves and surfaces has been discussed by derivatives, the mixed second derivative(s), and (in the 3D-case) the mixed third derivative of the function. and Thus, we specify 4 or 8 (depending on the dimension) ∂3d value and derivative data at each base vertex. Then, dijk (x)= (x)= ∂xi∂xj ∂xk 2 using Hermite interpolation by cubic polynomials, we ∂ pi − ∂x ∂x can compute these data of f(x) at the other vertices = j k + of the hierarchical T-mesh T . For any point x in any kx − pk T ∂pi ∂pi ∂pj cell C of , the value of f(x) is determined by the ∂x (xk − pk)+ ∂x (xj − pj )+ ∂x (xi − pi) + j k k + value and derivative data at the 4 (resp. 8) vertices of kx − pk3 the cell C. 3(x − p )(x − p )(x − p ) + i i j j k k . More precisely, we obtain a bicubic or tricubic poly- kx − pk5 nomial for each cell, which matches the value and (5) derivative data at the vertices. Thus, a PHT-spline de- Therefore, if we want to compute the required deriva- fined over a given T-mesh T can be determined by the tives of the signed distance function at a point x, we 2 value and derivative data at the base vertices of T . This only need to compute ∂pi and ∂ pi , where i 6= j, ∂xj ∂xj ∂xk is in agreement with the dimension formulas of PHT- j 6= k and k 6= i. spline given in [5]. In this paper we consider two representations of the curve or surface Γ. In the plane, we consider a closed planar parametric curve. In the three-dimensional space 3. THE SIGNED DISTANCE FUNCTION (SDF) we consider an implicitly defined surface, which is given as the zero set of scalar field. In both cases, we assume Consider a simple closed curve or surface Γ in the that the curve and surface is sufficiently smooth (C2 for Euclidean space Rn, where n = 2 or 3. (In the the planar case and C3 for the surface case), possibly remainder of this paper, we call a surface and assume Γ except for a few singular points. Thus, the derivatives of that this includes the case of curves.) The surface Γ the signed distance function are well defined for almost divides Rn into three parts: the interior Γ+, the exterior all points. Γ− and Γ. The signed distance function of Γ is a scalar The purpose of considering two different representa- function which is defined in Rn as: tions (implicitly defined surfaces vs. parametric curves) kx − pk if x ∈ Γ+ is to cover in this paper the evaluation of the deriva- tives of the signed distance function for both types of d(x)= −kx − pk if x ∈ Γ− , (1) representations. Clearly, our method can be extended 0 if x ∈ Γ to other representations of smooth curves and surfaces, such as parametric free-form surfaces and implicitly where p = (p1, ..., pn) is the closest point of x = defined planar curves. (x1, ..., xn) on the surface Γ. 3.2 The case of planar parametric curves ∗ 3.1 Derivatives Given a planar parametric curve c(u), if p = c(u ) denote the closest point of x = (x1, x2) in c(u), then For later reference we compute the derivatives of the we have: signed distance function. We assume that the point x ∂p dc ∂u ∗ + = (u ). (6) lies in Γ , i.e., ∂xj du ∂xj Computing the derivative of d(x)= kx − pk, (2) ∗ dc ∗ (c(u ) − x) · (u )=0 and that it possesses a unique closest point on Γ. Thus, du x is not on the medial axis. (Points in Γ− can be dealt with respect to x1 gives with analogously.) The computation of the derivatives of the signed distance function (2) leads to the following dc ∂u ∗ 1 dc ∗ (u ) − · (u )+ formulas: du ∂x 0 du 1 (7) d2c ∂u ∂d c ∗ x ∗ +( (u ) − ) · 2 (u )=0.