Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing

08/30/2015
by   Gary Pui-Tung Choi, et al.
0

Point cloud is the most fundamental representation of 3D geometric objects. Analyzing and processing point cloud surfaces is important in computer graphics and computer vision. However, most of the existing algorithms for surface analysis require connectivity information. Therefore, it is desirable to develop a mesh structure on point clouds. This task can be simplified with the aid of a parameterization. In particular, conformal parameterizations are advantageous in preserving the geometric information of the point cloud data. In this paper, we extend a state-of-the-art spherical conformal parameterization algorithm for genus-0 closed meshes to the case of point clouds, using an improved approximation of the Laplace-Beltrami operator on data points. Then, we propose an iterative scheme called the North-South reiteration for achieving a spherical conformal parameterization. A balancing scheme is introduced to enhance the distribution of the spherical parameterization. High quality triangulations and quadrangulations can then be built on the point clouds with the aid of the parameterizations. Also, the meshes generated are guaranteed to be genus-0 closed meshes. Moreover, using our proposed spherical conformal parameterization, multilevel representations of point clouds can be easily constructed. Experimental results demonstrate the effectiveness of our proposed framework.

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1 Introduction

Contemporary scanning technologies enable efficient acquisitions of 3D objects. Using modern 3D scanners, data points are sampled from the surfaces of 3D objects for further analyses and usages. Point clouds are widely applied in computer graphics, vision and many other engineering fields. However, the data points acquired by laser scanners are often complex and unorganized. Moreover, the absence of the connectivity information in point cloud data poses difficulties in understanding the underlying geometry of the 3D objects. This largely hinders the applications of the data. For instance, many applications in 3D printing [38, 27] and texture mapping [39, 24] are built upon mesh structures. With the rapid development of the computer industry, finding a high quality meshing framework for point cloud data is increasingly important.

One possible approach for mesh generation on point clouds is to parameterize a point cloud to a simpler domain with the corresponding genus, such as the unit sphere for genus-0 point clouds. Then, a triangulation or a quadrangulation can be created on the parameter domain instead of the original complicated point cloud. Finally, a mesh structure on the point cloud can be defined with respect to the structure on the parameter domain. The major difficulty of computing parameterizations of point-set surfaces is the extremely limited information they can provide. Most of the existing surface parameterization methods are developed on meshes only. In other words, besides the locations of the point data, a given connectivity is also required as an input. The connectivity information plays an important role in representing the surface structure as well as in approximating continuous operators to minimize certain distortions. As a result, most conventional mesh parameterization approaches fail to work on point clouds. Without the connectivity information, the underlying geometry of the point cloud data become more obscure. Hence, it is more challenging in developing parameterization schemes with good quality for point cloud data.

A good parameterization scheme of point cloud must satisfy certain criteria. In particular, it should retain the geometric information of a point cloud as complete as possible. In our case, one of the ultimate goals is to create a triangulation for a point cloud by finding a Delaunay triangulation on a simpler parameter domain. It is noteworthy that in general, a mesh structure with good quality on the parameter domain does not necessarily imply that the associated mesh structure of the original data points is satisfactory. In other words, meshing the parameter domain may provide meaningless results if the parameterization scheme is arbitrarily chosen. Note that the regularity of the mesh structures is related to the angle structure of the triangles and quadrilaterals. To ensure the regularity of the associated mesh structure on the point cloud, the parameterization should preserve the angle structure of the triangles and quadrilaterals on the parameter domain. This motivates us the use of conformal mappings.

For smooth surfaces, it is well known that the conformal parameterizations preserve angles and hence the local geometry of the surfaces. It is natural to consider the discrete analog of conformal parameterization for point cloud data. Since data points are sampled from real 3D surfaces, we can assume that every point cloud has an underlying geometry. Based on this important assumption, we consider finding conformal parameterizations of genus-0 point clouds. In [6], Choi et al. proposed a fast spherical conformal parameterization algorithm for genus-0 closed surfaces in two steps. In the first step, a Laplace equation is solved on a planar triangular domain and the inverse stereographic projection is applied to obtain an initial spherical parameterization. In the second step, quasi-conformal theories are applied to enhance the conformality of the spherical parameterization. The computation is linear and the conformality distortion of the parameterization is minimal. However, the algorithm is developed on triangular meshes only. In this work, we extend and improve the algorithm for point clouds with spherical topology.

The aforementioned algorithm in [6] developed on meshes involves solving a Laplace equation. To extend the algorithm for point clouds, we propose a new weight function for enhancing the accuracy of the approximation of the Laplace-Beltrami (LB) operators on point clouds. Using our improved approximation, the LB operator in the mentioned algorithm can be accurately computed on point clouds. Also, we replace a key step of solving for a quasi-conformal map in the mentioned algorithm by an iterative scheme, called the North-South reiteration, for improving the conformality of the parameterizations. Furthermore, we introduce a balancing scheme for enhancing the distribution of the parameterization results. Experimental results demonstrate the effectiveness of our proposed parameterization algorithm for genus-0 point clouds. Our algorithm achieves global spherical parameterizations with minimal conformality distortions. Furthermore, with the aid of our parameterization schemes, we can easily generate high quality triangulations and quadrangulations on point clouds. The meshes generated are guaranteed to be genus-0 closed meshes. Moreover, multilevel representations of the point clouds can also be easily computed with the aid of our spherical parameterization scheme.

The rest of the paper is organized as follows. The contribution of our work is highlighted in Section 2. In Section 3, we review the related previous works on point cloud parameterizations and approximations of differential operators on point clouds. In Section 4, we introduce the mathematical background of our work. In Section 5, we review a spherical conformal parameterization scheme for triangular meshes, which is closely related to our proposed framework for point clouds. In Section 6, we explain our proposed framework for spherical conformal parameterization and mesh generation of point clouds. In Section 7, we demonstrate the effectiveness of our proposed framework by numerous experiments. The paper is concluded in Section 8.

2 Contribution

In this work, we propose a framework for meshing using spherical conformal parameterizations of genus-0 point clouds. Our proposed method is advantageous in the following aspects:

  1. We extend and improve the spherical conformal parameterization algorithm on meshes in [6] for point clouds. An accurate approximation of the Laplace-Beltrami operator is achieved using the moving least square method [63, 25, 26] together with a new Gaussian-type weight function. A key step of the parameterization algorithm in [6] for computing a quasi-conformal map is replaced by solving a Laplace equation on the complex plane, followed by an iterative scheme called the North-South (N-S) reiteration. Also, the point distribution of the parameterization is enhanced by a balancing scheme for point clouds.

  2. Our spherical parameterization method is efficient and robust to complex geometric structures. The algorithm completes within a few minutes and can handle highly convoluted point clouds.

  3. Unlike most of the existing approaches, our algorithm specifically minimizes the conformality distortion of the parameterizations. Since the local geometry is preserved under the global spherical conformal parameterizations, we can create an almost-Delaunay triangulation on a point cloud by computing a Delaunay triangulation of its spherical conformal parameterization. The resulting triangulation on the point cloud preserves the regularity of that on the parameterization.

  4. High quality quad meshes can also be generated using our spherical conformal parameterization scheme.

  5. Unlike the conventional approaches for meshing, our method is topology preserving. The meshes produced using our proposed framework are guaranteed to be genus-0 closed meshes. No post-processing is required.

  6. Our method is stable under geometrical and topological noises on the input point clouds.

  7. With the aid of our spherical conformal parameterization scheme, multilevel representations of genus-0 point clouds can be easily constructed.

Methods Topology Parameter domain Local/global parameterization? Distortion to be minimized
Meshless parameterization [10, 11] Disk topology Plane Global /
Meshless parameterization for Spherical Topology [18] Genus-0 Planes Local /
Spherical embedding [50] Genus-0 Sphere Global Stretch
Discrete one-forms [44] Genus-1 Planes Local /
As-rigid-as-possible meshless parameterization [48] Disk topology Plane Global ARAP
Meshless quadrangulation by global parameterization [62] Unrestricted Plane Global Gradient and principal fields
Table 1: Several previous works on meshing point clouds using parameterization.

3 Previous Works

In this section, we describe some previous works closely related to our work.

Surface parameterization has been extensively studied by different research groups. For surveys on surface parameterization methods, please refer to [12, 13, 19, 42]. In particular, conformal parameterizations have been well established on meshes. For the recent works, Lai et al. [21] proposed a folding-free spherical conformal mapping scheme by the harmonic energy minimization. In [56], Aflalo et al. proved theoretical bounds of the conformal factor and proposed a method that minimizes the area distortion and avoids numerical errors of the conformal mapping. In [6], Choi et al. developed a linear algorithm for spherical conformal parameterizations of genus-0 closed meshes.

In the last few decades, numerous studies have been devoted to the parameterization of point cloud data. In [10, 11], Floater and Reimers proposed the meshless parameterization method for unorganized point sets. The point sets are parameterized onto a planar domain by solving a sparse linear system. In [50], Zwicker and Gotsmann presented a parameterization approach for a genus-0 point cloud using a -nearest neighborhood graph of the point cloud, followed by a spherical embedding method for planar graphs. In [1, 2, 3], Azariadis and Sapidis introduced the notion of dynamic base surfaces and suggested a parameterization scheme by orthogonally projecting a point cloud onto the dynamic base surface. Guo et al. [16] computed a global conformal parameterization of point-set surfaces, based on Riemann surface theory and Hodge theory. In [44], Tewari et al. proposed a doubly-periodic global parameterization of point cloud sampled from a closed surface of genus 1 to the plane, with the aid of discrete harmonic one-forms. Wang et al. [45] suggested a parameterization method for genus-0 cloud data. A point cloud is first mapped onto its circumscribed sphere, then the sphere is mapped onto an octahedron and finally unfolded to a 2D image. In [48], Zhang et al. presented an as-rigid-as-possible parameterization approach for point cloud data. A point cloud with disk topology is mapped onto the plane by a local flattening step and a rigid alignment. In [25], Liang et al. constructed spherical conformal mappings of genus-0 point clouds by adapting the harmonic energy minimization algorithm in [21]. Meng et al. [34]

proposed a neural network based method for point cloud parameterization. An adaptive sequential learning algorithm is applied to dynamically adjust the parameterization.

The use of parameterization of point cloud is widespread in computer science and engineering. One of the major applications of point cloud parameterization is mesh generation. Instead of a convoluted point cloud, mesh reconstruction is usually completed on a simpler parameter domain. In [10, 11], Floater and Reimers applied their proposed parameterization scheme for meshing point clouds with disk topology. In [18], Hormann and Reimers extended the parameterization method in [11] for surface reconstruction of point clouds with spherical topology. In [50], Zwicker and Gotsmann used their proposed parameterization method for mesh reconstruction of genus-0 point clouds. Tewari et al. [44] performed surface reconstruction using their proposed doubly-periodic global parameterization. Li et al. [62] proposed a meshless quadrangulation scheme by global parameterization. The input point cloud is cut to be with disk topology and parameterized onto the plane for meshing. Zhang et al. [48] suggested a mesh reconstruction method of point cloud data by meshless denoising and their proposed parameterization scheme. Table 1 compares several previous works on meshing point clouds using parameterizations. The above previous works reflect the importance of parameterization in surface reconstruction of point cloud data.

Finding a conformal parameterization involves solving differential equations. In particular, for conformal parameterizations of point clouds, it is necessary to build a discrete analog of the differential operators on point clouds. Numerous works on approximating differential operators on point cloud have been reported. In [35], Nayroles et al.

described a diffuse approximation method for estimating the derivatives at a given set of points. In

[57], Belkin and Niyogi established a theoretical foundation for the Laplace-Beltrami operator on point clouds. Belkin et al. [4] proposed the PCD Laplace operator for approximating the LB operator using an integral approximation. The moving least square (MLS) method [43, 22] is widely used for the approximation. A number of algorithms for the approximation of derivatives are developed based on the MLS method [28, 23, 36, 5]. In [63], Lange and Polthier proposed a point set analogue of the Laplace-Beltrami and shape operator using the MLS method. In [25, 26], Liang et al. approximated the LB operator on point clouds by the MLS method with a special weighting function. In [20], Lai et al.

presented a local mesh approach for solving PDEs on point clouds. A local mesh structure is constructed at each point using local principal component analysis (PCA). Macdonald

et al. [33] computed reaction-diffusion processes on point clouds. In [31], Lozes et al. proposed a method to solve PDEs on point clouds for image processing using partial difference operators on weighted graphs.

4 Mathematical background

In this section, we introduce some basic mathematical concepts closely related to our work. For more details, readers are referred to [40, 41, 55].

4.1 Conformal maps

An altas of a manifold is said to be conformal if all of its transition maps are biholomorphic. A conformal structure is the maximal conformal altas, and a surface with a conformal structure is called a Riemann surface. Suppose , are two Riemann surfaces with local coordinate systems and , where

are vector-valued functions. The first fundamental forms of

and are respectively defined by

(1)

where , . Consider . In local coordinate systems, can be regarded as , with . The pull-back metric defined on , induced by and , is the metric

(2)

is said to be conformal if there exists a positive scalar function , called the conformal factor, such that . An immediate consequence of the above is that every conformal map preserves angles and hence the local geometry of the surface.

4.2 Harmonic maps

By the uniformization theorem, every genus-0 closed surface is conformally equivalent to . Hence, it is natural to consider mappings between a genus-0 closed surface and the unit sphere. The Dirichlet energy for a map is defined as

(3)

In the space of mappings, the critical points of are called harmonic mappings. For genus-0 closed surfaces, conformal maps are equivalent to harmonic maps [55]. Hence, the problem of finding a conformal map is equivalent to an energy minimization problem.

4.3 Quasi-conformal maps

Quasi-conformal maps are a generalization of conformal maps. Mathematically, is a quasi-conformal map if it satisfies the Beltrami equation:

(4)

for some complex-valued function satisfying and is non-vanishing almost everywhere. Here, the complex partial derivatives are defined by

(5)

is called the Beltrami coefficient of the quasi-conformal map . Note that the quasi-conformal map is conformal around a small neighborhood of if and only if , as Equation (4) becomes the Cauchy-Riemann equation in this situation.

Suppose and are quasi-conformal maps with the Beltrami coefficients and respectively. Then, the Beltrami coefficient of the composition map is explicitly given by

(6)

Quasi-conformal maps are also defined between two Riemann surfaces and . A Beltrami differential on is an assignment to each chart of an complex-valued function defined on the local parameter such that on the domain also covered by another chart , where and . An orientation preserving diffeomorphism is called quasi-conformal associated with if for any chart on and any chart on , the mapping is quasi-conformal associated with . Readers are referred to [14] for more details of quasi-conformal maps.

4.4 Stereographic projection

In our work, we frequently make use of the stereographic projection. Mathematically, the stereographic projection is a conformal map with

(7)

The inverse stereographic projection is a conformal map with

(8)

Similarly, we define the south-pole stereographic projection by

(9)

The inverse south-pole stereographic projection is the map with

(10)

4.5 Point cloud and local system

A point cloud is a set of sample points representing a Riemann surface . Because of the absence of the connectivity information, we construct a local coordinate system for on each point and approximate the derivatives. To achieve this, We define an atlas for each point , where is an open cover and is the associated local coordinate function. is formed using the collection of all neighboring points of , denoted by . Specifically, we apply the -Nearest-Neighbors (-NN) algorithm to define the neighborhood. The -nearest neighborhood of is a set with the distinct elements in (including ) closest from under the Euclidean 2-norm. In this work, we apply the KD-tree implementation by Lin [65] for the computation. We denote with . Then, one common approach for constructing a local coordinate system is to define the normal vector as the -axis, which is more convenient for further computation. There are various methods to obtain the tangent planes and the normal vectors for point clouds, such as the principal component analysis (PCA) method [17]. Using the PCA method for , we obtain three vectors which form an orthonormal basis of .

Then, we project to the plane spanned by by . Now we have the projection and the local coordinates , where and for . Therefore, we can define by . Also, the neighborhood can be regarded as a graph of its projection , that is, .

5 An overview of the fast spherical conformal parameterization algorithm for triangular meshes

In this section, we briefly describe the approach in [6] for computing a spherical conformal parameterization of a genus-0 closed triangular mesh . This approach motivates our proposed parameterization scheme for genus-0 point clouds.

To compute a conformal mapping , it suffices to solve Equation (3). This can be achieved by solving the Laplace equation subject to , where is the tangential component of on the tangent plane of . This tangential approach was applied by Oberknapp and Polthier in [64]. Note that this problem is nonlinear because of the constraint . In [53, 54], Angenent et al. linearize this problem by solving the equation on the complex plane:

(11)

given three boundary constraints , where such that the triangle and the triangle are with the same angle structures. Note that since the target domain is now . As the nonlinear constraint is removed, the above problem becomes linear and can be solved using the cotangent formula [37].

After solving Equation (11), the inverse stereographic projection is applied for obtaining a spherical parameterization. However, unlike in the continuous case, the spherical parameterization in the discrete case is with large conformality distortion at the north pole of the sphere due to the discretization and the approximation errors. Hence, Choi et al. [6] proposed to apply the south-pole stereographic projection to map the sphere to a planar domain . Note that the region with large distortion is the innermost region of while the outermost region of is with negligible distortion. Denote the above steps by a map . To correct the distortion of , Choi et al. made use of the quasi-conformal theory.

Let be the Beltrami coefficient of the map . Fixing the outermost region on , Choi et al. [6] composed the map with a quasi-conformal map with the associated Beltrami coefficient . Let and . Specifically, by considering the Beltrami Equation (4), each pair of the partial derivatives and can be expressed as linear combinations of the other [61],

(12)

where . Since and , the map can be constructed by solving the following equations

(13)

where . In the discrete case, the above elliptic PDEs (13) can be discretized into sparse symmetric positive definite linear systems as described in [32, 6]. In [58], Jones and Mahadevan derived the system (12) from the conjugate Beltrami equation and proposed an alternative approach for solving the system. Specifically, the authors considered minimizing the following functional

(14)

using a Euler-Lagrange variational approach. Despite the different implementations, both of the two abovementioned methods effectively solve for a quasi-conformal map on triangulated meshes. Then, by Equation (6), the composition map is with the Beltrami coefficient and hence is conformal. Readers are referred to [6] for more details.

Note that a key step above is the computation of the quasi-conformal map for improving the conformality, which is guaranteed by the composition formula (6). However, the Beltrami coefficients in the above algorithm are approximated on the triangular faces of a mesh. Hence, the above algorithm cannot be directly applied for point clouds. Moreover, even if we can define the discrete Beltrami coefficients on point clouds, Equation (6) may not hold anymore. Therefore, we need to replace this key step by a new method suitable for point clouds.

6 Meshing genus-0 point clouds using spherical conformal parameterization

In this section, we discuss our proposed framework for meshing genus-0 point clouds. The main steps involved include solving a series of Laplace equations on the complex plane for the spherical conformal parameterization of a genus-0 point cloud, and creating a mesh structure with the aid of the global parameterization.

6.1 Approximation of the Laplace-Beltrami operator

Weight Formula of
Constant weight
Exponential weight
Inverse of squared distance weight
Wendland weight [46, 47]
Special weight [25]
Table 2: Some common weighting functions for the MLS approximation.

In this subsection, we explain our approximation scheme for the Laplace-Beltrami operator in the Laplace Equation (11) on a point cloud by the moving least-square method. The moving least-square method is widely used for approximation [28, 23, 36, 5, 63, 25, 26]. In particular, Liang et al. [25, 26] approximated the LB operator on point clouds using the MLS method with a special weight function. Our approximation scheme is built upon the method in [63, 25, 26]. In this work, we propose a new weight function to achieve a more accurate approximation of the LB operator.

First, we discuss our approximation method for the derivatives on the point cloud . To simplify the discussion, we only consider the approximation on the patch of a point . Recall that can be regarded as a graph of its projection , that is, . Denote the derivatives of along the -direction and the -direction by and respectively. We select a set of basic functions as a basis and write , where are some coefficients to be determined. In our work, we use as the basis of the space of all polynomials with second order or below, which means . We add a remark here that is an appropriate choice for our approximation. Since second derivatives are considered in approximating the LB operator, polynomials with at least second order are needed. On the other hand, if we fit a polynomial with third order () or higher, it will be too sensitive to noises and the approximation gets worse. Therefore, is a suitable dimension for our approximation.

In the approximation, we aim to minimize

(15)

where for some weighting function . The weight function significantly affects the accuracy and robustness of the approximation. Hence, one must carefully choose a suitable weight function. Table 2 lists some common weighting functions.

Note that the information provided by the data points near the center point should be more reliable than that of the data points distant from . The closer the data points are to , the more reliable they are. Hence, it is natural to consider a smooth weight function which concentrates at . This motivates us to use of a weight function of the Gaussian type. As a remark, in [57, 4], Belkin et al. used a Gaussian weight function in the form of for integral approximation. In our MLS approximation, we propose another Gaussian-type weight function:

(16)

where is the maximum distance from in . Numerical experiments are demonstrated in Section 7 to support our proposed weight function with the specific factor inside the exponent. It can be observed that our proposed weight results in more accurate approximations of the LB operator on point clouds.

With the proposed weight function, we now solve the minimization problem (15). Denote and .

Let , , , and . The minimization problem in (15) can be written as follows:

(17)

We can solve it using the least-square method, namely solving

(18)

Next, for any function defined on the neighborhood , we can approximate it by a combination of :

(19)

Similarly, the coefficients can be approximated. Let , , , and . We can find the coefficients by solving the following least-square problem

(20)

Since we know the explicit formula of the derivatives of each , we can compute the approximated derivatives of , such as

(21)

Now, we are ready to introduce the construction of the LB operator of a smooth function on . For any smooth real-valued function on the , the LB operator of is given by

(22)

where is a point in , is the metric of the surface at , , and .

Since and is a graph of , we have

(23)

where .

We use Equation (21) to calculate the first order partial derivatives of . Then, we proceed to compute . Since we have a closed form of and the LB operator is a second order differential operator, by differentiating Equation (22), we get

(24)

where are coefficients which depend on partial derivatives of . This completes our approximation scheme for the LB operator on point clouds. With this approximation, we are now ready to describe our proposed spherical conformal parameterization algorithm for genus-0 point clouds.

6.2 Spherical conformal parameterization of genus-0 point clouds

In this subsection, we introduce our proposed method for the spherical conformal parameterizations of genus-0 point clouds.

Given a point cloud sampled from a genus-0 closed surface , our goal is to find a conformal map which effectively resembles the conformal map . By the previous section, we can approximate the LB operator on . Denote the approximated LB operator on the point cloud by . The approximation allows us to solve the Laplace equation (11) on point clouds for a map . More specifically, we solve the following equation

(25)

subject to the constraints for , where . The choice of the three boundary points affects the conformality of the map . In the case of triangular meshes, are chosen to be the three vertices of the most regular triangle among all triangles on the input mesh [6]. Here, the regularity of a triangle is defined by

(26)

where and are the three angles in the triangle . However, in the case of point clouds, we do not have the required connectivity information. Hence, we choose the three points in a different way.

Recall that in approximating the LB operator, it is necessary to find the nearest neighboring data points for each point on the point cloud . We consider forming a triple using and two other neighboring points and , where . Different combinations of and result in different triples . Then, we propose to choose the three boundary points in the constraint of Equation (25) by considering

(27)

among all combinations of , and .

After solving Equation (25) with our proposed boundary constraints, we apply the inverse stereographic projection on to obtain a spherical point cloud. Recall that the conformality distortion around the north pole is large due to the approximation error in the stereographic projection. Note that the key step in the method in [6] for correcting the distortion via a composition of quasi-conformal maps does not work for the case of point clouds. Now, we propose a new method to correct the conformality distortion by solely using the LB operator.

We begin with the south-pole stereographic projection to project the spherical point cloud back onto the complex plane. Under the projection, the North pole of the sphere, which corresponds to the outermost region of , is mapped to the innermost region on the complex plane. It follows that the outermost region is now with very low distortion while the innermost region is with large distortion. We use the outermost low-distortion data points as the boundary constraints and solve the Laplace equation again:

(28)

subject to the boundary constraints for all data points in the outermost low-distortion region. The low-distortion boundary constraints provide us with a more accurate result in the inner part of the planar region. Finally, we apply the inverse south-pole stereographic projection and obtain a composition map

(29)

This step effectively replaces the step in the mesh parameterization algorithm in [6] which involves computing a quasi-conformal map.

Altogether, by solving Equation (25) and Equation (28) and using a number of projections, we can obtain a conformal map . Note that the method in [6] is based on certain manipulations of Beltrami coefficients and quasi-conformal maps. In contrast, our new method only involves solving Laplace equations. The equivalence between the two approaches can be explained as follows.

In the first step, the conformality distortion of the spherical parameterization is due to the error in the stereographic projection. Then in the approach in [6], the entire initial parameterization result is used in Equation (13) for computing a quasi-conformal map in order to cancel the distortion. The method is theoretically guaranteed by the composition formula (6) of quasi-conformal maps. In contrast, in our new approach, we only make use of the most accurate part in the initial parameterization result. More explicitly, we use the southern-most regions as the boundary constraints and compute the remaining part of the spherical parameterization again, with the aid of the LB operator. The replacement of the south-pole step in [6] by our new south-pole step can be justified by the following theorem.

Let and be two Riemann surfaces, and is a prescribed Beltrami differential on . Then, the map solved by Equation (13) is a harmonic map between and . Consequently, solving the Laplace equation (28) is equivalent to solving Equation (13).

Let be the coordinates of with respect to the distorted metric . Then, the harmonic map between and is a critical point of the following energy

(30)

On the other hand, by definition of the Beltrami equation (4), the solution to Equation (13) is the critical point of the following energy functional

(31)

It is shown in [60] that the above two energy functionals have the same set of critical points. Hence, solving the generalized Laplace equation (13) for a quasi-conformal map is equivalent to solving the Laplace equation (28) for a harmonic map under the distorted metric. Then, the conformality of our approach is again guaranteed by the composition formula (6) of quasi-conformal maps.

Therefore, in the continuous case, under suitable boundary conditions in Equation (13) and Equation (28), both of the two methods are theoretically guaranteed for producing a conformal map.

However, in the discrete case, the two methods perform differently. For the case of triangular meshes, the Beltrami coefficients can be accurately computed and the composition formula (6) of quasi-conformal maps is accurate under the discretization. In this situation, the method in [6] can be effectively applied. Yet, for the case of point clouds, we only have an approximation of the LB operator but not the Beltrami coefficients, and there is no guarantee about the composition formula (6) of quasi-conformal maps. Consequently, it is more suitable to use our proposed method as it only involves solving the Laplace Equation (28). However, since the accuracy of our proposed method depends on the boundary constraints in solving the Laplace Equation (28), the boundary constraints obtained from the initial parameterization result may contain small error and hence slightly affect the result in solving Equation (28). Therefore, it is desirable to perform some more iterations for obtaining a more accurate result.

It is noteworthy that in the parameterization algorithm in [6] for triangular meshes, no further steps are required after the second step. However, because of the abovementioned issue about the boundary constraints in the Laplace Equation (28), further iterations are necessary for enhancing the parameterization result. We call them the North-South (N-S) reiterations. In each N-S reiteration, two Laplace equations are solved again after the north-pole stereographic projection and the south-pole stereographic projection respectively. For solving each Laplace equation, we fix the outermost points on to ensure the existence of the solution.

More specifically, in each N-S reiteration, we first project the previous spherical parameterization result onto the complex plane using the north-pole stereographic projection. Next, we compute a harmonic map by solving the Laplace equation

(32)

with the boundary constraints for the outermost of the data points on . After obtaining , the inverse north-pole stereographic projection is again applied, followed by the south-pole stereographic projection. Then, we compute another harmonic map by solving the Laplace equation

(33)

with the boundary constraints for the outermost of the data points on . We then define the updated spherical parameterization by the composition map

(34)

We check whether the above updated parameterization result is close to the previous parameterization result . If yes, then the parameterization is stable and we complete the algorithm. If no, we apply another N-S reiteration on the updated parameterization point by repeating the procedures and so on. In practice, we choose . Our proposed spherical conformal parameterization scheme is outlined in Algorithm 1.

To explain the motivation of our proposed N-S reiteration scheme, we define the N-S Dirichlet energy by

(35)

where is the Dirichlet energy. It follows that the minimum of is attained if and only if and are minimized, which implies that is minimized and is conformal. Therefore, to find a conformal , we can consider minimizing the N-S Dirichlet energy . More specifically, we aim to minimize both and . Note that these two energies are respectively minimized by solving the Laplace equations (32) and (33) in our proposed N-S reiteration. Introducing the N-S reiteration for minimizing the energies is advantageous for two reasons. Firstly, it linearizes the computation as we only need to solve linear systems on . Secondly, it avoids the error induced by the stereographic projection as we consider both the north-pole step and the south-pole step in each reiteration.

Note that theoretically we only need to take away the two points of infinity, and at least 3 points on are required to be fixed to guarantee the existence of the solution of the two Laplace equations. However, in terms of the numerical computations, the large matrix equations may be ill-posed if only 3 points are fixed as boundary constraints. Therefore, we consider fixing the outermost of data points in solving Equations (32) and (33). Nevertheless, fixing these extra points may not affect the accuracy of the solution too much. This can be explained with the aid of the Beltrami holomorphic flow [14] as follows.

[Beltrami holomorphic flow on [14]] There is a one-to-one correspondence between the set of quasi-conformal diffeomorphisms of that fix the points and and the set of smooth complex-valued functions on with . Furthermore, the solution to the Beltrami equation (4) depends holomorphically on . Let be a family of Beltrami coefficients depending on a real or complex parameter . Suppose also that can be written in the form

(36)

for , with suitable in the unit ball of such that as . Then, for all ,

(37)

locally uniformly on as , where

(38)

In our case, as the conformality distortion of the outermost region is negligible, is compactly supported around origin. Hence, it can be deduced from Equation (38) in the above theorem that the data points located farther away from the origin will be associated with a smaller flow , as the denominator in the integral becomes larger. Therefore, in each iteration, the outermost points will remain almost unchanged, while the innermost points (which have the largest conformality distortion) will be adjusted and improved. In other words, fixing more outermost points for the numerical stability does not affect the solution much. Numerical experiments are presented in Section 7 to verify the convergence of our N-S reiteration scheme. Intuitively, the boundary constraints in Equations (32) and (33) are adjusted to the positions associated with a conformal map by the iterations. They are observed to eventually stabilize and hence we obtain the desired conformal map by solving Equations (32) and (33) with these boundary constraints.

Finally, we make an important remark about our proposed spherical conformal parameterization algorithm for genus-0 point clouds. In addition to genus-0 po