Fairing-PIA: progressive-iterative approximation for fairing curve and surface generation

Abstract

The fairing curves and surfaces are used extensively in geometric design, modeling, and industrial manufacturing. However, the majority of conventional fairing approaches, which lack sufficient parameters to improve fairness, are based on energy minimization problems. In this study, we develop a novel progressive-iterative approximation method for the fairing curve and surface generation (fairing-PIA). Fairing-PIA is an iteration method that can generate a series of curves (surfaces) by adjusting the control points of B-spline curves (surfaces). In fairing-PIA, each control point is endowed with an individual weight. Thus, the fairing-PIA has many parameters to optimize the shapes of curves and surfaces. Not only a fairing curve (surface) can be generated globally through fairing-PIA, but also the curve (surface) can be improved locally. Moreover, we prove the convergence of the developed fairing-PIA and show that the conventional energy minimization fairing model is a special case of fairing-PIA. Finally, numerical examples indicate that the proposed method is effective and efficient.

Appendix

In this section, we show the convergence of the fairing-PIA method when the diagonal elements of matrix $\varvec{\Omega }$ are equal to a constant value $\omega$, and coefficient matrix $\varvec{B}$ is a positive semidefinite matrix.

Lemma Appendix 1

Let the diagonal elements of matrix $\varvec{\Lambda }$ satisfy $0<\mu _i<\frac{2}{\lambda _{\text {max}}\left( \varvec{B}\right) },\;i=1,2,\cdots ,n$, where $\lambda _{\text {max}}$ is the largest eigenvalue of $\varvec{B}$. The eigenvalues $\beta$ of the matrix $\varvec{\Lambda B}$(9) are all real and satisfy $0\le \beta <2$.

Proof

Matrix $\varvec{B}$ is a positive semidefinite matrix. Thus, real orthogonal matrix $\varvec{U}$ and diagonal matrix $\varvec{S}=\text {diag}\left( s_1,s_2, \cdots ,s_n\right)$, $s_i\ge 0,\; i=1,2,\cdots ,n$ exist, such that $\varvec{B} = \varvec{USU}^T$. We obtain the following by denoting $\varvec{C}=\varvec{S}^{\frac{1}{2}}\varvec{U}^T$:

$$\begin{aligned} \varvec{B} = \varvec{C}^T\varvec{C}. \end{aligned}$$

(26)

Suppose that $\beta$ is an arbitrary eigenvalue of matrix $\varvec{\Lambda C}^T\varvec{C}$ w.r.t. eigenvector $\varvec{v}$, i.e.,

$$\begin{aligned} \varvec{\Lambda C}^T\varvec{Cv} = \beta \varvec{v}. \end{aligned}$$

(27)

In this case, we obtain

$$\begin{aligned} \varvec{C\Lambda C}^T\left( \varvec{Cv}\right) = \beta \left( \varvec{Cv}\right) , \end{aligned}$$

by multiplying both sides of Eq. (27) by $\varvec{C}$, indicating that $\beta$ is also an eigenvalue of the matrix $\varvec{C\Lambda C}^T$ w.r.t. eigenvector $\varvec{Cv}$. Moreover, for all nonzero vectors $\varvec{x}\in R^{n}$, it holds

$$\begin{aligned} \varvec{x}^T\varvec{C\Lambda C}^T\varvec{x} = \left( \varvec{x}^T\varvec{C\Lambda }^{\frac{1}{2}}\right) \left( \varvec{x}^T\varvec{C\Lambda }^{\frac{1}{2}}\right) ^T\ge 0. \end{aligned}$$

Therefore, we conclude that the matrix $\varvec{C\Lambda C}^T$ is a positive semidefinite matrix. Thus, all its eigenvalues are nonnegative real numbers, i.e., $\beta \ge 0$. According to Eq. (27), the eigenvalues of matrix $\varvec{\Lambda B}=\varvec{\Lambda C}^T\varvec{C}$ are also all nonnegative real numbers.

On the other hand, given that $0<\mu _i<\frac{2}{\lambda _{\text {max}}\left( \varvec{B}\right) },\;i=1,2,\cdots ,n$, the eigenvalues of $\varvec{\Lambda B}$ meet

$$\begin{aligned} 0\le & {} \lambda \left( \varvec{\Lambda B}\right)< \left\| \varvec{\Lambda B}\right\| _2 \le \left\| \varvec{\Lambda }\right\| _2 \left\| \varvec{B}\right\| _2\\= & {} \lambda _{\text {max}}\left( \varvec{\Lambda }\right) \lambda _{\text {max}}\left( \varvec{B}\right) < 2. \end{aligned}$$

$\square$

Remark Appendix 2

Let $n_0$ denote the dimension of the zero eigenspace of $\varvec{B}$ (10). Given that $\varvec{\Lambda }$ is nonsingular, the ranks of the matrix $\varvec{B}$ and $\varvec{\Lambda B}$ are the same, i.e.,

$$\begin{aligned} \text {rank}\left( \varvec{\Lambda B}\right) =\text {rank}\left( \varvec{B}\right) =n-n_0. \end{aligned}$$

Moreover, in the proof of Theorem Appendix 5, a lemma similar to Lemma 2.5 in [47] is used as follows:

Lemma Appendix 3

The algebraic multiplicity of the zero eigenvalue of matrix $\varvec{\Lambda B}$ is equal to its geometric multiplicity.

Proof

Based on Eq. (26), matrix $\varvec{\Lambda B}$ can be written as $\varvec{\Lambda C}^T\varvec{C}$. Then, the proof of this lemma is similar to that of Lemma 2.5 provided in [47]. $\square$

Lemma Appendix 4

The Jordan canonical form of the matrix $\varvec{\Lambda B}$(9) can be represented as

(28)

where

is called a Jordan block of size $n_i$ with eigenvalue $\beta _i$, and $0<\beta _i<2$ and $i=1,2,\cdots ,k$ are the nonzero eigenvalues of $\varvec{\Lambda B}$.

Proof

Based on Lemma Appendix 1, the eigenvalues of $\varvec{\Lambda B}$ are all nonnegative real numbers. Thus, the Jordan canonical form of $\varvec{\Lambda B}$ can be written as

where $\varvec{J}_{n_i}(\beta _i,1)$ is a Jordan block of size $n_i$ with nonzero eigenvalue $\beta _i$ of $\varvec{\Lambda B}$, $i=1,2,\cdots ,k$, and $\varvec{J}_{m_j}(0,1)$ is a Jordan block of size $m_i$ with zero eigenvalue of $\varvec{\Lambda B}$, $j=1,2,\cdots ,l$.

Moreover, Lemma Appendix 3 states that the algebraic multiplicity of the zero eigenvalue of the matrix $\varvec{\Lambda B}$ is equal to its geometric multiplicity. This finding means that the Jordan blocks w.r.t the zero eigenvalues of the matrix $\varvec{\Lambda B}$ are all equal to the matrix $(0)_{1\times 1}$. $\square$

Theorem Appendix 5

If $\varvec{B}$ is a positive semidefinite matrix, then iterative method (9) is convergent.

Proof

A positive semidefinite matrix $\varvec{B}$ can be decomposed as

$$\begin{aligned} \varvec{B}=\varvec{V}\text {diag}(\lambda _1,\lambda _2,\cdots ,\lambda _{n-n_0},\underset{n_0}{\underbrace{0,\cdots ,0}})\varvec{V}^T, \end{aligned}$$

where $\varvec{V}$ is an orthogonal matrix, and $\lambda _i$ and $i=1,2,\cdots ,n-n_0$ are both the nonzero eigenvalues and nonzero singular values of $\varvec{B}$. Then, the Moore-Penrose (M-P) inverse of $\varvec{B}$ has the corresponding decomposition as follows:

$$\begin{aligned} \varvec{B}^{+}=\varvec{V}\text {diag}(\frac{1}{\lambda _1},\frac{1}{\lambda _2},\cdots ,\frac{1}{\lambda _{n-n_0}},\underset{n_0}{\underbrace{0,\cdots ,0}})\varvec{V}^T. \end{aligned}$$

Thus,

$$\begin{aligned} \varvec{B}^{+}\varvec{B} = \varvec{V}\text {diag}(\underset{n-n_0}{\underbrace{1,1,\cdots ,1}},\underset{n_0}{\underbrace{0,\cdots ,0}})\varvec{V}^T. \end{aligned}$$

(29)

Based on Lemma Appendix 4, the Jordan canonical form of the matrix $\varvec{\Lambda B}$ is $\varvec{J}$ (28). We obtain the following by using the Jordan decomposition: $\varvec{\Lambda B}=\varvec{T}^{-1}\varvec{JT}$, where $\varvec{T}$ is an invertible matrix. Hence,

Based on Lemma Appendix 1, $-1<1-\beta _i<1$, $i=1,2,\cdots ,k$. Then, we obtain the following by combining Eq. (29):

$$\begin{aligned}&\lim _{h\rightarrow \infty }\left( \varvec{I}-\varvec{\Lambda B} \right) ^h\nonumber \\&\quad = \varvec{T}^{-1}\text {diag}( \underset{n-n_0}{\underbrace{0,0,\cdots ,0}},\underset{n_0}{\underbrace{1,\cdots ,1}})\varvec{T}\nonumber \\&\quad = \varvec{T}^{-1}(\varvec{I}-\text {diag}(\underset{n-n_0}{\underbrace{1,1,\cdots ,1}},\underset{n_0}{\underbrace{0,\cdots ,0}}))\varvec{T}\nonumber \\&\quad = \varvec{I}-\left( \varvec{VT}\right) ^{-1}\varvec{B}^{+}\varvec{B}\left( \varvec{VT}\right) . \end{aligned}$$

(30)

On the other hand, the coefficient matrix $\varvec{B}$ is singular; thus, the linear system in Eq. (17) has its solution if and only if

$$\begin{aligned} \left( 1-\omega \right) \varvec{BB}^{+}\varvec{N}^T\varvec{Q}=\left( 1-\omega \right) \varvec{N}^T\varvec{Q}. \end{aligned}$$

Therefore, we obtain the following by subtracting $\left( 1-\omega \right) \varvec{B}^{+}\varvec{N}^T\varvec{Q}$ from both sides of iterative scheme (9):

$$\begin{aligned} \begin{aligned}&\varvec{P}^{[k+1]}-\left( 1-\omega \right) \varvec{B}^{+}\varvec{N}^T\varvec{Q}\\&\quad = \left( \varvec{I}-\varvec{\Lambda B}\right) \varvec{P}^{[k]}+\left( 1-\omega \right) \varvec{\Lambda R}^T\varvec{Q}\\&\qquad - \left( 1-\omega \right) \varvec{B}^{+}\varvec{N}^T\varvec{Q}\\&\quad = \left( \varvec{I}-\varvec{\Lambda B}\right) \varvec{P}^{[k]}\\&\qquad + \left( 1-\omega \right) \varvec{\Lambda }\varvec{BB}^{+}\varvec{N}^T\varvec{Q} - \left( 1-\omega \right) \varvec{B}^{+}\varvec{N}^T\varvec{Q} \\&\quad = \left( \varvec{I}-\varvec{\Lambda B}\right) \varvec{P}^{[k]} - \left( 1-\omega \right) \left( \varvec{I}-\varvec{\Lambda B}\right) \varvec{B}^{+}\varvec{N}^T\varvec{Q} \\&\quad = \left( \varvec{I}-\varvec{\Lambda B}\right) \left( \varvec{P}^{[k]}-\left( 1-\omega \right) \varvec{B}^{+}\varvec{N}^T\varvec{Q}\right) \\&\quad = \left( \varvec{I}-\varvec{\Lambda B}\right) ^{k+1}\left( \varvec{P}^{[0]}-\left( 1-\omega \right) \varvec{B}^{+}\varvec{N}^T\varvec{Q}\right) . \end{aligned} \end{aligned}$$

From Eq. (30), when $k\rightarrow \infty$, the above equation tends to

$$\begin{aligned}{} & {} \varvec{P}^{[\infty ]} - \left( 1-\omega \right) \varvec{B}^{+}\varvec{N}^T\varvec{Q}\\{} & {} \quad = \left( \varvec{I}-\left( \varvec{VT}\right) ^{-1}\varvec{B}^{+}\varvec{B}\left( \varvec{VT}\right) \right) \\{} & {} \quad \left( \varvec{P}^{[0]}-\left( 1-\omega \right) \varvec{B}^{+}\varvec{N}^T\varvec{Q}\right) . \end{aligned}$$

Consequently,

$$\begin{aligned} \begin{aligned} \varvec{P}^{[\infty ]}&= \left( \varvec{I}-\left( \varvec{VT}\right) ^{-1}\varvec{B}^{+}\varvec{B}\left( \varvec{VT}\right) \right) \\&\quad \left( \varvec{P}^{[0]}-\left( 1-\omega \right) \varvec{B}^{+}\varvec{N}^T\varvec{Q}\right) \\&\quad + \left( 1-\omega \right) \varvec{B}^{+}\varvec{N}^T\varvec{Q} \\&= \left( 1-\omega \right) \left( \varvec{VT}\right) ^{-1}\varvec{B}^{+}\varvec{B}\left( \varvec{VT}\right) \varvec{B}^{+}\varvec{N}^T\varvec{Q}\\&\quad + \left( \varvec{I}-\left( \varvec{VT}\right) ^{-1}\varvec{B}^{+}\varvec{B}\left( \varvec{VT}\right) \right) \varvec{P}^{[0]}, \end{aligned} \end{aligned}$$

where $\varvec{P}^{[0]}$ is the initial control point, which can be randomly selected. When $\varvec{P}^{[0]}=0$, iterative method (9) converges to

$$\begin{aligned} \varvec{P}^{[\infty ]} = \left( 1-\omega \right) \left( \varvec{VT}\right) ^{-1}\varvec{B}^{+}\varvec{B}\left( \varvec{VT}\right) \varvec{B}^{+}\varvec{N}^T\varvec{Q}. \end{aligned}$$

(31)

$\square$

Remark Appendix 6

If matrix $\varvec{V}$ is the inverse matrix of $\varvec{T}$, that is, $\varvec{VT}=\varvec{TV}=\varvec{I}$, then Eq. (31) becomes

$$\begin{aligned} \varvec{P}^{[\infty ]} = \left( 1-\omega \right) \varvec{B}^{+}\varvec{N}^T\varvec{Q}, \end{aligned}$$

where $\varvec{B}^{+}$ is the M-P pseudo-inverse of the matrix $\varvec{B}$. Then iterative method (9) converges to the solution of classical energy minimization Eq. (17).