Integration over curves and surfaces defined by the closest point mapping
- Catherine Kublik^{1}Email author and
- Richard Tsai^{2, 3}
DOI: 10.1186/s40687-016-0053-1
© Kublik and Tsai 2016
Received: 12 May 2015
Accepted: 10 November 2015
Published: 14 April 2016
Abstract
We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wishes to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation—initially derived to integrate over manifolds of codimension one—to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.
Keywords
Boundary integrals Closest point mapping Level set methods1 Introduction
This paper is motivated by the recent success in the closest point methods and the Dynamic Surface Extension method [23], for evolving interfaces and solving partial differential equations on surfaces [11–13, 19], by the need to process data sets that contain unstructured points sampled from some underlying surfaces, and targets applications where manifolds are not defined by patches of explicit parameterizations and may evolve drastically due to some coupled processes; see, e.g., free boundary problems [9]. Our work provides a convenient way to formulate boundary integral methods in such applications without conversion to local parameterizations. If the manifolds are defined by explicit parameterizations, it is natural and typically more accurate to use conventional methods such as Nyström methods using quadratures on the parameter space or Boundary Element Methods with weak formulations, see, e.g., [1]. Additionally, for applications involving fluid–structure interactions, we mention the immersed boundary method which involves accurate discretizations of Dirac delta measures [14, 17].
Finally, closest point mappings can also be computed easily from dense and unorganized point sets that are acquired directly from an imaging device (e.g., LIDAR). This paper lays the foundation of a numerical scheme for computing integrals over surfaces sampled by unstructured point clouds.
2 Integration using the closest point mapping
In this section, we relate the Jacobian J in (2) to the singular values of the Jacobian matrix of the closest point mapping from \(\mathbb {R}^2\) or \(\mathbb {R}^3\) to \(\Gamma \), where \(\Gamma \) denotes the curves or surfaces on which integrals are defined. We assume that in three dimensions, if \(\Gamma \) is not closed, it has smooth boundaries. For clarity of the exposition in the rest of the paper, we will now denote the distance function simply by d.
2.1 Codimension 1
Lemma 1
Proof
Proposition 2
Proof
Theorem 3
Proof
If \(\Gamma \) is closed we combine Eq. (2) with the result \(J(\mathbf {x})=\Sigma (\mathbf {x})\) from Eq. (10) of Proposition 2.
If \(\Gamma \) is open there is a little more to show since Eq. (2) was only derived for closed manifolds. Before we state the result, it is necessary to understand how \(\Gamma _{\eta }\) defined in (9) (an \(\eta -\)level set of d) looks like for an open curve in two dimensions and for a surface with boundaries in three dimensions.
In two dimensions, \(\Gamma _{\eta }\) consists of a flat tubular part on either side of the curve and two semi-circles at the two ends of the curve. See Fig. 1.
In three dimensions \(\Gamma \) is in general made up of three distinct parts: the interior part, the edges of the boundary and the corners. If we assume that \(\Gamma \) has N edges then we can write \(\Gamma =\Gamma ^{o}\cup (\cup _{i=1}^{N}E_{i})\cup (\cup _{i=1}^{N}C_{i})\), where \(\Gamma ^{o}\) is the interior of \(\Gamma \), \(E_{i}\) is the i-th edge of the boundary of \(\Gamma \) and \(C_{i}\) is its i-th corner. In that setting we can write \(\Gamma _{\eta }=I_{\eta }\cup (\cup _{i=1}^{N}T_{i}^{\eta })\cup (\cup _{i=1}^{N}S_{i}^{\eta })\), where \(I_{\eta }\) is the inside portion of \(\Gamma _{\eta }\), \(T_{i}^{\eta }\) is the cylindrical part of \(\Gamma _{\eta }\) representing the set of points located at a distance \(\eta \) from the i-th edge \(E_{i}\), and finally \(S_{i}^{\eta }\) is the spherical part of \(\Gamma _{\eta }\) representing the set of points located at a distance \(\eta \) from the i-th corner \(C_{i}\). See Fig. 2.
In both cases, we need to integrate over \(\Gamma _{\eta }\) and then subtract the two semi-circles at the two end points of the curve (in two dimensions) or subtract the portions of sphere at the corners of the surface and the portions of cylinders at the edges of the surface (in three dimensions). However, it turns out that the subtraction is unnecessary since \(\Sigma (\mathbf {x})=0\) on each of the subtracted pieces as shown below.
Two dimensions. On the semi-circle around the end point of a curve, the closest point mapping is constant since all points on the semi-circle \(\Gamma _{\eta }\) map to the end point. As a result, the singular values of the Jacobian matrix of the closest point mapping are all zeros and thus \(\Sigma (\mathbf {x})=0\) on the semi-circles around the end points of a curve.
Three dimensions. As in two dimensions, on the portions of sphere around a corner point of a surface, the closest point mapping is constant and thus \(\Sigma (\mathbf {x})=0\). On the portion of cylinders, the closest point mapping is constant along the radial dimension (one of the principal directions or singular vector) resulting of the singular value along that direction to be zero. Since \(\Sigma (\mathbf {x})\) is the product of the singular values, it follows that \(\Sigma (\mathbf {x})=0\) on the portion of cylinders as well. Consequently, Eq. (16) holds for any \(C^{2}\) curve or surface with \(C^2\) boundaries of codimension 1. \(\square \)
2.2 Codimension 2
Proposition 4
Consider a single \(C^{2}\) curve \(\Gamma \) in \(\mathbb {R}^{3}\) parameterized by \(\gamma (s)\) where s is the arclength parameter, and let d be the distance function to \(\Gamma \). We define \(K_{\epsilon }\) to be a \(C^{1}\) averaging kernel compactly supported in \([0,\epsilon ]\) and \(P_{\Gamma }:\mathbb {R}^{3}\mapsto \Gamma \) to be the closest point mapping to \(\Gamma \).
Now if we consider a \(C^{2}\) curve in three dimensions and let \(P_{\Gamma }\) be its closest point mapping, we have the following proposition:
Theorem 5
Proof
Since \(K_{\epsilon }\) is compactly supported in \([0,\epsilon ]\) it is sufficient to consider points in the tubular neighborhood of the curve \(\Gamma \). Thus, for \(\mathbf {x}\) in the tubular neighborhood, there exists \(0 \le \eta \le \epsilon \) such that \(\mathbf {x}\in \Gamma _{\eta }\).
Case 1: \(\mathbf {x}\) is on the spherical part of \(\Gamma _{\eta }\) corresponding to the \(\eta \)-distance to either of the two end points of the curve \(\Gamma \). WLOG we assume that \(\mathbf {x}\) is at a distance \(\eta \) from the first end point \(C_{1}\) parameterized by \(\gamma (0)\). The result is the same if \(\mathbf {x}\) is on the other sphere, i.e., at a distance \(\eta \) from the other end point \(C_{2}.\) In that case, \(P_{\Gamma }(\mathbf {x)}=\gamma (0)\) for all \(\mathbf {x}\) on the spherical part so that the Jacobian matrix \(P'_{\Gamma }=0\). Therefore, for \(\mathbf {x}\) on the spherical part of \(\Gamma _{\eta }\), all singular values of the Jacobian matrix are zero.
3 Numerical simulations
In this section, we investigate the convergence of our numerical integration using simple Riemann sums over uniform Cartesian grids. Unless stated otherwise, the singular values are computed from the matrix, the elements of which are computed by the standard central difference approximations of the Jacobian matrix \(P'_{\Gamma }.\) In other words, the Jacobian matrix \(P'_{\Gamma }\) is computed by using finite differences to evaluate the partial derivatives of each component of \(P_\Gamma (\mathbf {x})\); more precisely, if \(P_\Gamma (\mathbf {x}) ~= ~(p_{1}(\mathbf {x}), p_{2}(\mathbf {x}), p_{3}(\mathbf {x}))\), and \(\mathbf {x}~=~(x_{1},x_{2},x_{3})\) we use finite difference to approximate \(\frac{\partial p_{j}}{\partial x_{k}}\) for \(1 \le j,k \le 3\). We do not evaluate the expressions that involve the partial derivatives of the distance function.
Errors for a portion of circle
n | Relative error | Order |
---|---|---|
64 | \(2.7994\times 10^{-4}\) | – |
128 | \(7.0665\times 10^{-5}\) | 1.99 |
256 | \(1.7187\times 10^{-5}\) | 2.04 |
512 | \(4.2719\times 10^{-6}\) | 2.01 |
1024 | \(1.0636\times 10^{-6}\) | 2.01 |
2048 | \(2.6567\times 10^{-7}\) | 2.00 |
4096 | \(6.6045\times 10^{-8}\) | 2.01 |
8192 | \(1.6513\times 10^{-8}\) | 2.00 |
3.1 Integration of codimension one surfaces
We tested our numerical integration on two different portions of circle, a torus, a quarter sphere and a three quarter sphere. We computed their respective lengths or surface areas by integrating the constant 1 over the curve or surface. Each of these tests were designed to exhibit the convergence rate of our formulations on cases with varying difficulty. In particular, the convergence rate of our formulation depends on the smoothness of the closest point mapping inside the tubular neighborhood of the curve or surface.
Errors for a tilted portion of circle
n | Relative error | Order |
---|---|---|
64 | \(3.7159\times 10^{-5}\) | – |
128 | \(2.5786\times 10^{-7}\) | 7.17 |
256 | \(4.2361\times 10^{-6}\) | \(-4.04\) |
512 | \(3.2246\times 10^{-6}\) | 0.39 |
1024 | \(1.8876\times 10^{-6}\) | 0.77 |
2048 | \(1.0132\times 10^{-7}\) | 0.90 |
4096 | \(5.2372\times 10^{-7}\) | 0.95 |
8192 | \(2.6615\times 10^{-7}\) | 0.98 |
Errors for a torus
n | RE \(_{\varvec{\infty }}\) | Order | RE \(_\mathbf{4 }\) | Order | RE \(_\mathbf{1 }\) | Order |
---|---|---|---|---|---|---|
32 | \(6.2030\times 10^{-3}\) | \(-\) | \(1.1699\times 10^{-2}\) | \(-\) | \(5.8000\times 10^{-2}\) | \(-\) |
64 | \(1.8073\times 10^{-4}\) | 5.10 | \(1.0169\times 10^{-3}\) | 3.52 | \(1.4456\times 10^{-2}\) | 2.00 |
128 | \(6.6838\times 10^{-6}\) | 4.76 | \(1.3568\times 10^{-5}\) | 6.23 | \(3.9830\times 10^{-3}\) | 1.86 |
256 | \(4.1530\times 10^{-7}\) | 4.01 | \(7.1567\times 10^{-7}\) | 4.24 | \(1.4391\times 10^{-3}\) | 1.47 |
512 | \(5.0379\times 10^{-8}\) | 3.04 | \(6.1982\times 10^{-8}\) | 3.53 | \(5.1463 \times 10^{-4}\) | 1.48 |
Errors for a quarter sphere
n | Relative error | Order |
---|---|---|
32 | \(9.2825\times 10^{-3}\) | \(-\) |
64 | \(1.8365\times 10^{-3}\) | 2.34 |
128 | \(2.7726\times 10^{-4}\) | 2.73 |
256 | \(7.1886\times 10^{-5}\) | 1.95 |
512 | \(1.4811\times 10^{-5}\) | 2.30 |
Errors for a three quarter sphere
n | Relative error | Order |
---|---|---|
32 | \(1.1726\times 10^{-2}\) | \(-\) |
64 | \(1.1733\times 10^{-3}\) | 3.32 |
128 | \(9.1325\times 10^{-4}\) | 0.36 |
256 | \(3.8238\times 10^{-4}\) | 1.26 |
512 | \(7.8308\times 10^{-5}\) | 2.29 |
3.2 Integrating along curves in three dimensions
Errors for a coil
n | Relative Error | Order |
---|---|---|
60 | \(5.5078\times 10^{-3}\) | \(-\) |
120 | \(1.1476\times 10^{-3}\) | 2.63 |
240 | \(2.3409\times 10^{-4}\) | 2.29 |
480 | \(3.7166\times 10^{-5}\) | 2.66 |
3.3 One-sided discretization of the Jacobian matrix
Here for completeness, we describe the one-sided discretization used in computing results reported in Table 5. For simplicity we provide the explanation in \(\mathbb R^2\). The discretization generalizes easily to 3D.
4 Summary
In this paper, we presented a new approach for computing integrals along curves and surfaces that are defined either implicitly by the distance function to these manifolds or by the closest point mappings. We are motivated by the abundance of discrete point sets sampled from surfaces using devices such as LIDAR, the need to compute functionals defined over the underlying surfaces, as well as many applications involving the level set method or the use of closest point methods.
Contrary to most other existing approximations using either smeared out Dirac delta functions or locally obtained parameterized patches, we derive a volume integral in the embedding Euclidean space which is equivalent to the desired surface or line integrals. This allows for easy construction of higher-order numerical approximations of these integrals. The key components of this new approach include the use of singular values of the Jacobian matrix of the closest point mapping, which can be computed easily to high order even by simple finite differences.
Declarations
Acknowledgements
Kublik’s research was partially funded by a University of Dayton Research Council Seed Grant and Tsai’s research is partially supported by Simons Foundation, NSF Grants DMS-1318975, DMS-1217203, and ARO Grant No. W911NF-12-1-0519.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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