Curvature estimation for point cloud 2-manifolds based on the heat kernel

1.   Introduction

With the advancement of 3D acquisition technology, the point-based representation of complex objects and environmental point clouds has been extensively utilized in various shape modeling, graphical rendering, and engineering design and manufacturing applications, as cited in ^[1]. With the growth of available data, significant attention has been given to analyzing the original point clouds.

Due to the uncertainty of the exact shape represented by point clouds, curvature estimation, including the two principal curvatures and their related directions, mean curvature and Gaussian curvature, has numerous applications in computer aided design, computer graphics, and other associated fields, and it involves many applications such as surface segmentation (as mentioned in ^[2,3,4]), point cloud simplification^[5], point cloud registration^[6], surface reconstruction^[7,8], and feature extraction ^[9,10]. More generally, a simple and accurate curvature estimation method for point cloud surfaces would ensure reliable numerical behavior for a large number of applications, which is a hot and challenging issue.

In this paper, we presented a series of differential geometry properties for point cloud surfaces based on the heat kernel function. At the core of them is a novel mean curvature normal operator that is based on the discrete Laplace-Beltrami operator derived from the heat kernel function. The proposed method does not need to fit and interpolate point clouds as a local surface to indirectly approximate the geometry properties of point clouds, and it does not need to rely on any local mesh or Voronoi-based map. They only involve the coordinates of point cloud surfaces and are truly mesh-free methods.

1.1.   Related works

Before delving into our contributions, we initially reviewed some previous work, confining our exposition to discretization methods rather than surveying the numerous applications in which they are employed. Surprisingly, though, these study topics have witnessed very distinct developments and we reviewed the existing ones separately.

1.1.1.   Curvature estimation on meshes

The estimation of curvature properties for the mesh surface have been studied for many years and is a crucial topic^[11]. The method commonly applied is to define a concept of curvature for meshes and to study how well they approximate the actual curvature of the underlying surface. Taubin ^[12] estimated the principal curvatures and directions of a triangular mesh surface based on the Euler formulas. Chen^[13] defined some intrinsic surface properties from triangular surface, which is based on the Meusnier and Euler theorem. Dong^[14] presented an algorithm to estimate principal curvatures by simplifying the Chen's method. A classical approximating strategy was presented by Meyer^[15] and they proposed a unified and consistent set of flexible operators to approximate important geometric attributes, including normal vectors and curvatures (Gaussian, mean, and principal) on any triangular mesh. Goldfeather^[16] gave a cubic algorithm to approximate principal direction vectors on mesh surfaces. Goes^[17] constructed a class of discrete differential operators for arbitrary polygon mesh. Based on the local approximation of mesh vertices and associated normals by a quadratic surface, Makovník^[18] introduced a method to estimate the mean and Gaussian curvature and several related quantities for a polygonal mesh.

1.1.2.   Curvature estimation on point clouds

Recently, several methods have been put forward to estimate curvatures and feature information on point cloud surfaces. These schemes can roughly be divided into two categories: Continuous approximation and discrete estimates.

Continuous approximation: These methods are the simplest and give rise to plenty of references. Their first step typically is to reconstruct the point cloud surface into a continuous implicit or parametric form and then compute curvature attributes with classical curvature formulas in differential geometry. Douros^[19] locally fitted the point cloud surface with an analytic representation described by quadric surface patches and then assessed curvature properties at each point. Chen^[20] first made use of an implicit quadric surface to fit the point cloud data and then calculated the curvature according to the curvature formulas of the implicit surface. Spek^[21] presented a fast method by iteratively fitting a parabolic quadric surface to compute principal curvatures from range images. Cheng^[22] fitted the point clouds into a parametric surface to study the point cloud simplification method. Based on principal component analysis, Asao^[23] fitted a quadratic hypersurface by minimizing the square error. ^[24] selected 8 neighbors ring from a point cloud for each point and then they estimated the surface curvatures by fitting the point cloud surface into a quadratic surface.

Discrete estimates: These schemes generally utilize a series of local neighbors of point clouds, such as ball neighbors, $K$-neighbors, or Voronoi cell, to approximate the original point clouds. Based on an analytic energy function in ^[25,26], Yang^[27] directly computed Gasussian and mean curvatures with the curvature formulas in^[28] derived by an implicit function. Similarly, Tian^[29] adopted the same energy function and drew a similar conclusion as Yang^[27]. Moreover, this energy function referred to in ^[25,26] was also used by Miao ^[30] to roughly estimate the curvatures of discrete point clouds. Zhang^[31] presented a robust algorithm to estimate principal curvatures and his basic idea was to locally fit each normal section circle. A more detailed introduction can be found in his other study^[32]. Wang^[33] proposed two different curvature estimation methods with local differential properties called the Voronoi method and MLS projection. Then, Yao^[6] applied Wang's method to develop a point cloud registration algorithm. Based on the Voronoi cells, Quentin^[34] presented an efficient and robust method for extracting curvature information of point clouds. His method is integral in nature and adopts convolved covariance matrices of Voronoi cells. Lachaud^[35] estimated curvature tensor information by generating random triangles. Based on the Weiangarten map, Cao^[36] proposed a direct and efficient method to estimate the curvature. He automatically acquired curvatures from the Weingarten map with a least square fitting method. Lange^[37] also presented a method for anisotropic fairing of a point-sampled surface by applying an anisotropic geometric mean curvature flow.

1.2.   Contributions

Our main intention is to deduce the geometrical properties of point cloud surfaces, including the mean $\kappa_M$, Gaussian $\kappa_G$, two principal curvatures $\kappa_{1, 2}$, and two principal directions $\mathcal{T}_{1, 2}$. Overall, our main contributions can be subdivided into three aspects:

(ⅰ) A new discretized Laplace-Beltrami operator applying an inherent distance parameter for point cloud surfaces is proposed. This operator serves as the basis for accurately estimating the mean curvature.

(ⅱ) We introduce a new computational method aimed at the area of point cloud surfaces. The provided scheme does not rely on any mesh or Voronoi-based map and can simply transform the complex problem of computing the area into solving a series of linear equations.

(ⅲ) A strong constraint condition is introduced to estimate the principal curvatures and directions by finding an optimal ellipse. The results are better than those in ^[31,32] which do not have any constraints.

1.3.   Outlines

The remainder of this paper is structured as follows: Based on the work in Appendix A, Section 2 illustrates the concrete contexts for estimating mean curvatures over a sampled point cloud surface, see Algorithm 1. Section 3 mainly discusses a simple method to find two principal curvatures and related principal directions by finding an optimal ellipse. The corresponding Gaussian curvature is acquired instantly. Some pratical experimental cases are presented in Section 4 to demonstrate the efficiency and accuracy of the proposed methods. Some important conclusions are drawn in Section 5.

2.   Mean curvature

Our method for estimating the mean curvature can be purely depicted in term of the operations carried out on the smooth point cloud surface $\mathcal{M}$ which is sampled by a series of points ${\rm{P}} = \left\{ {{{\rm{x}}_1}, {{\rm{x}}_2}, \cdots, {{\rm{x}}_n}} \right\}$ and embedded within the Euclidean space $\mathbb{R}^3$. Let $\Delta$ be the negative semi-definite Laplace-Beltrami operator that acts on the differential real-valued functions $f:\mathcal{M} \to \mathbb{R}$ over the point cloud surface $\mathcal{M}$. The estimation of mean curvatures with the heat kernel functions involves the following fundamental steps.

Let us commence by considering the approximation of the Laplace-Beltrami operator $\Delta_{pc}$ for the fixed heat diffuse time $t$ over the point cloud surface $\mathcal{M}$. A sufficiently differentiable function $u({\rm{x}}, t)$ satisfies the heat equation if

Given an initial heat distribution $u({\rm{x}}, 0) = f(\rm{x})$, the heat distribution^[38,39,40]

is the solution of the heat equation at heat diffuse time $t$, where the heat kernel function is usually taken as a Gaussian function^[41]:

Subsequently, an approximating operator which has been proved to approximate the actual Laplace-Beltrami operator is constructed ^[42]:

Based on the formula $d{{\rm{x}}_j} = \frac{1}{{\mathcal{A}({\mathcal{M}})}}{\mathcal{A}}({{\rm{x}}_j})$ and Theorem 3.1 in ^[42], we discretized the above expression using a series of neighbor points over a point cloud surface $\mathcal{M}$ with the empirical version of the integrals involved:

where $B(\rm{x})$ is the Ball neighbor of the point $\rm{x} \in \rm{P}$, it satisfies

where $r = \alpha \cdot h(3.5 \le \alpha \le 4.5)$ is the ball radius (see Figure 1) and it usually remains constant for all points ${\rm{p}} \in \rm{P}$ to ensure that the Laplace-Beltrami operator $\Delta_{pc}$ is symmetric. Additionally, $h$ is an inherent distance parameter for a given point cloud surface and it will be further discussed in the subsequent subsection. The above approximating operator also can be rewritten into a matrix form

where the matrix ${{\bf{L}}_{pc}^t} \in {\mathbb{R}^{n \times n}}$ is a Laplace matrix relative to the heat diffuse time $t$ over a point cloud surface $\mathcal{M}$.

2.1.   Redefining the Laplace-Beltrami operator $\Delta_{pc}$ with $h$

The key to the success of Algorithm 1 lies in finding a stable heat diffuse time $t$. Eq 2.3 indicates that a smaller heat diffuse time $t$ will yield a better approximation with respect to the Laplace-Beltrami operator $\Delta_{pc}$ and area $\mathcal{A}$. However, decreasing the quantity of $t$ does not necessarily enhance the approximating accuracy for a fixed sampled point cloud surface $\mathcal{M}$. Therefore, the motivation for step 1 of Algorithm 1 is to search an optimal heat diffuse time $t$ that is neither too large nor too small. In practice, determining an optimal and stable $t$ is difficult and we here propose a simple searching case that works well, that is, $t = mh^2$, where $m > 0$ is a constant and $h$ is a distance parameter that is highly sensitive to the distribution of the sampled point clouds $\rm{P}$.

Typically, to acquire a symmetric Laplace matrix ${{\bf{L}}_{pc}^t}$, $h$ is taken as the mean spacing distance between all points and their nearest ones, which works well in performance for a point cloud surface sampled uniformly and it can be seen as a global distance parameter, that is

But in practice, most of the sampled point cloud surfaces are non-uniform. In this situation, a series of local and self-adaption parameters $h$ for any $\rm{x} \in \rm{P}$ are available, which are taken as the linear combination of the spacing distance between their nearest point and farthest point among their neighbors, that is

where $\gamma \in [0, 1]$ is a linear parameter. That means that the distance parameter $h$ for different points ${\rm{x}}_i \in \rm{P}$ does not necessarily have to been taken as the same value as in Eq 2.6. At this time, the $K$-nearest neighbor is available and it surely causes a non-symmetric Laplace matrix ${{\bf{L}}_{pc}^t}$. Fortunately, this non-symmetric property does not affect the accuracy of Algorithm 1. Moreover, the experimented and convincing constant $m = \frac{1}{4}$ with a high accuracy is adopted (see Appendix A) and then the identity concerning the heat diffuse time $t$ and distance parameter $h$,

is available in all examples. Therefore, by discarding the unstable heat diffuse time $t$, we redefined the approximating Laplace-Beltrami operator ${\Delta _{pc}}f({\rm{x}})$ in Eq 2.4 relative to the inherent and consistent distance parameter $h$ referred to in Eqs 2.6 and 2.7 for a given point cloud surface $\mathcal{M}$:

Meanwhile, the rewritten form $\Delta _{pc}^h$ generates a new Laplace matrix ${{\bf{L}}_{pc}^h} \in {\mathbb{R}^{n \times n}}$ and its symmetric property depends on weather the global neighbors $B_r(\rm{x})$ and distance parameter $h$ are searched.

2.2.   Computation of area

The area $\mathcal{A}$, an essential property in the discretized Laplace-Beltrami operator^[15], is defined based on the physical attributes associated with the underlying problem. In this paper, only the coordinates of the sampled point clouds $\rm{P}$ are available and thus area must be estimated to make the discretization process feasible. However, existing point-based Laplace-Beltrami operator techniques, such as ^[43], resort to the local Voronoi map or tangent spaces at the sampled point clouds $\rm{P}$ as the basic tool to acquire local areas. In contract, our method from step 2 of Algorithm 1 does not require any auxiliary tool to compute each local area $\mathcal{A}_{i}$ aiming at point ${\rm{x}}_{i} \in \rm{P}$, which makes it truly meshless. The reasoning is to estimate ${\mathcal{A}_i}(i = 1, 2, \cdots, n)$ by solving an optimization problem of a linear system derived from the unitary property^[41] of heat kernel functions in Eq 2.2, that is

By utilizing the identity in Eq 2.8 and Appendix A, the above integral can be empirically discretized as

More specifically, this property yields a linear system

for all points ${\rm{x}}_i \in \rm{P}$, where ${H_{ij}} = \frac{1}{{\pi {h^2}}}{e^{ - \frac{{{{\left\| {{x_i} - {x_j}} \right\|}^2}}}{{{h^2}}}}} \ge 0$ is an element of the matrix $\bf{H}$ relative to the neighbor point ${\rm{x}}_j$ of ${\rm{x}}_i$ and $b_i = 1$ is an element of the vector $\bf{b}$. The rationality of applying this linear system to approximate the actual local area can be referred to in Appendix A.

2.3.   Mean curvature normal

In this subsection, we compute the integral of the mean curvature normal over the point cloud surface $\mathcal{M}$. From step 3 of Algorithm 1, we define an operator that maps a point $\rm{x} \in \rm{P}$ to the vector

where $\rm{K}_{pc}$ is the mean curvature normal operator and is also an approximation of the Laplace-Beltrami operator for the point cloud surface $\mathcal{M}$, and $\kappa_{M}$ is the mean curvature. Based on the Laplacian ${\Delta _{u, v}}{\rm{x}} = - {{\rm{x}}_{uu}} - {{\rm{x}}_{vv}}$^[44], it follows that

We intended to discretize the interval of the mean curvature normal operator $\rm{K}_{pc}$. By considering Eq 2.9, a pratical discretization at a point ${\rm{x}}_i \in \rm{P}$ is defined by

Based on this formula, we can easily compute the mean curvature $\kappa_M$ by taking half of the magnitude of this last expression, which is

where the sign of mean curvature $\kappa_{M}$ is determined by the directions of the mean curvature normal $\rm{K}_{pc}$ and normal vector $\bf{n}$. If ${\rm{K}_{pc}} \cdot \bf{n} > 0$, the $\kappa_{M}$ is taken as positive. On the contrary, it takes as negative. For the normal vector $\bf{n}$, principal component analysis (PCA)^[45] is adopted in this paper.

2.4.   Mean curvature as a quadrature

To determine the two principal curvatures and directions at a point ${\rm{x}}_i \in \rm{P}$, we first show the mean curvature $\kappa_M$ from Eqs 2.10 and 2.11, which can be interpreted as a quadrature of normal curvatures:

where $\kappa _{i, j}^N$ is defined as

The radius $R$ of the osculating circle passing through two points ${\rm{x}}_i$ and ${\rm{x}}_j$ can be easily found with the estimated mean curvature in Eq 2.12, see Figure 2. Since we were obliged to have a right angle at the neighbor point ${\rm{x}}_j$ of ${\rm{x}}_i$, we have $({{\rm{x }}_i} - {{\rm{x}}_j}) \cdot ({{\rm{x}}_i} - {{\rm{x}}_j} - 2R\bf{n}) = 0$. This implies

It further proves that $\kappa _{i, j}^N$ is a normal curvature estimated in the direction of edge ${\rm{x}}_i{\rm{x}}_j$. Therefore, Eq 2.11 can be interpreted as a quadrature of normal curvature with weights $w_{ij}$:

where ${w_{ij}} = \frac{1}{{\pi {h^4}}}{e^{ - \frac{{{{\left\| {{{\rm{x}}_i} - {{\rm{x}}_j}} \right\|}^2}}}{{h^2}}}}{\left\| {{{\rm{x}}_i} - {{\rm{x}}_j}} \right\|^2}{\mathcal{A}_j}$ is a weight function relative to the edge ${{\rm{x}}_i}{{\rm{x}}_j}$.

3.   Principal curvature and Gaussian curvature

To calculate the two principal curvatures $\kappa_{1, 2}$ and their related principal directions $\mathcal{T}_{1, 2}$ on a sampled point cloud surface $\mathcal{M}$, finding the optimal fitting ellipse is adopted in this paper, which has been referred to in ^[12,31,32]. Since the mean curvatures $\kappa_M$ have been accurately estimated in the previous subsection, we can utilize the property of the mean curvatures involved, that is, $\kappa_1 + \kappa_2 = 2\kappa_M$, as a strong constraint condition to solve the two eigenvalues and eigenvectors of a symmetric curvature tensor matrix $\bf{B}$. This matrix is composed of three unknown coefficients $a, b$, and $c$:

According to the eigenvalue properties of matrices as shown in ^[46], it can be seen that ${\kappa _1} + {\kappa _2} = a + c$ and ${\kappa _1} \cdot {\kappa _2} = ac - {b^2}$. Moreover, the matrix $\bf{B}$ can be employed to estimate the normal curvatures in any direction on the tangent plane $\mathcal{T}({\rm{x}}_i)$, as seen in Figure 2. In other words, we have that

where $\chi_{i, j} \in \mathbb{R}^2$ is a unit vector relative to the projection $\mathcal{T}_{i, j} \in \mathbb{R}^3$ of the edge ${\rm{x}}_i{{\rm{x}}_j}$ on the tangent plane $\mathcal{T}({\rm{x}}_i)$, which is usually acquired by projecting all the points in $B(\rm{x})$ onto the tangent plane $\mathcal{T}({\rm{x}}_i)$:

The three unknown coefficients $a$, $b$, and $c$ in the matrix $\bf{B}$ can be found by minimizing the error function

along with three constraint conditions:

where $\varepsilon > 0$ is a tiny constant. Subsequently, the two principal curvatures $\kappa_{1, 2}$ of a point ${\rm{x}}_i \in \rm{P}$ can be directly acquired through the two eigenvalues of the tensor matrix $\bf{B}$. Its two eigenvectors in 2D typically correspond to the actual principal directions $\mathcal{T}_{1, 2}$. Eventually, one can directly compute the Gaussian curvature $\kappa_G$ with the conventional differential geometry formula:

4.   Results and applications

Appendix A has already verified the validity of the mean curvatures with the heat kernel function in some analytic models (like the Saddle, Arch, and Paraboloid) and all the results in Figures 3–6 show that the absolute errors of the mean curvatures $\kappa_M$ and the areas $\mathcal{A}$ of the point cloud surfaces $\mathcal{M}$ are lower than $1\%$ in the case of $m = 1/4$. Therefore, unless otherwise specified, $m = 1/4$ will be applied in all examples. Moreover, the linear parameter $\gamma$ in Eq 2.7 belonging to $0.1 \sim 0.5$ will work well in practice in terms of performance. The proposed method can also be adopted to compute mean curvatures on some non-analytic models discretized by large-scale scan point clouds, as seen in Figure 7.

To further prove the validity of the proposed method with the heat kernel function, we tested the Gaussian curvature $\kappa_G$, principal curvatures $\kappa_{1, 2}$, and their directions $\mathcal{T}_{1, 2}$ on some analytic models whose differential properties can be computed exactly with some ready-made curvature formulas. We first sampled a torus function ${(\sqrt {{x^2} + {y^2}} - r_2)^2} + {z^2} = {r_1^2}$ with $r_1 = 1$ and $r_2 = 2.3701$. It cannot be denied that finding an optimal fitting ellipse directly to acquire the Gaussian curvature $\kappa_G$, principal curvatures $\kappa_{1, 2}$, and principal directions $\mathcal{T}_{1, 2}$, such as ^[12,31,32], is fairly robust. But we have provided the mean curvatures $\kappa_{M}$ with a relatively low absolute error (see Figures 5 and 6), which will be adopted to serve as a strong constraint, that is $\kappa_1 + \kappa_2 = 2\kappa_M$ (see Eq 3.1). This constraint is perfectly valid for smooth surfaces, especially closed ones described by point clouds. Figures 8(b) and (c) show the visualized cases for the Gaussian curvature and principal directions in the above sampled Torus model. For the case in this research without the constraint in Eq 3.1, the principle is similar to ^[12,31,32]. Therefore, the comparisions for non-constrained examples are ignored here. But for open ones, we sampled three implicit surfaces and tested the principal directions in Figure 9. As expected, the computed and exact directions match very closely away from the boundary of the point cloud surfaces. Near the boundary of point cloud surfaces, the principal directions estimated by applying the heat kernel function follow the edges of a surface. We here provide another error mechanism aimed at the principal directions $\mathcal{T}_{1, 2}$:

This mechanism is mainly obtained from the average included angle between the exact principal direction $\chi^*$ and the computed ones $\chi$. Table 1 provides the estimating errors of the principal directions $\chi_{1, 2}$ concerning the methods that find an optimal fitting ellipse, which will infer our advantages of possessing a strong constraint $\kappa_1 + \kappa_2 = 2\kappa_M$.

Moreover, to demonstrate the advantages of the proposed methods, some classical schemes, classified as continuous approximation and discrete estimates, respectively, are selected as our comparators. Table 2 has already demonstrated with detail the comparisons of the average absolute error for $\kappa_M$, $\kappa_1$, and $\kappa_{G}$ on implicit open cases referred to in Figure 9, such as the Arch, Saddle, and Paraboloid models with different density $n$. Here, Eq 9.1 is applied. It is not difficult for us to find that the proposed method in the heat kernel has a higher accuracy than the compared references. This is because the references ^[27,29] highly depend on a series of precise normal vectors and they should constantly search for an "optimal" parameter $\mu$ in approximating the actual model curvatures. Usually, finding a suitable $\mu$ is difficult. Besides, the references ^[31,32] estimated the point cloud surface properties by finding a fitting ellipse without any constraint. Therefore, they have a bad performance even on some analytic models.

Since the heat kernel method in Algorithm 1 provides a mean curvature normal operator, it basically provides a precise estimating method for mean curvatures, see Eq 2.11. This strategy can also be used to detect sharp edges. In order to better understand the effect of edge sharpness on the quality of feature detection, we sampled a folded square surface made of two planar rectangular patches joined by a common edge, where the density of the point cloud is $n = 961$. Figure 10 illustrates that feature estimation with the mean curvature is available in detecting sharp edges whose interior angle is equal to $1^{\circ}$. Our method is also applicable to large-scale scanning point clouds. As shown in Figure 11, a comparison of different mean curvature methods for feature detection on a large-scale scanning model (block model) is presented. It is not difficult to find that the proposed method also has a good performance in the application of feature detection.

5.   Conclusions

Based on the heat kernel function, geometry properties such as the Gaussian curvature $\kappa_G$, mean curvature $\kappa_M$, principal curvature $\kappa_{1, 2}$, and their principal directions $\mathcal{T}_{1, 2}$ for sampled point cloud surfaces $\mathcal{M}$ are estimated. The proposed method does not need to locally approximate and fit the point clouds and it should not apply any mesh or Voronoi-based map, which is a truly mesh-free method. Experiments show that the provided curvature estimation method has higher accuracy than the existing methods and has great potential in practical applications. In particular, the mean curvature can be used to detect the features of point clouds even for very small interior angles. Although the mean curvatures using the heat kernel function are accurate, the Gaussian curvature is still acquired by finding an optimal ellipse. In the future, we would like to develop another new operator using the Gauss-Bonnet theorem to derive the Gaussian curvature on point cloud surfaces.

Author contributions

Kai Wang: Designed project, wrote the first draft, provided figures and revised the manuscript; Xiheng Wang: Designed the project and revised the manuscript; Xiaoping Wang: Conceptualized and supervised the project, revised the manuscript, and acquired funding. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

This research was supported by National Natural Science Foundation of China (No. 51575266 and No. 52075258).

Conflict of interest

The authors have no competing interests to declare that are relevant to the content of this article.

Appendix A: Choice of parameter $m$

To find an optimal heat diffuse time $t$ for the point cloud surface $\mathcal{M}$, we suggested representing the heat diffuse time $t$ as a quadratic function $t = mh^2$ in relation to the distance parameter $h$ that is closely related to the sampled distribution of the point clouds themselves. Then, the problem of choosing a parameter $t$ can be transformed into evaluating the constant $m$. This assessment criterion can also be discovered in computing geodesic map with the heat method ^[47], which performs extremely well in practice.

Influence of the constant $m$ on area $\mathcal{A}$

To estimate an ideal $m$, an error evaluating mechanism with respect to illustrating absolute errors of the area $\mathcal{A}$ is defined first:

where the symbol $\mathcal{A}^*$ represents the exact solution and $\mathcal{A}$ is the area sum of the points in $\mathcal{M}$. This mechanism is suitable for point cloud surfaces $\mathcal{M}$ where the superficial area can be easily acquired, which means that the given models have analytic equations, as see in Figure 4. Moreover, Figure 5 shows the visualization for the influences on the estimated area for different parameters $m$ and it reveals that $m = 1/4$ is applicable in computing the area for the point cloud surface $\mathcal{M}$.

Influence of $m$ on mean curvature $\kappa_M$

Our goal is to accurately compute the mean curvatures $\kappa_M$ on the point cloud surfaces $\mathcal{M}$ first and then we need to expand these results to estimate some extra key properties, such as the principal curvatures $\kappa_{1, 2}$, the principal directions $\mathcal{T}_{1, 2}$, and even the Gaussian curvature $\kappa_{G}$. Similar to the area $\mathcal{A}$, the mean curvatures $\kappa_{M}$ are sensitive to $m$. Therefore, we develop another error evaluating mechanism aimed at the mean curvatures $\kappa_M$ of the point cloud surfaces:

where $\kappa_i$ represents the estimated curvature of the point ${\rm{x}}_i \in \rm{P}$ and $\kappa^{*}$ is its related exact solution. This mechanism can also be adopted to evaluate the Gaussian curvature $\kappa_G$, mean curvature $\kappa_M$, and two principal curvature $\kappa_{1, 2}$.

Using some formulas in Example 4.1 in ^[28], some models with analytic mean curvatures, such as the Saddle, Arch, and Paraboloid, are adopted to analyze the influence of the parameter $m$. Figure 5 illustrates some absolute error curves with respect to $m$ and Figure 6 shows the visualization of mean curvature $\kappa_M$ with a different $m$ on a unit sphere. All the experiments in this section suggest that $m = 1/4$ is a continuously stable parameter for the pending function $t = m{h^2}$ and it generates lower errors in approximating the actual area sum and mean curvature, as seen in Figures 3–6. To sum up, $m = 1/4$ is available and the inherent distant parameter $h$ will replace the heat diffuse time $t$ in subsequent computations.