Approximation of Offset Surfaces Using Generalized Wendland Radial Basis Functions

Abstract

We introduce a new methodological approach for the approximation of generalized offset surfaces using smoothing radial basis functions (RBFs). Existing offset surface generation methods often exhibit limitations such as self-intersections and singularities, which affect their robustness and accuracy. To address these issues, we employ generalized Wendland radial basis functions, which are compactly supported and provide enhanced stability. We establish the existence and uniqueness of the solution of the proposed method, analyze its computational aspects, and prove convergence results. Finally, numerical experiments are presented to demonstrate its effectiveness as well as to compare it with an existing method from the literature.

1. Introduction

Offset surfaces play a fundamental role in geometric modeling and computer-aided design (CAD), where they are used to generate surfaces at a prescribed distance from a given reference geometry. They are widely applied in tool path generation, tolerance analysis, and shape optimization. However, the construction of offset surfaces for complex geometries often leads to significant analytical and numerical difficulties, including singularities and self-intersections [1,2]. A detailed discussion of these challenges can be found in [3].

In addition to these traditional CAD applications, scattered data approximation has become increasingly important in contemporary research and industry. In particular, surface reconstruction from 3D point clouds, obtained from scanners or imaging devices, is a fundamental problem in reverse engineering, medical imaging, and industrial inspection. These data sets are often irregularly distributed and corrupted by noise, making classical mesh-based or globally supported methods less effective. This highlights the necessity for efficient, robust, and meshfree approaches capable of handling scattered data in realistic scenarios.

Classical approaches for offset surface approximation are mainly based on spline techniques. Hoschek and Schneider investigated spline approximation of offset curves and surfaces, while Akhrif et al. [1] proposed bicubic splines for generalized offset surfaces. Although effective, these methods rely on mesh-based representations and may lack flexibility when dealing with scattered or irregular data.

Radial basis functions (RBFs) offer an alternative framework for surface approximation. In particular, compactly supported RBFs introduced by Wendland [4] provide computational efficiency due to sparsity. The extension of these functions to even-dimensional spaces was addressed by Schaback [5], ensuring their applicability in a broader setting.

Several works highlight the flexibility and robustness of RBFs in various applications beyond offset surfaces. Narkawich et al. [6] established Sobolev bounds for scattered zeros, Bilal et al. [7] proposed meshless collocation for discontinuous coefficients, Fortes et al. [8] applied RBFs to 3D data sets with volume constraints, and González-Rodelas et al. [9] addressed fuzzy data approximation, other references following the same line can be consulted such as [10,11]. Although these works do not directly focus on offset surfaces, they demonstrate the adaptability and stability of RBF-based approaches in handling irregular or noisy data.

In this context, the choice of the basis functions plays a crucial role. Generalized Wendland $RBFs$ combine compact support, locality, and strong approximation properties. Their use leads to sparse systems, improved numerical stability, and efficient handling of scattered data. Moreover, their well-defined construction in even dimensions ensures the theoretical consistency of the approximation framework.

The present work proposes a smoothing approach for the approximation of generalized offset surfaces using compactly supported generalized Wendland $RBFs$. The method combines meshfree flexibility with improved computational efficiency. A convergence analysis is provided, and numerical experiments illustrate the effectiveness of the approach. Furthermore, the estimated approximation errors are compared with those obtained in [1], allowing a quantitative evaluation of the proposed method.

To position the proposed approach within the context of existing research, we next review relevant contributions on offset surface approximation using both spline-based and radial basis function methods.

2. Related Work

The approximation of offset surfaces has been extensively studied in geometric modeling and computer-aided design. Early contributions focused on spline-based techniques. Hoschek and Schneider [12] investigated the approximation of offset curves and surfaces using classical spline methods, highlighting both their effectiveness and limitations in the presence of singularities and complex geometries. Akhrif et al. [1] proposed bicubic spline techniques for generalized offset surfaces, while Kouibia and Pasadas [2] developed biquadratic spline methods to reconstruct offset surfaces with holes.

Radial basis functions (RBFs) provide a meshfree alternative for surface approximation. Iske [13] studied scattered data approximation using positive definite kernels, emphasizing stability and convergence properties. Wendland [4] introduced compactly supported RBFs, which improve computational efficiency by generating sparse system matrices. Schaback [5] extended these constructions to even-dimensional spaces, and Wendland [14] provided error estimates supporting their theoretical reliability.

Recent work explored generalized Wendland functions for surface approximation. Kouibia et al. [15] proposed bivariate generalized Wendland RBFs with strong convergence properties. Kosari et al. [16] developed hybrid RBF and B-spline methods to improve stability, while Paulo et al. [17] introduced Hermitian RBF techniques incorporating geometric constraints. Kouibia and Pasadas [18] emphasized smoothing variational approaches for handling noisy data.

Other studies demonstrate the flexibility of $RBFs$ beyond offset surfaces. Narkawich et al. [6], Bilal et al. [7], Fortes et al. [8], and González-Rodelas et al. [9] illustrate their effectiveness in multidimensional approximation, discontinuous problems, and noisy data sets. These contributions support the use of RBFs as a robust and adaptable framework.

Despite these advances, several challenges remain. Spline-based methods depend on mesh generation, while globally supported $RBFs$ may lead to dense and computationally expensive systems. Efficient and stable approximation of generalized offset surfaces, particularly in the presence of noise and complex geometries, remains an open problem. The approach proposed in this work addresses these issues by combining smoothing techniques with compactly supported generalized Wendland radial basis functions, leading to a meshfree, efficient, and stable framework.

The rest of this work is organized as follows. Section 2 presents the notation and preliminary necessary for the problem formulation. Section 3 is devoted to the analysis of generalized compactly supported Wendland radial basis functions. In Section 4, we formulate and study the smoothing variational spline problem using generalized Wendland functions. In Section 5, we compute the explicit form of the resulting function, whereas a convergence theorem is proved in Section 6. The final section provides numerical and graphical examples illustrating the effectiveness of the proposed approach, together with a comparison with an existing method in the literature.

3. Notations and Formulation of the Global Problem

Given an open, convex, and bounded set $\mathrm{\Omega }\subset {\mathbb{R}}^2$, let $H^3(\mathrm{\Omega },{\mathbb{R}}^3)$ denote the usual Sobolev space of order 3. This space is equipped with the inner semi-products

$${(\mathit{u},\mathit{v})}_{\mathcal{l}}=\sum _{\left|\mathit{\beta }\right|=l}{\int }_{\mathrm{\Omega }}{\langle D^{\mathit{\beta }}\mathit{u}(\mathit{p}),D^{\mathit{\beta }}\mathit{v}(\mathit{p})\rangle }_{3}d\mathit{p},0\le \mathcal{l}\le 3,$$

the semi-norms ${\left|\mathit{u}\right|}_{\mathcal{l}}={(\mathit{u},\mathit{u})}_{\mathcal{l}}^{{\displaystyle \frac{1}{2}}}$, for $0\le \mathcal{l}\le 3$, and the norm ${\parallel \mathit{u}\parallel }_3={\left({\sum }_{\mathcal{l}=0}^3{\left|\mathit{u}\right|}_{\mathcal{l}}^2\right)}^{{\displaystyle \frac{1}{2}}}$.

Let ${\mathbb{R}}^{n,3}$ be the space of real matrices with n rows and 3 columns, equipped with the inner product

$${\langle A,B\rangle }_{n,3}=\sum _{i=1}^n\sum _{j=1}^3{a}_{ij}{b}_{ij},\forall A={(a_{ij})}_{{\displaystyle \genfrac{}{}{0pt}{}{1\le i\le n}{1\le j\le 3}}},B={(b_{ij})}_{{\displaystyle \genfrac{}{}{0pt}{}{1\le i\le n}{1\le j\le 3}}}\in {\mathbb{R}}^{n,3}$$

and the corresponding norm ${\displaystyle {\langle A\rangle }_{n,3}={\langle A,A\rangle }_{n,3}^{\frac{1}{2}}}$.

For a regular parametric surface $\mathit{s}(u,v)=(x(u,v),y(u,v),z(u,v))$, its two unit tangent vectors in the directions of u and v and its normal vector are given by

$${\mathit{e}}_1={\displaystyle \frac{{\mathit{s}}_u(u,v)}{{\langle {\mathit{s}}_u(u,v)\rangle }_3}},{\mathit{e}}_2={\displaystyle \frac{{\mathit{s}}_v(u,v)}{{\langle {\mathit{s}}_v(u,v)\rangle }_3}},\mathit{n}(u,v)={\displaystyle \frac{{\mathit{s}}_u(u,v)\times {\mathit{s}}_v(u,v)}{{\langle {\mathit{s}}_u(u,v)\times {\mathit{s}}_v(u,v)\rangle }_3}},$$

respectively.

In this situation, the generalized offset surface ${\mathit{s}}^0(u,v)$ with the variable offset distance and direction determined by $d_1(u,v){\mathit{e}}_1(u,v)$, $d_2(u,v){\mathit{e}}_2(u,v)$ and $d_3(u,v)\mathit{n}(u,v)$ is defined by

$$\begin{array}{c}{\mathit{s}}^0(u,v)=\mathit{s}(u,v)+d_1(u,v){\mathit{e}}_1(u,v)+d_2(u,v){\mathit{e}}_2(u,v)+d_3(u,v)\mathit{n}(u,v).\end{array}$$

Let $\gamma \subset {\mathbb{R}}^3$ be a surface defined by a parametrization $\mathit{s}\in C^2(\mathrm{\Omega };{\mathbb{R}}^3)$. For each $n\in {\mathbb{N}}^{★}$, let $A=\{{\mathit{a}}_1,\dots ,{\mathit{a}}_n\}$ be a finite subset of n distinct points of $\mathrm{\Omega }$.

Now, let $d_1(u,v){\mathit{e}}_1(u,v)$, $d_2(u,v){\mathit{e}}_2(u,v)$ and $d_3(u,v)\mathit{n}(u,v)$ be some given offset variable distances and directions.

For any $i=1,\dots ,n$, let

$${\mathit{s}}_i=\mathit{s}({\mathit{a}}_i)+d_1({\mathit{a}}_i){\mathit{e}}_1({\mathit{a}}_i)+d_2({\mathit{a}}_i){\mathit{e}}_2({\mathit{a}}_i)+d_3({\mathit{a}}_i)\mathit{n}({\mathit{a}}_i).$$

4. Generalized Wendland Compactly Supported Radial Basis Functions

Definition 1. 

Let ϕ be a continuous function $\varphi :{\mathbb{R}}_0^{+}\to \mathbb{R}$, a set $\mathrm{\Omega }\subset {\mathbb{R}}^2$, and a finite set $C_N=\{{\mathit{\theta }}_1,\dots ,{\mathit{\theta }}_N\}$ of points of Ω. The linear space generated by the functions set

$$R_N=\{\varphi ({\langle ·-{\mathit{\theta }}_1\rangle }_2),\dots ,\varphi ({\langle ·-{\mathit{\theta }}_N\rangle }_2)\}$$

(1)

is called the radial basis functions space relative to the function ϕ and the centers set $C_N$.

Definition 2. 

We consider a function $u\in C(\mathrm{\Omega })$. The $RBFs$ interpolant associated with u is defined by:

$$R_{u,C_N}(\mathit{x})=\sum _{i=1}^N{\alpha }_i\varphi ({\langle \mathit{x}-{\mathit{\theta }}_i\rangle }_2),\mathit{x}\in \mathrm{\Omega },$$

(2)

where ${\alpha }_1,\dots ,{\alpha }_N\in \mathbb{R}$ are determined by the interpolating conditions

$$R_{u,C_N}({\mathit{\theta }}_i)=u({\mathit{\theta }}_i),1\le i\le N.$$

(3)

$$\left(\begin{array}{ccc}\varphi (\parallel {\theta }_1-{\theta }_1\parallel )& \dots & \varphi (\parallel {\theta }_1-{\theta }_N\parallel )\\ ⋮& \ddots & ⋮\\ \varphi (\parallel {\theta }_N-{\theta }_1\parallel )& \dots & \varphi (\parallel {\theta }_N-{\theta }_N\parallel )\end{array}\right)\left(\begin{array}{c}{\alpha }_1\\ ⋮\\ {\alpha }_N\end{array}\right)=\left(\begin{array}{c}u({\theta }_1)\\ ⋮\\ u({\theta }_N)\end{array}\right).$$

Remark 1. 

The radial basis function interpolating $R_{u,C_N}$ exists and is unique if and only if the interpolation matrix is invertible:

$$\det ({(\varphi ({\langle {\mathit{\theta }}_i-{\mathit{\theta }}_j\rangle }_2))}_{1\le i,j\le N})\ne 0.$$

The algorithm for constructing the generalized Wendland functions in even dimensions $2m$ is developed in [5].

$${\varphi }_{2m,{\displaystyle \frac{2\mathcal{l}-1}{2}}}(r)=r^{2\mathcal{l}}{p}_{m,\mathcal{l}}(r^2)L(r)+q_{m,\mathcal{l}}(r^2)S(r),r\in [0,1],$$

for integers $m,\mathcal{l}\ge 0$, with

$$L(r)={\displaystyle \frac{\log r}{1+\sqrt{1-r^2}}},S(r)=1-r^2,$$

and $p_{m,\mathcal{l}}$, $q_{m,\mathcal{l}}$ are polynomials of degree $m-1$ and $m-1+\mathcal{l}$, respectively.

Remark 2. 

The function $L(r)$ is not defined at $r=0$. However, the overall $RBFs$ remains well-defined at this point by continuity, and the value at $r=0$ is obtained by taking the limit as $r\to 0$, which yields a finite value. Therefore, the function is extended at $r=0$ by continuity.

For the Wendland function ${\varphi }_{d,k}$ in dimension d pair with parameter k, the native Sobolev space is:

$${\mathcal{N}}_{{\varphi }_{d,k}}=H^{k+{\displaystyle \frac{d}{2}}+{\displaystyle \frac{1}{2}}}({\mathbb{R}}^d).$$

• k = Wendland smoothness parameter ($k=0,1,2,\dots$).
• The native space = Sobolev space of order $s=k+{\displaystyle \frac{d}{2}}+{\displaystyle \frac{1}{2}}$, see the specific cases in

  Table 1. Generalized Wendland functions ${\varphi }_{2,k+{\displaystyle \frac{1}{2}}}$ for $k=0,1,2$ and their native space.

  Generalized Wendland Functions	Native Space
  ${\displaystyle {\varphi }_{2,1/2}(r)={\displaystyle \frac{\sqrt{2}}{3\sqrt{\pi }}}\left(3r^{2}L(r)+(2r^2+1)S(r)\right)}$	$H^2$
  ${\displaystyle {\varphi }_{2,3/2}(r)=-{\displaystyle \frac{\sqrt{2}}{60\sqrt{\pi }}}\left(15r^{4}L(r)+(8r^4+9r^2-2)S(r)\right)}$	$H^3$
  ${\displaystyle {\varphi }_{2,5/2}(r)={\displaystyle \frac{\sqrt{2}}{2520\sqrt{\pi }}}\left(105r^{6}L(r)+(48r^6+87r^4-38r^2+8)S(r)\right)}$	$H^4$

  for $k=0,1,2$.

Theorem 1. 

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be an open set, $C_N=\{{\mathit{\theta }}_1,\dots ,{\mathit{\theta }}_N\}\subset \mathrm{\Omega }$ a set of centers, and let $n,k\in \mathbb{N}$ with $k\ge 0$. Let $R_{f,C_N}$ denote the RBF interpolating of $f\in H^{k+2}(\mathrm{\Omega })$ relative to $C_N$ using ${\varphi }_{2,k+1/2}$.

Define the fill distance of $C_N$ in Ω by

$$h=\underset{\mathit{x}\in \mathrm{\Omega }}{\sup }\underset{1\le i\le N}{\min }{\langle \mathit{x}-{\mathit{\theta }}_i\rangle }_2$$

Then, for all $j=0,\dots ,k+2$, we have

$$|f-R_{f,C_N}{|}_j\le C{h}^{k+2-j}{\parallel f\parallel }_{k+2},$$

(4)

where C is independent of f.

Proof. 

Analogously to the proof of Theorem 1 in [15].    □

Remark 3. 

By applying the preceding theorem to the case $k=1$, we obtain the following estimate, for all $f\in H^3(\mathrm{\Omega })$

$$|f-R_{f,C_N}{|}_j\le C{h}^{3-j}{\parallel f\parallel }_3,\forall j=0,\dots ,3.$$

Let us consider $f=(f_1, f_2, f_3)\in H^3(\mathrm{\Omega };{\mathbb{R}}^3),$ with $f_{\mathcal{l}}\in H^3(\mathrm{\Omega }),\mathcal{l}=1,2,3.$

The RBF interpolating of f is defined component-wise as

$$R_{f,C_N}=\left(R_{f_1,C_N}, R_{f_2,C_N}, R_{f_3,C_N}\right),$$

where $R_{f_{\mathcal{l}},C_N}$ denotes the scalar RBF interpolating of $f_{\mathcal{l}}$ with respect to the same centers set $C_N$.

Vector Sobolev semi-norm

By definition of the vector-valued Sobolev space $H^j(\mathrm{\Omega };{\mathbb{R}}^3)$, the associated semi-norm satisfies

$${\left|\mathbf{v}\right|}_{H^j(\mathrm{\Omega };{\mathbb{R}}^3)}^2=\sum _{\mathcal{l}=1}^3{\left|v_{\mathcal{l}}\right|}_{H^j(\mathrm{\Omega })}^2,\forall \mathbf{v}=(v_1,v_2,v_3).$$

Component-wise application of the scalar estimate

Applying inequality $(4)$ to each component $f_{\mathcal{l}}$, $\mathcal{l}=1,2,3,$

$$|f_{\mathcal{l}}-R_{f_{\mathcal{l}},C_N}{|}_{H^j(\mathrm{\Omega })}\le C{h}^{k+2-j}{\parallel f_{\mathcal{l}}\parallel }_{H^{k+2}(\mathrm{\Omega })}.$$

Squaring and summing over $\mathcal{l}=1, 2, 3,$ we obtain

$$\sum _{\mathcal{l}=1}^3|f_{\mathcal{l}}-R_{f_{\mathcal{l}},C_N}{|}_{H^j(\mathrm{\Omega })}^2\le C^2{h}^{2(k+2-j)}\sum _{\mathcal{l}=1}^3{\parallel f_{\mathcal{l}}\parallel }_{H^{k+2}(\mathrm{\Omega })}^2.$$

We deduce

$$|f-R_{f,C_N}{|}_{H^j(\mathrm{\Omega };{\mathbb{R}}^3)}^2\le C^2{h}^{2(k+2-j)}{\parallel f\parallel }_{H^{k+2}(\mathrm{\Omega };{\mathbb{R}}^3)}^2.$$

Taking the square root, and for $k=1$ yields the desired estimate

$$|f-R_{f,C_N}{|}_{H^j(\mathrm{\Omega };{\mathbb{R}}^3)}\le C{h}^{3-j}{\parallel f\parallel }_{H^3(\mathrm{\Omega };{\mathbb{R}}^3)}.$$

(5)

5. Smoothing Variational Splines by Generalized Wendland Functions

Given a function $f\in H^3(\mathrm{\Omega },{\mathbb{R}}^3)$ and a set of finite points $A=\{{\mathit{a}}_1,\dots ,{\mathit{a}}_n\}\subset \mathrm{\Omega }$, we consider the functional $\beta :H^3(\mathrm{\Omega },{\mathbb{R}}^3)\to {\mathbb{R}}^{n,3}$ given by

$$\beta v={(v({\mathit{a}}_i))}_{1\le i\le n}\in {\mathbb{R}}^{n,3}$$

and let ${\mathrm{\Gamma }}_{\epsilon }^n$ be the functional defined on $H^3(\mathrm{\Omega },{\mathbb{R}}^3)$ by

$${\mathrm{\Gamma }}_{\epsilon }^n(\mathit{v})=\sum _{i=1}^n{\langle \mathit{v}({\mathit{a}}_i)-{\mathit{s}}_i\rangle }_3^2+\epsilon {\left|\mathit{v}\right|}_3^2.$$

We consider the following minimization problem: Find ${\mathit{\sigma }}_{\epsilon }^n\in R_N$ such that

$$\forall \mathit{v}\in R_N,{\mathrm{\Gamma }}_{\epsilon }^n({\mathit{\sigma }}_{\epsilon }^n)\le {\mathrm{\Gamma }}_{\epsilon }^n(\mathit{v}).$$

(6)

Suppose that A is a ${\mathbb{P}}_2$-unisolvent set, that is,

$$\ker \beta \cap {\mathbb{P}}_2(\mathrm{\Omega },{\mathbb{R}}^3)=\left\{0\right\}$$

(7)

and suppose that

$$\underset{\mathit{x}\in \mathrm{\Omega }}{\sup }\underset{\mathit{a}\in A}{\min }{\langle \mathit{x}-\mathit{a}\rangle }_2=o({\displaystyle \frac{1}{n}}),n\to +\infty .$$

(8)

Theorem 2. 

Problem (6) has a unique solution, called the smoothing variational spline in $R_N$ relative to A, β, $d_1$, $d_2$, $d_3$ and ε, which is also the unique solution of the following variational problem: Find ${\sigma }_{\epsilon }^n\in R_N$ such that

$$\forall v\in R_N,{\langle \beta {\mathit{\sigma }}_{\epsilon }^n,\beta \mathit{v}\rangle }_{n,3}+\epsilon {({\mathit{\sigma }}_{\epsilon }^n,\mathit{v})}_3=\sum _{i=1}^n\langle {\mathit{s}}_i,\mathit{v}({\mathit{a}}_i){\rangle }_3.$$

(9)

Proof. 

Let us define the following application by

$$v\to [[v]]={\left(\sum _{i=1}^n{\langle \mathit{v}({\mathit{a}}_i)\rangle }_3^2+\epsilon {\left|\mathit{v}\right|}_3^2\right)}^{{\displaystyle \frac{1}{2}}}.$$

By the reasoning of Proposition 1.1 in [19], we obtain that the linear form $[[v]]$ is a norm in $H^3(\mathrm{\Omega },{\mathbb{R}}^3)$, which is equivalent to the norm $\parallel ·\parallel$.

Since

$$\beta \in \mathcal{L}\left(H^3(\mathrm{\Omega }),{\mathbb{R}}^3\right),$$

the operator

$$\beta :H^3(\mathrm{\Omega },{\mathbb{R}}^3)⟶{\mathbb{R}}^{n,3},\beta \mathit{v}={(v_1(a_i),v_2(a_i),v_3(a_i))}_{1\le i\le n},$$

is linear and continuous.

We define the bilinear form

$$a(\mathit{u},\mathit{v})=2({\langle \beta \mathit{u},\beta \mathit{v}\rangle }_{n,3}+\epsilon {(\mathit{u},\mathit{v})}_3).$$

Continuity:

For all $\mathit{u},\mathit{v}\in H^3(\mathrm{\Omega },{\mathbb{R}}^3)$, we have

$$|a(\mathit{u},\mathit{v})|\le \max (1,\epsilon )[[v]][[u]].$$

Coercivity:

For all $\mathit{v}\in H^3(\mathrm{\Omega },{\mathbb{R}}^3)$, it holds that

$$a(\mathit{v},\mathit{v})={\parallel \beta \mathit{v}\parallel }^2+\epsilon {\left|\mathit{v}\right|}_{3,\mathrm{\Omega }}^2\ge \min (1,\epsilon ){[[v]]}^2.$$

The linear form defined from $R_N$ into $\mathbb{R}$ by

$$g(v)=2(\sum _{i=1}^n{\langle {\mathit{s}}_i,\mathit{v}({\mathit{a}}_i)\rangle }_3)$$

is continuous on $H^3(\mathrm{\Omega },{\mathbb{R}}^3)$.

Therefore, by the Lax–Milgram theorem (see [20]), there exists a unique solution   ${\sigma }_{\epsilon }^n\in H^3(\mathrm{\Omega },{\mathbb{R}}^3)$ to Problem (10).

Furthermore, ${\sigma }_{\epsilon }^n$ is the minimizer of the functional

$$G(v)=\frac{1}{2}a(v,v)-g(v)={\mathrm{\Gamma }}_{\epsilon }^n(v)-{\langle \beta s_i\rangle }_{n,3}^2.$$

As a result, ${\sigma }_{\epsilon }^n$ is the unique solution of Problem (6).      □

6. Computing the Solution

In this section, we describe how to compute the generalized offset surfaces’ RBF spline in $R_N$. The solution ${\sigma }_{\epsilon }^n$ can be expressed by:

$${\displaystyle {\mathit{\sigma }}_{\epsilon }^n=\sum _{i=1}^N{\alpha }_i{\mathit{\varphi }}_{2,k+\frac{1}{2}}({\langle \mathit{\theta }-{\mathit{\theta }}_i\rangle }_2),\forall \mathit{\theta }\in \mathrm{\Omega }}$$

such that   ${\alpha }_1,\dots ,{\alpha }_N\in {\mathbb{R}}^3$.

Let $v_i$ be the function:

$$v_i(\mathit{\theta })={\varphi }_{2,k+{\displaystyle \frac{1}{2}}}({\langle \mathit{\theta }-{\mathit{\theta }}_i\rangle }_2),\forall \mathit{\theta }\in \mathrm{\Omega },$$

Change of Basis

Let $({\mathit{e}}_1,{\mathit{e}}_2,{\mathit{e}}_3)$ denote the canonical basis of ${\mathbb{R}}^3$. Starting from the vector-valued $RBF$ approximation

$${\sigma }_{\epsilon }^n(\mathit{\theta })=\sum _{i=1}^N{\mathit{\alpha }}_i{v}_i(\mathit{\theta }),{\mathit{\alpha }}_i\in {\mathbb{R}}^3,$$

we introduce the vector basis functions

$$\{B_1,\dots ,B_{3N}\}\subset R_N$$

defined, for $\mathcal{l}=1,\dots ,3N$, by

$$B_{\mathcal{l}}(\mathit{\theta })=v_i(\mathit{\theta }){\mathit{e}}_j,$$

where the indices i and j are given by

$$i=⌊{\displaystyle \frac{\mathcal{l}-1}{3}}⌋+1,j=\mathcal{l}-3(i-1).$$

With this change of basis, the approximation can be rewritten as

$${\sigma }_{\epsilon }^n=\sum _{\mathcal{l}=1}^{3N}{\alpha }_{\mathcal{l}}{B}_{\mathcal{l}},\mathit{\alpha }={({\alpha }_1,\dots ,{\alpha }_{3N})}^{\mathrm{T}}\in {\mathbb{R}}^{3N}.$$

where these coefficients are given as the solution of the linear system:

$$({\mathcal{A}}^t\mathcal{A}+\epsilon \mathcal{R})\mathit{\alpha }=\mathit{b}$$

such as:

$$\begin{array}{c}\mathcal{A}={\left(\beta {\mathit{B}}_l({\mathit{a}}_i)\right)}_{\begin{array}{c}1\le l\le 3N\\ 1\le i\le n\end{array}},\mathcal{A}\in {\mathbb{R}}^{3n\times 3N},\hfill \\ \mathcal{R}={\left({({\mathit{B}}_l,{\mathit{B}}_m)}_3\right)}_{1\le l,m\le 3N},\mathcal{R}\in {\mathbb{R}}^{3N\times 3N},\hfill \\ \mathit{\alpha }={({\alpha }_1,\dots ,{\alpha }_{3N})}^T,\\ \mathit{b}={({\sum }_{i=1}^n{\langle {\mathit{s}}_i,{\mathit{B}}_l({\mathit{a}}_i)\rangle }_3)}_{1\le l\le 3N},\mathit{b}\in {\mathbb{R}}^{3N}.\end{array}$$

Now, by adapting the notations and reasoning as the proof of ([3], Proposition 7), we obtain the following result.

Remark 4. 

The coercivity of the bilinear form $a(u,v)$ implies that the system matrix ${\mathcal{A}}^t\mathcal{A}+\epsilon \mathcal{R}$ is symmetric positive definite and therefore invertible.

7. Convergence

The aim of this section is to prove that the offset RBF spline in $R_N$ relative to $n, A, d_1, d_2, d_3$ and $\epsilon$ converge to $s^0$ as n tends to $+\infty$.

Theorem 3. 

Suppose hypotheses (7) and (8) hold and that

$$\epsilon =o(1),n\to +\infty$$

(10)

and

$${\displaystyle \frac{n{h}^6}{\epsilon }}=o(1),n\to +\infty .$$

(11)

Then

$$\underset{n\to +\infty }{\lim }{\parallel {\sigma }_{\epsilon }^n-s^0\parallel }_2=0.$$

Proof. 

Step 1. First, we have

$${\mathrm{\Gamma }}_{\epsilon }^n({\sigma }_{\epsilon }^n)\le {\mathrm{\Gamma }}_{\epsilon }^n({\mathbf{R}}_{s^0,C_N})$$

with $R_{s^0,C_N}$ being the interpolation RBFs function of $s^0$ relative to $C_N$ from ${\varphi }_{k+2,k+1/2}$. Then:

$${\langle \beta {\sigma }_{\epsilon }^n-\beta s^0\rangle }_{n,3}^2+\epsilon |{\sigma }_{\epsilon }^n{|}_3^2\le {\langle \beta R_{s^0,C_N}-\beta s^0\rangle }_{n,3}^2+\epsilon {|R_{s^0,C_N}|}_3^2.$$

(12)

which implies that:

$${\displaystyle {\left|{\sigma }_{\epsilon }^n\right|}_3^2\le \frac{1}{\epsilon }{\langle \beta R_{s^0,C_N}-\beta s^0\rangle }_{n,3}^2+{\left|R_{s^0,C_N}\right|}_3^2.}$$

(13)

Hence

$${\displaystyle {\left|{\sigma }_{\epsilon }^n\right|}_3^2\le \frac{1}{\epsilon }\sum _{i=1}^n{〈R_{s^0,C_N}(a_i)-s_i〉}_3^2+{\left|R_{s^0,C_N}\right|}_3^2.}$$

It follows from (5) that

$${\left|R_{s^0,C_N}\right|}_3^2\le K_1{\parallel s^0\parallel }_{H^3(\mathrm{\Omega };{\mathbb{R}}^3)}^2.$$

and again applying (5) for $j=0$, there exists $K_2>0$ such that, for each $i=1,\dots ,n$, we have

$$\sum _{i=1}^n{〈R_{s^0,C_N}(a_i)-s_i〉}_3^2\le nK{h}^6{\parallel s^0\parallel }_3^2.$$

(14)

Thus, from (13) and (14), we obtain:

$$|{\sigma }_{\epsilon }^n{|}_3^2\le {\displaystyle \frac{1}{\epsilon }}{\langle \beta R_{s^0,C_N}-\beta s^0\rangle }_{n,3}^2+|R_{s^0,C_N}{|}_3^2\le ({\displaystyle \frac{n{h}^6}{\epsilon }}+1)K{\parallel s^0\parallel }_3^2$$

and, from (11), we conclude that there exists $C_1>0$ and $n_1\in \mathbb{N}$ such that

$$|{\sigma }_{\epsilon }^n{|}_3^2\le C_1,\forall n\ge n_1.$$

(15)

In addition, from (12)–(14), we have that

$${\langle \beta {\sigma }_{\epsilon }^n-\beta s^0\rangle }_{n,3}^2\le (n{h}^6+\epsilon )C{\parallel s^0\parallel }_3^2$$

and, from (10) and (11), there exists $C_2>0$ and $n_2\in \mathbb{N}$ such that

$${\langle \beta {\sigma }_{\epsilon }^n-\beta s^0\rangle }_{n,3}^2\le C_2,\forall n\ge n_2.$$

(16)

It follows that:

$${\langle \beta {\sigma }_{\epsilon }^n\rangle }_{n,3}=\sum _{i=1}^n{\langle {\sigma }_{\epsilon }^n(a_i)\rangle }_3\le C$$

(17)

Hence, from (15)–(17), it follows that there exist a constant $M>0$ and an integer $n_0\in \mathbb{N}$ such that

$${\parallel {\sigma }_{\epsilon }^n\parallel }_3\le M,n\ge n_0,$$

Since the family ${\left({\sigma }_{\epsilon }^n\right)}_{n\ge n_0}$ is bounded in $H^3(\mathrm{\Omega };{\mathbb{R}}^3)$, it follows that there exists a sub-sequence ${\left({\sigma }_{{\epsilon }_m}^{n_m}\right)}_{m\in \mathbb{N}}$, with ${\epsilon }_m=o(1)$, $\underset{n_m\to +\infty }{\lim }{n}_m=+\infty$, and an element $s^{*}\in H^3(R;{\mathbb{R}}^3)$ such that

$${\sigma }_{{\epsilon }_m}^{n_m} \mathrm{converges} \mathrm{weakly} \mathrm{to} s^{*}in{H}^3(\mathrm{\Omega };{\mathbb{R}}^3).$$

(18)

Step 2. We suppose that $s^0\ne s^{*}$

From the continuous injection of $H^3(\mathrm{\Omega };{\mathbb{R}}^3)$ into $C(\mathrm{\Omega };{\mathbb{R}}^3)$, it follows that there exist $\eta >0$ and a non-empty open $\omega \subset \mathrm{\Omega }$ such that

$$\forall a\in \omega ,{〈s^{*}(a)-s^0(a)〉}_3>\eta .$$

As this injection is compact, from (18),

$${\displaystyle \exists m_0\in \mathbb{N},\forall m\ge m_0,\forall a\in \omega ,〈{\sigma }_{{\epsilon }_m}^{n_m}(a)-s^{*}(a)〉\le \frac{\eta }{2}·}$$

Hence, for all $m\ge m_0$ and all $a\in \omega$, we have

$${\displaystyle {〈{\sigma }_{{\epsilon }_m}^{n_m}(a)-s^0(a)〉}_3\ge {〈s^{*}(a)-s^0(a)〉}_3-{〈{\sigma }_{{\epsilon }_m}^{n_m}(a)-s^{*}(a)〉}_3>\frac{\eta }{2}·}$$

(19)

For sufficiently large m and using the density (8), we deduce that there exists a point $a^{n_m}\in A^n\cap \omega$ such that

$${\sigma }_{{\epsilon }_m}^{n_m}(a^{n_m})=s^0(a^{n_m}),$$

which is a contradiction with (19). Consequently $s^{*}=s^0.$

• Step 3. As $H^3(\mathrm{\Omega };{\mathbb{R}}^3)$ is compactly injected in $H^2(\mathrm{\Omega };{\mathbb{R}}^3)$, using (18) and taking into account that $s^{*}=s^0$, we have

$$s^0=\underset{m\to +\infty }{\lim }{\sigma }_{{\epsilon }_m}^{n_m} \mathrm{in} H^2(\mathrm{\Omega };{\mathbb{R}}^3).$$

Then,

$$\underset{m\to +\infty }{\lim }{(({\sigma }_{{\epsilon }_m}^{n_m},s^0))}_2={∥s^0∥}_2^2.$$

(20)

Thus, for any $m\in \mathbb{N}$, we have

$${∥{\sigma }_{{\epsilon }_m}^{n_m}-s^0∥}_2^2={∥{\sigma }_{{\epsilon }_m}^{n_m}∥}_2^2+{∥s^0∥}_2^2-{(({\sigma }_{{\epsilon }_m}^{n_m},s^0))}_2.$$

(21)

Therefore

$$\underset{m\to +\infty }{\lim }{∥{\sigma }_{{\epsilon }_m}^{n_m}-s^0∥}_2=0.$$

(22)

Step 4. We suppose now that ${∥{\sigma }_{\epsilon }^n-s^0∥}_2$ does not tend to 0 as n tends to infinity. In this case, there will exist $\alpha >0$ and a sequence ${\sigma }_{{\epsilon }_m^{\prime }}^{n_m^{\prime }}$, such that

$$\underset{m\to +\infty }{\lim }{∥{\sigma }_{{\epsilon }_m^{\prime }}^{n_m^{\prime }}-s^0∥}_2>\alpha \forall m^{\prime }\in \mathbb{N}.$$

(23)

The sequence ${\left({\sigma }_{{\epsilon }_m^{\prime }}^{n_m^{\prime }}\right)}_{m^{\prime }\in \mathbb{N}}$ is bounded in $H^3(\mathrm{\Omega };{\mathbb{R}}^3)$. Hence, by following the same way of steps (1), (2) and (3), we deduce that from such a sequence, we can extract a sub-sequence that converges towards $s^0$, which produces a contradiction with (23).   □

8. Numerical and Graphical Examples

To illustrate the proposed method, we consider the following example. Let $\Sigma$ be a regular surface, its graph appears in Figure 1, parameterized by

$\mathit{s}\in C^4(\mathrm{\Omega };{\mathbb{R}}^3)$, with $\mathrm{\Omega }=(0,1)\times (0,1)$ and

$$\mathit{s}(x,y)=\left(cos\left(\pi x-{\displaystyle \frac{3\pi }{2}}\right)y, sin\left(\pi x-{\displaystyle \frac{3\pi }{2}}\right)y,x\right).$$

In order to evaluate the performance of the method using Spline RBF, two relative error estimates were computed as follows:

$$E_I=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^{1000}{(R_{s,C_N}(a_i)-s(a_i))}^2}{{\sum }_{i=1}^{1000}s{(a_i)}^2}}},E_{SR}=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^{1000}{({\sigma }_{\epsilon }^n(a_i)-s(a_i))}^2}{{\sum }_{i=1}^{1000}s{(a_i)}^2}}},$$

with $\{a_1,\dots ,a_{1000}\}\subset \mathrm{\Omega }$ being one thousand distinct random points used to estimate the relative errors of $R_{s,C_N}$ and ${\sigma }_{\epsilon }^n$ with respect to s in the $L^2(\mathrm{\Omega };{\mathbb{R}}^3)$ norm.

By Theorems 1 and 3, the error estimates $E_I$ and $E_{SR}$ converge to zero as $n\to +\infty$ under suitable conditions.

The discrete space employed to compute the approximated solution ${\sigma }_{\epsilon }^n$, its corresponding graph appears on the top part of the Figure 2, is the $RBF$ space generated by the generalized Wendland function ${\mathrm{\Psi }}_{2,3/2}$ associated with the chosen set of centers:

$$C_N=\left\{\left({\displaystyle \frac{i}{n_x-1}},{\displaystyle \frac{j}{n_y-1}}\right),i=0,\dots ,n_x-1,j=0,\dots ,n_y-1\right\}.$$

The total number of centers is therefore $N=n_x{n}_y$.

Table 2. Computed relative error $E_{SR}$ for $n=1500$ approximation points and $N=200$ for different values of $\epsilon$. $E_I=6.17087\times {10}^{-3}$.

$\mathit{\epsilon }$	${\mathit{E}}_{\mathit{S}\mathit{R}}$
${10}^{-1}$	$4.87525\times {10}^{-2}$
${10}^{-2}$	$5.19658\times {10}^{-2}$
${10}^{-3}$	$1.73149\times {10}^{-1}$
${10}^{-4}$	$1.28889\times {10}^{-2}$
${10}^{-5}$	$1.98264\times {10}^{-1}$
${10}^{-6}$	$2.51422\times {10}^{-1}$
${10}^{-7}$	$7.37446\times {10}^{-3}$
${10}^{-8}$	$4.14123\times {10}^{-3}$
${10}^{-9}$	$9.16208\times {10}^{-4}$
${10}^{-10}$	$8.60467\times {10}^{-4}$
${10}^{-11}$	$8.50724\times {10}^{-4}$
${10}^{-12}$	$8.49649\times {10}^{-4}$
${10}^{-13}$	$8.49540\times {10}^{-4}$
${10}^{-14}$	$8.49529\times {10}^{-4}$
${10}^{-15}$	$8.49528\times {10}^{-4}$

presents the relative error estimate $E_{SR}$ obtained with $N=200$ centers and $n=1500$ approximation points for different values of $\epsilon$. As observed, when the parameter $\epsilon$ decreases, the relative error progressively decreases and stabilizes at the order of ${10}^{-4}$.

Table 3. Computed relative error $E_{SR}$ with $N=100$ and $\epsilon ={10}^{-12}$ for different values of n approximation points.

n	${\mathit{E}}_{\mathit{S}\mathit{R}}$
500	$1.2249\times {10}^{-3}$
1000	$5.24961\times {10}^{-4}$
1500	$4.28259\times {10}^{-4}$
4500	$3.64319\times {10}^{-4}$

Presents the relative error estimates $E_{SR}$, for $\epsilon ={10}^{-12}$ and $N=100$, for various values of n. The results show that $E_{SR}$ decreases as n increases.

Table 4. Computed relative errors $E_{SR}$ and $E_I$ with $n=1000$ and $\epsilon ={10}^{-9}$ for different values of N.

N	${\mathit{E}}_{\mathit{S}\mathit{R}}$	${\mathit{E}}_{\mathit{I}}$
50	$3.88952\times {10}^{-3}$	$5.40413\times {10}^{-3}$
100	$1.20101\times {10}^{-3}$	$1.60605\times {10}^{-3}$
150	$9.77687\times {10}^{-4}$	$1.63735\times {10}^{-3}$
250	$9.63553\times {10}^{-4}$	$2.60284\times {10}^{-3}$

Presents $E_{SR}$ and $E_I$ for $n=1000$ and $\epsilon ={10}^{-9}$, for various values of N. It can be observed that $E_{SR}$ decreases monotonically as N increases. In contrast, $E_I$ initially decreases but increases for larger values of N, likely due to ill-conditioning of the interpolation matrix. Furthermore, $E_{SR}$ remains consistently lower than $E_I$ for all considered values of N.

Figure 3 shows the original surface (the bottom part) and its $RBF$ approximation of the generalized offset surface for $n=500$, $N=132$ and $\epsilon ={10}^{-14}$, its graph is shown on the top part, with the variable offset direction and offset distance $0.3sin(x+y+0.5)e_1+0.3e_2-0.3cos(0.5(x+y+1))n$. In this case the computation of the relative error is $E_{SR}=4.30542\times {10}^{-4}$.

8.1. Normal Perturbation

Let $n(x,y)\in {\mathbb{R}}^3$ denote the exact unit normal to the surface at a point $(x,y)$.

8.1.1. Addition of Gaussian Noise

A perturbed normal is defined as:

$$\tilde{n}(x,y)=n(x,y)+\alpha (x,y)$$

(24)

where

$$\alpha (x,y)\sim \mathcal{N}(0,{\sigma }^2{I}_3)$$

(25)

is an isotropic Gaussian noise in dimension 3.

Here, the list of $\sigma$ values, listeSigmaNormal in the code, corresponds to the different standard deviations used for the Gaussian noise in the experiments:

$$\mathtt{listeSigmaNormal}=\{0.01,0.05,0.1\}.$$

The parameter $\sigma$ controls the intensity of the perturbation applied to the normals.

8.1.2. Renormalization

The perturbed normal is then normalized to preserve unit length:

$$\widehat{n}(x,y)={\displaystyle \frac{\tilde{n}(x,y)}{\parallel \tilde{n}(x,y)\parallel }}$$

(26)

This operation introduces a directional (angular) perturbation of the normal.

8.1.3. Perturbed Surface

After perturbing the normals, the surface becomes:

$$s_{\sigma }(x,y)=s(x,y)+0.3\left(e_1(x,y)+e_2(x,y)-\widehat{n}(x,y)\right)$$

(27)

Thus, only the normal component is affected by the noise, leading to a geometric deformation of the surface mainly in the normal direction.

Numerical experiments are conducted for different values of the parameter $\sigma$, allowing us to assess the robustness of the method with respect to noise on the normals (see

Table 5. Computed relative error $E_{SR}$ for $n=1500$, $N=200$ and $ϵ={10}^{-14}$ for different values of Gaussian noise ${\sigma }_{\mathrm{Normal}}$.

${\mathit{\sigma }}_{\mathbf{Normal}}$	${\mathit{E}}_{\mathit{S}\mathit{R}}$
0.01	$1.41694\times {10}^{-3}$
0.05	$3.38417\times {10}^{-3}$
0.1	$6.62664\times {10}^{-3}$

).

8.2. Example Torus

Let $\mathrm{\Omega }=(0,1)\times (0,1)$. We consider the regular surface, which parameterizes a torus (its graph appears on the left side of Figure 4), defined by:

$$\mathbf{r}(x,y)=\left(\begin{array}{c}(1+0.35cos(2\pi y))cos(2\pi x)\\ (1+0.35cos(2\pi y))sin(2\pi x)\\ 0.35sin(2\pi y)\end{array}\right),(x,y)\in (0,1)\times (0,1).$$

(28)

The generalized offset surface is defined by:

$${\mathbf{r}}^0(x,y)=\mathbf{r}(x,y)+0.15\left(0.5{\mathbf{e}}_1(x,y)+0.3{\mathbf{e}}_2(x,y)+\mathbf{n}(x,y)\right).$$

(29)

Its graph is shown in Figure 4 right side.

8.3. Comparison with Other Method

The work we can compare to ours is the one [1] where the authors there consider using the same approximation tool as ours for the example given by

$$\mathit{s}(x,y)=\left(cos\left(\pi x-{\displaystyle \frac{3\pi }{2}}\right)y, sin\left(\pi x-{\displaystyle \frac{3\pi }{2}}\right)y,x\right),$$

but in the space of bicubic splines. As is known, they are piecewise polynomials, which makes studying the theory easy in one way or another, and especially reduces the time cost of computing the approximation results. In this context, we can compare the following results. In [1], the estimation of the relative error, given by

$$E_r={\left({\displaystyle \frac{{\sum }_{i=1}^{1000}{\langle {\sigma }_N^n(b_i)-r^0(b_i)\rangle }_3^2}{{\langle r^0(b_i)\rangle }_3^2}}\right)}^{{\displaystyle \frac{1}{2}}},$$

is of the order $E_r=5.29\times {10}^{-4}$ obtained with the bicubic spline method, for $n=500$ approximation points and $\epsilon ={10}^{-7}$, variable offset direction and offset distance $0.3sin(x+y+0.5)e_1+0.3e_2-0.3cos(0.5(x+y+1))n$, while for a constant offset direction $0.3(e_1+e_2-n)$, the computation of such an estimation is $E_r=4.45\times {10}^{-4}$ obtained with the bicubic spline method for $n=500$ approximation points and $\epsilon ={10}^{-7}$. In this manuscript, we have obtained that the estimation of the degree of approximation, given by $E_{SR}$, is $E_{SR}=3.2233\times {10}^{-4}$ for $N=132$ and $\epsilon ={10}^{-14}$ and $n=500$. We can conclude that the order of the estimation of the relative error is the same in both manuscripts; we can even observe that ours is slightly better, although our method uses $RBF$ approximation, so it is assumed that the study in this case is more complicated.

8.4. Conclusions and Perspectives

In this manuscript, we have presented firstly some analysis of the generalized compactly supported Wendland radial basis functions. After this, we have formulated and studied the smoothing variational spline problem by generalized Wendland functions. We have described how to compute the generalized offset surfaces’ RBF spline in the space $R_N$. Furthermore, the convergence theorem is proved, which shows that the offset RBF spline in $R_N$ relative to $n, A, d_1, d_2, d_3$ and $\epsilon$ converges to $s^0$ as n tends to $+\infty$. The end of this manuscript provided numerical and graphical examples illustrating the effectiveness of the proposed approach, together with a comparison with an existing method in the literature.

From the study of the Table 2, Table 3, Table 4 and Table 5, we can observe that as the value of the parameter $\epsilon$ (its introduction is so important in any approximation, since this parameter avoids any oscillation in the approximation method) decreases, the estimation of the error decreases; furthermore, as the data points increase, the estimation of the error decreases. Hence, we can conclude the compatibility between the theory of the convergence results and the numerical ones.

In short, from the analysis presented in the Figure 2, Figure 3 and Figure 4, the studies of the Table 2, Table 3, Table 4 and Table 5, and the last paragraph comparing with other methods that exist in the literature, we can conclude the effectiveness and the validity of our approach as an approximation method.

As future research work, we propose conducting the same study with other alternative approaches to filling and detecting the holes.