Keywords

1 Introduction

Currently, enterprises engaged in the design and manufacture of complex equipment are actively using CAD/CAM/CAE/PDM systems to control the quality of products. In these systems, increased attention is paid to improving the technology of geometric three-dimensional modeling. The main problem is not the modeling process itself, but the methods of modifying and optimizing the created geometric models, which is very critical during the iterative mode of the designer. Therefore, today, the problems of improving the methods of geometric modeling of three-dimensional objects using the mathematical apparatus standard for CAD/CAE/CAM systems, as well as the adaptation of these methods for specific industrial applications, are relevant. Recently, wavelets have been used in many problems of geometric modeling, mainly in computer graphics [1, 2]. In this article, the inverse discrete wavelet transform and convolution transform are applied to finding computational algorithms for locally approximating splines [3], and then applying them to the development of a CAD system for manufacturing structures from composite materials by automated calculation.

2 Veivlets on the Segment and Rectangle

For geometric applications we will consider actual spaces \(L_{2} (R)\) and \(L_{2} [a;b]\). Let’s take a look at the actual features defined on the segment \([a;b]\). Let the function \(\varphi \in L_{2} (R)\) satisfy the large-scale ratio [4].

$$ \varphi (x) = \sqrt 2 \sum\limits_{k \in Z} u_{k} \varphi (2x - k),u_{k} \in R $$

and has a compact medium. Let’s denote \(\varphi_{jk} (x) = \varphi (2^{j} x - k)\), \(x \in [a;b]\), \(j,k \in Z\). It is clear that for everyone \(j\) different from zero on the segment \([a;b]\) will be only the final number of such functions. Let these be functions for certainty \(\varphi_{j,0} ,\varphi_{j,1} , \ldots ,\varphi_{{j,n_{j} - 1}}\).

Let’s take a look [4] at the sequence \(V_{0} \subset V_{1} \subset \ldots\) of space subspaces \(L_{2} [a;b]\).

$$ V_{j} = lin\left\{ {\varphi_{j,0} ,\varphi_{j,1} , \ldots ,\varphi_{{j,n_{j} - 1}} } \right\} = \left\{ {\sum\limits_{s = 0}^{{n_{j} - 1}} a_{s} \varphi_{j,s} :\,\,\,\,\,a_{s} \in R,s = 0,1, \ldots ,n_{j} - 1} \right\},\dim V_{j} = n_{j} $$

Because \(V_{j - 1} \subset V_{j}\), then \(\varphi_{j - 1,k} = \sum\limits_{s = 0}^{{n_{j} - 1}} p_{s,k}^{j} \varphi_{j,s}\). Let’s introduce the designations [4].

$$\varPhi _{j} (x) = \left( {\begin{array}{*{20}c} {\varphi_{j,0} (x)} & {\varphi_{j,1} (x)} & \ldots & {\varphi_{{j,n_{j} - 1}} (x)} \\ \end{array} } \right),P_{j} = (p_{s,k}^{j} )_{s = 0,k = 0}^{{n_{j} - 1,\,\,n_{j - 1} - 1}} $$

Then \(\varPhi_{j - 1} = \varPhi_{j} P_{j}\). Let’s designate the symbol \(W_{j - 1}\) as an orthogonal addition to space \(V_{j - 1}\) in space \(V_{j}\) because \(V_{j} = V_{j - 1} \oplus W_{j - 1}\) and \(W_{j - 1} \subset V_{j}\), then \(W_{j - 1}\) the ultimate space. If \(W_{j} = lin\left\{ {\psi_{j,0} ,\psi_{j,1} , \ldots ,\psi_{{j,m_{j} - 1}} } \right\}\), \(\dim W_{j} = m_{j}\), then \(\psi_{j - 1,k} = \sum\limits_{s = 0}^{{n_{j} - 1}} q_{s,k}^{j} \varphi_{j,s}\). Functions \(\psi_{j,k}\) are called veilets, and spaces \(W_{j}\) are called veilet spaces [4].

$$ \begin{array}{*{20}l} {\varPsi_{j} (x) = \left( {\begin{array}{*{20}c} {\psi_{j,0} (x)} & {\psi_{j,1} (x)} & \ldots & {\psi_{{j,m_{j} - 1}} (x)} \\ \end{array} } \right),} \hfill \\ { Q_{j} = (q_{s,k}^{j} )_{s = 0,k = 0}^{{n_{j} - 1,m_{j - 1} - 1}} .} \hfill \\ \end{array} $$

Then \(\varPsi_{j - 1} = \varPhi_{j} Q_{j}\). It should be noted that. \(n_{j} + m_{j} = n_{j + 1}\).

Let it be \(f \in L_{2} [a;b]\) and \(\varPi_{j} :L_{2} [a;b] \to V_{j}\) projector. Then the approximation \(\varPi_{j} f\) can be decomposed into a rougher approximation \(\varPi_{j - 1} f\) and clarifying the stoic \(\varPi_{j - 1}^{W} f\).

$$ \varPi_{j} f = \sum\limits_{k = 0}^{{n_{j} - 1}} c_{j,k} \varphi_{j,k} = \varPi_{j - 1} f + \varPi_{j - 1}^{W} f = \sum\limits_{k = 0}^{{n_{j - 1} - 1}} c_{j - 1,k} \varphi_{j - 1,k} + \sum\limits_{k = 0}^{{m_{j - 1} - 1}} d_{j - 1,k} \psi_{j - 1,k} . $$

Let’s introduce two vectors of coefficients into the review.

$$ C_{j} = \left( {\begin{array}{*{20}c} {c_{j,0} } & \ldots & {c_{{j,n_{j} - 1}} } \\ \end{array} } \right)^{T} ,D_{j} = \left( {\begin{array}{*{20}c} {d_{j,0} } & \ldots & {d_{{j,n_{j} - 1}} } \\ \end{array} } \right)^{T} . $$

The first vector describes the approximation of a function \(f\), and the second vector is a veilet-coefficient that characterizes deviation \(\varPi_{j - 1} f\) from \(\varPi_{j} f\) [4].

$$ C_{j} = P_{j} C_{j - 1} + Q_{j} D_{j - 1} . $$

On this equality it is possible to restore \(\varPi_{j} f\) the approach on a rougher approach \(\varPi_{j - 1} f\) and veilet-coefficents. Because line operators (projectors) \(V_{j} \to V_{j - 1}\) are \(V_{j} \to W_{j - 1}\) defined by some matrix \(A_{j}\), \(B_{j}\), then \(C_{j - 1} = A_{j} C_{j}\), \(D_{j - 1} = B_{j} C_{j}\). By the function’s \(f\) vevlet transformation, we will understand the location of vectors \(C_{0} ,D_{0} ,D_{1} , \ldots ,D_{j - 1}\). The relationship [4] between the matrix \(A_{j} ,B_{j}\) and the \(P_{j} ,Q_{j}\).

$$ \left( {\begin{array}{*{20}c} {A_{j} } \\ {B_{j} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {P_{j} } & {Q_{j} } \\ \end{array} } \right)^{ - 1} . $$

The matrix in \({\text{Q}}_{j}\) the article [4] is defined from a homogeneous system of linear equations \({\text{T}}_{j} {\text{Q}}_{j} = 0,\) where \({\text{T}}_{j} = {\text{P}}_{j}^{T} [(\varPhi_{j} ,\varPhi_{j} )]\), as well \([(\varPhi_{j} ,\varPhi_{j} )] = ((\varphi_{j,i} ,\varphi_{j,s} ))_{i,s = 0}^{{n_{j} - 1}}\) - the matrix of scalar works. Matrices \({\text{Q}}_{j}\) and \({\text{P}}_{j}\) are known as synthesis filters. Matrices \({\text{A}}_{j}\) and \({\text{B}}_{j}\) are known as analysis filters. The set \(\{ {\text{P}}_{j} ,{\text{Q}}_{j} ,{\text{A}}_{j} ,{\text{B}}_{j} \}\) is called a filter bank.

About the above-written approach to building a veilet system on a segment in the article [5] is applied to the case when the function \(\varphi (x)\) is chosen B-spline of arbitrary order \(n\). Define the B-splines of the order as \(n\) a convolution [6]

$$ N_{n} = N_{n - 1} * N_{0} , N_{0} (x) = \left\{ \begin{gathered} 1,\,\,\,\,\,x \in [0;1), \hfill \\ 0,\,\,\,\,x \notin [0;1). \hfill \\ \end{gathered} \right. $$

Note some of the well-known properties of B-splines [6]. Second, \(N_{n} (x) \ge 0\) as shown \(x\) in the feature \({\text{supp}}\,\,N_{n} (x) = [0;n + 1]\), As shown in [6], the function \(N_{n} (x)\), satisfies the scale ratio

$$ N_{n} (x) = \sum\limits_{k = 0}^{n + 1} \frac{{p_{k} }}{{2^{n} }}N_{n} (2x - k), p_{k} = C_{n + 1}^{k} = \frac{(n + 1)!}{{k!(n + 1 - k)!}}. $$
(1)

In addition, the function \(N_{n} (x)\) satisfies the ratio of [6]

$$ N_{n} (x) = \frac{x}{n}N_{n - 1} (x) + \frac{n + 1 - x}{n}N_{n - 1} (x - 1) $$
(2)

and justly equality [6].

$$ N^{\prime}_{n + 1} (x) = N_{n} (x) - N_{n} (x - 1). $$
(3)

In article [5], the filter bank is built for the case \(\varphi (x) = N_{n} (x)\).

Consider now the use of veilet systems on the segment to build two-dimensional veiwelets on a rectangular area. Let the sequences \(V_{0,i} \subset V_{1,i} \subset \ldots V_{j,i} \subset\) of the final subspaces of space \(L_{2} [a_{i} ;b_{i} ]\) scale functions \(\varphi^{i}\) and banks of filters \(P_{j,i}\), \(Q_{j,i}\), \(A_{j,i}\), \(B_{j,i}\), \(i = 1,2\). The standard approach [7] to the construction of multidimensional wavelet systems is to take tensor products of basis functions from \(V_{j,i}\). Define subspaces \(V_{j}^{2} = V_{j,1} \otimes V_{j,2} = lin\left\{ {f_{1} \otimes f_{2} : f_{1} \in V_{j,1} , f_{2} \in V_{j,2} } \right\}\), where the function \(f_{1} \otimes f_{2}\) is defined by the rule \(f_{1} \otimes f_{2} (x,y) = f_{1} (x)f_{2} (y).\) In addition, we define spaces \(W_{j}^{2}\) as follows \(V_{j}^{2} = V_{j - 1}^{2} \oplus W_{j - 1}^{2}\). Then, if \(f \in L_{2} ([a_{1} ;b_{1} ] \times [a_{2} ;b_{2} ])\) and \(\varPi_{j} :L_{2} ([a_{1} ;b_{1} ] \times [a_{2} ;b_{2} ]) \to V_{j}^{2}\) is a projector, then

$$ \begin{gathered} \varPi_{j} f = \sum\limits_{m = 0}^{{n_{j,1} - 1}} \,\,\,\sum\limits_{l = 0}^{{n_{j,2} - 1}} c_{m,l}^{j} \varphi_{j,m}^{(1)} \otimes \varphi_{j,l}^{(2)} = \sum\limits_{m = 0}^{{n_{j,1} - 1}} \,\,\,\sum\limits_{l = 0}^{{n_{j,2} - 1}} \left( {\sum\limits_{k = 0}^{{n_{j - 1,1} - 1}} \,\,\,\sum\limits_{s = 0}^{{n_{j - 1,2} - 1}} } \right.c_{k,s}^{j - 1} p_{m,k}^{j,1} p_{l,s}^{j,2} \hfill \\ + \sum\limits_{k = 0}^{{m_{j - 1,1} - 1}} \,\,\sum\limits_{s = 0}^{{n_{j - 1,2} - 1}} r_{k,s}^{j - 1} q_{m,k}^{j,1} p_{l,s}^{j,2} + \sum\limits_{k = 0}^{{n_{j - 1,1} - 1}} \,\,\,\sum\limits_{s = 0}^{{m_{j - 1,2} - 1}} h_{k,s}^{j - 1} p_{m,k}^{j,1} q_{l,s}^{j,2} + \sum\limits_{k = 0}^{{m_{j - 1,1} - 1}} \,\,\,\sum\limits_{s = 0}^{{m_{j - 1,2} - 1}} \left. {d_{k,s}^{j - 1} q_{m,k}^{j,1} q_{l,s}^{j,2} } \right)\varphi_{j,m}^{(1)} \otimes \varphi_{j,l}^{(2)} . \hfill \\ \end{gathered} $$
(4)

If you introduce the matrix \(C_{j} = (c_{m,l}^{j} )_{m,l = 0}^{{n_{j,1} - 1,n_{j,2} - 1}}\), \(R_{j} = (r_{k,s}^{j} )_{k,s = 0}^{{m_{\,j,1} - 1,\,\,\,n_{j,2} - 1}}\), \(H_{j} = (h_{k,s}^{j} )_{k,s = 0}^{{n_{j,1} - 1,\,\,\,m_{j,2} - 1}}\), \(D_{j} = (d_{k,s}^{j} )_{k,s = 0}^{{m_{j,1} - 1,\,\,m_{j,2} - 1}}\), into consideration (1), we get it from equality [5]

$$ C_{j} = P_{j,1} C_{j - 1} P_{j,2}^{T} + Q_{j,1} R_{j - 1} P_{j,2}^{T} + P_{j,1} H_{j - 1} Q_{j,2}^{T} + Q_{j,1} D_{j - 1} Q_{j,2}^{T} $$
(5)

It is also obvious [5] that

$$ C_{j - 1} = A_{j,1} C_{j} A_{j,2}^{T} ; R_{j - 1} = B_{j,1} C_{j} A_{j,2}^{T} ; H_{j - 1} = A_{j,1} C_{j} B_{j,2}^{T} ; D_{j - 1} = B_{j,1} C_{j} B_{j,2}^{T} $$
(6)

Formulas (6) give the veilet decomposition of the approximate \(\varPi_{j} f\) function of two arguments, and formula (5) gives a veilet-resolution.

3 Computational Algorithms for Locally Approximates

In this section, we obtained some computational formulas for local approximation spline in narrow grids based on wavelet recovery. The obtained formulas will be used in the approximate calculation of integral data obtained in various grids [8].

Consider the following B-splines

$$ N_{l,m,i} (x) = N_{m - 1} (2^{l} x + m - i),\;l, i \in Z,\;m = 1,2, \ldots $$

Note some of the obvious properties of these features.

Lemma 1.

There is equality

$$ N_{l,m,i} (x) = N_{l,m,m} (x + 2^{ - l} (m - i)) $$
(7)

Proof.

We have

$$ N_{l,m,m} (x + 2^{ - l} (m - i)) = N_{m - 1} (2^{l} (x + 2^{ - l} (m - i))) = N_{m - 1} (2^{l} x + m - i) = N_{l,m,i} (x). $$

Lemma 2.

There is equality

$$ N_{0,m,m} (2^{l} x + k) = N_{l,m,m - k} (x) $$
(8)

Proof.

We have

$$ N_{l,m,m - k} (x) = N_{m - 1} (2^{l} x + k) = N_{0,m,m} (2^{l} x + k). $$

Lemma 3.

Functions \(N_{l,m,i} (x)\) satisfy the following Cox – de Boer relation [8].

$$ {N_{l,m,i} (x) = \frac{{i - 2^{l} x}}{m - 1}N_{l,m - 1,i} (x) + \frac{{2^{l} x - i + m}}{m - 1}N_{l,m - 1,i - 1} (x)}; $$
(9)
$$ N_{l,1,i} \left( x \right) = \left\{ {\frac{{1,x \in \left[ {\frac{i - 1}{{2^{l} }},\frac{i}{{2^{l} }}} \right);}}{{0,x \notin \left[ {\frac{i - 1}{{2^{l} }},\frac{i}{{2^{l} }}} \right).}}} \right. $$

Fair equality

$$ N^{\prime}_{l,m,i} (x) = 2^{l} \cdot N_{l,m - 1,i - 1} (x) - 2^{l} \cdot N_{l,m - 1,i} (x). $$
(10)

Proof.

From equality (2) we get

$$ \begin{gathered} N_{l,m,i} (x) = N_{m - 1} (2^{l} x + m - i) = \frac{{2^{l} x + m - i}}{m - 1}N_{m - 2} (2^{l} x + m - i) \hfill \\ + \frac{{m - (2^{l} x + m - i)}}{m - 1}N_{m - 2} (2^{l} x + m - i - 1). \hfill \\ \end{gathered} $$

Given that \(N_{m - 2} (2^{l} x + m - i) = N_{l,m - 1,i - 1} (x)\), \(N_{m - 2} (2^{l} x + m - i - 1) = N_{l,m - 1,i} (x)\) we get the approval of the lemma. Equality (10) follows from equality (3).

Lemma 4.

The function \(N_{0,m,m}\) satisfies a large-scale ratio.

$$ N_{0,m,m} (x) = \sum\limits_{k = 0}^{m} \frac{{p_{k} }}{{2^{m - 1} }}N_{0,m,m} (2x - k),\;p_{k} = \frac{m!}{{k!(m - k)!}} $$
(11)

Proof.

From (1) follows

$$ N_{0,m,m} (x) = N_{m - 1} (x) = \sum\limits_{k = 0}^{m} \frac{{p_{k} }}{{2^{m - 1} }}N_{m - 1} (2x - k) = \,\sum\limits_{k = 0}^{m} \frac{{p_{k} }}{{2^{m - 1} }}N_{1,m,m + k} (x) $$

From equality (8) we get \(N_{1,m,m + k} (x) = N_{0,m,m} (2^{l} x - k)\).

Lemma 5.

The function \(N_{l,m,i} (x)\) satisfies the ratio

$$ N_{l,m,i} (x) = \sum\limits_{k = 0}^{m} \frac{{p_{k} }}{{2^{m - 1} }}N_{l + 1,m,2i + k - m} (x). $$
(12)

Proof.

From (8) follows. \(N_{l,m,i} (x) = N_{0,m,m} (2^{l} x + m - i)\) from here, on the basis (11), conclude

$$ \begin{gathered} N_{l,m,i} (x) = \sum\limits_{k = 0}^{m} \frac{{p_{k} }}{{2^{m - 1} }}N_{0,m,m} (2(2^{l} x + m - i) - k) = \sum\limits_{k = 0}^{m} \frac{{p_{k} }}{{2^{m - 1} }}N_{0,m,m} (2^{l + 1} x + (2m - 2i - k)) \hfill \\ = \sum\limits_{k = 0}^{m} \frac{{p_{k} }}{{2^{m - 1} }}N_{l + 1,m,2i + k - m} (x). \hfill \\ \end{gathered} $$

Let \(M:Z^{2} \to R\). We will also \(M(k_{1} ,k_{2} )\) use the designation for values \(M_{{k_{1} ,k_{2} }}\). Let’s define the function with \(M^{ - } :Z^{2} \to R\) equality \(M_{{k_{1} ,k_{2} }}^{ - } = M_{{ - k_{1} , - k_{2} }}\).

Definition 1.

Let and \(M:Z^{2} \to R\), \(F:Z^{2} \to R\) – two functions with compact carriers. The convolution \(M*F\) of these functions is defined by the equality.

$$ (M*F)_{\lambda ,\rho } = \sum\limits_{{k_{1} ,k_{2} = - \infty }}^{ + \infty } {M_{{k_{1} ,k_{2} }} F_{{\lambda - k_{1} ,\,\,\rho - k_{2} }} } $$

We will call \({\text{M}}\) the function the core of the bundle.

Consider the case \(m = 4\). Then the ratio (12) will take the form

$$ N_{l,4,i} (x) = \frac{1}{8}N_{l + 1,4,2i - 4} (x) + \frac{1}{2}N_{l + 1,4,2i - 3} (x) + \frac{3}{4}N_{l + 1,4,2i - 2} (x) + \frac{1}{2}N_{l + 1,4,2i - 1} (x) + \frac{1}{8}N_{l + 1,4,2i} (x) $$

Suppose the values \(f_{{k_{1} ,k_{2} }} = f\left( {u_{{j,k_{1} ,k_{2} }} } \right)\), \(k_{1} ,k_{2} \in Z\) functions \(f(u_{1} ,u_{2} )\) are known in nodes from the site.

$$ u_{{j,k_{1} ,k_{2} }} = \left( {2^{ - j} k_{1} ,2^{ - j} k_{2} } \right), $$
$$ \begin{gathered} k_{1} = k_{j,1,0} - 2,k_{j,1,0} - 1, \ldots ,k_{j,1,1} + 2; \hfill \\ k_{2} = k_{j,2,0} - 2,k_{j,2,0} - 1, \ldots ,k_{j,2,1} + 2. \hfill \\ \end{gathered} $$

The local approximation spline \(\tilde{f}\) is determined by the equality [9].

$$ \tilde{f}(u_{1} ,u_{2} ) = \sum\limits_{{\lambda = k_{j,1,0} + 1}}^{{k_{j,1,1} + 3}} {\sum\limits_{{\rho = k_{j,2,0} + 1}}^{{k_{j,2,1} + 3}} {r_{\lambda ,\rho } N_{j,4,\rho } (u_{2} )N_{j,4,\lambda } (u_{1} )} } ,\;u_{i} \in [2^{ - j} k_{j,i,0} ;2^{ - j} k_{j,i,1} ],\;i = 1,2 $$

where is

$$ \begin{gathered} r_{\lambda ,\rho } = \frac{1}{36}f_{\lambda - 3,\rho - 3} - \frac{2}{9}f_{\lambda - 3,\rho - 2} + \frac{1}{36}f_{\lambda - 3,\rho - 1} - \frac{2}{9}f_{\lambda - 2,\rho - 3} + \frac{16}{9}f_{\lambda - 2,\rho - 2} - \frac{2}{9}f_{\lambda - 2,\rho - 1} + \hfill \\ + \frac{1}{36}f_{\lambda - 1,\rho - 3} - \frac{2}{9}f_{\lambda - 1,\rho - 2} + \frac{1}{36}f_{\lambda - 1,\rho - 1} . \hfill \\ \end{gathered} $$
(13)

denote

$$ R_{j} = (r_{\lambda ,\rho } )_{{\lambda = k_{j,1,0} + 1,\,\,\rho = k_{j,2,0} }}^{{k_{j,1,1} + 3, k_{j,2,1} + 1}} ; $$
(14)
$$ \begin{gathered} \varPhi_{j,1} (u_{1} ) = \left( {\begin{array}{*{20}c} {N_{{j,4,k_{j,1,0} + 1}} (u_{1} )} & \ldots & {N_{{j,4,k_{j,1,1} + 3}} (u_{1} )} \\ \end{array} } \right), \hfill \\ \varPhi_{j,2} (u_{2} ) = \left( {\begin{array}{*{20}c} {N_{{j,4,k_{j,2,0} + 1}} (u_{1} )} & \ldots & {N_{{j,4,k_{j,2,1} + 3}} (u_{2} )} \\ \end{array} } \right) \hfill \\ \end{gathered} $$

Then the function \(\tilde{f}\) can be rewritten in the form of [10].

$$ \begin{gathered} \tilde{f}(u_{1} ,u_{2} ) = \varPhi_{j,1} (u_{1} ) \cdot R_{j} \cdot (\varPhi_{j,2} (u_{2} ))^{T} , \hfill \\ u_{i} \in [2^{ - j} k_{j,i,0} ;2^{ - j} k_{j,i,1} ], \,\,i = 1,2. \hfill \\ \end{gathered} $$

Let’s introduce the kernel of the bundle (the values on the media are specified \([ - 1;1] \times [ - 1;1] \cap {\mathbf{Z}}^{2}\))

$$ \tilde{F} = \frac{1}{36}\left( {\begin{array}{*{20}c} 1 & { - 8} & 1 \\ { - 8} & {64} & { - 8} \\ 1 & { - 8} & 1 \\ \end{array} } \right) $$
(15)

Then, by definition, there is equality \(r_{\lambda ,\rho } = (\tilde{F}^{ - } *f)_{\lambda - 2,\rho - 2}\). Let it be

$$ P_{j + 1,i} = \frac{1}{8}\left( \begin{gathered} 4\,\,\,4\,\,\,0\,\,\,...\,\,\,0 \hfill \\ 1\,\,\,6\,\,\,\,1\,\,\,...\,\,\,\,0 \hfill \\ 0\,\,4\,\,\,\,4\,\,...\,\,\,\,0 \hfill \\ 0\,\,\,1\,\,\,\,6\,\,...\,\,\,\,0 \hfill \\ 0\,\,\,0\,\,\,4\,\,...\,\,\,\,0 \hfill \\ 0\,\,\,0\,\,\,1\,\,...\,\,\,\,0 \hfill \\ ................ \hfill \\ 0\,\,\,0\,\,\,0\,\,...\,\,\,\,4 \hfill \\ \end{gathered} \right) $$

The size matrix, \((2(k_{j,i,1} - k_{j,i,0} ) + 3) \times (k_{j,i,1} - k_{j,i,0} + 3)\), \(i = 1,2\). Then \(\varPhi_{j,i} = \varPhi_{j + 1,i} \cdot P_{j + 1,i}\).

Consequently,

$$ \tilde{f}(u_{1} ,u_{2} ) = \varPhi_{j + 1,1} (u_{1} ) \cdot (P_{j + 1,1} R_{j} \cdot P_{j + 1,2}^{T} ) \cdot (\varPhi_{j + 1,2} (u_{2} ))^{T} ,\,u_{i} \in [2^{ - j} k_{j,i,0} ;2^{ - j} k_{j,i,1} ],i = 1,2, $$

Where in \(k_{j + 1,i,0} = 2k_{j,i,0}\), \(k_{j + 1,i,1} = 2k_{j,i,1}\), \(i = 1,2\). With this transformation, we obtain approximation functions \(f\) on a finer mesh [11]. As follows from formulas (5), in this transformation there is a special case of wavelet recovery. In fig. The 1st example of such a recovery using the example of a test function \(f(u_{1} ,u_{2} ) = e^{{ - \frac{{u_{1}^{2} + u_{2}^{2} }}{8}}}\), \(u_{1} ,u_{2} \in [ - 10;10]\).

Denote

$$ R_{j + l} = (r_{\lambda ,\rho }^{(j + l)} )_{{\lambda = k_{j + l,1,0} ,\,\,\rho = k_{j + l,2,0} }}^{{k_{j + l,1,1} ,\,\,\,k_{j + l,2,1} }} = P_{j + l,1} \cdot \ldots \cdot P_{j + 1,1} \cdot R_{j} \cdot P_{j + 1,2}^{T} \ldots P_{j + l,2}^{T} . $$
(16)

Then the locally approximate splint on the grid \(\left\{ {(2^{ - j - l} k_{1} ,2^{ - j - l} k_{2} )} \right\}_{{(k_{1} ,k_{2} ) \in Z^{2} }}\) has the appearance.

$$ \tilde{f}(u_{1} ,u_{2} ) = \varPhi_{j + l,1} (u_{1} ) \cdot R_{j + l} \cdot (\varPhi_{j + l,2} (u_{2} ))^{T} ,u_{i} \in [2^{ - j - l} k_{j,i,0} ;2^{ - j - l} k_{j,i,1} ],i = 1,2 $$

We now obtain formulas for calculating the values of the locally approximating spline and its partial derivatives at grid nodes [12, 13]. From the Cox - de Boer relation (9) we find

$$ \begin{gathered} N_{l,4,i} \left( {\frac{i - 3}{{2^{l} }}} \right) = \frac{1}{6}; N_{l,4,i} \left( {\frac{i - 2}{{2^{l} }}} \right) = \frac{2}{3}; N_{l,4,i} \left( {\frac{i - 1}{{2^{l} }}} \right) = \frac{1}{6}; \hfill \\ N_{l,3,i} \left( {\frac{i - 2}{{2^{l} }}} \right) = N_{l,3,i} \left( {\frac{i - 1}{{2^{l} }}} \right) = \frac{1}{2}, N_{l,2,i} \left( {\frac{i - 1}{{2^{l} }}} \right) = 1. \hfill \\ \end{gathered} $$
Fig. 1.
figure 1

Vailet-recovery of local-approximation splint.

Full size image

Therefore, based on the formula (10) we get [14, 15].

$$ \begin{gathered} N_{l,4,i}^{\prime } \left( {\frac{i - 3}{{2^{l} }}} \right) = 2^{l - 1} , N_{l,4,i}^{\prime } \left( {\frac{i - 2}{{2^{l} }}} \right) = 0, N_{l,4,i}^{\prime } \left( {\frac{i - 1}{{2^{l} }}} \right) = - 2^{l - 1} ; \hfill \\ N_{l,4,i}^{\prime \prime } \left( {\frac{i - 3}{{2^{l} }}} \right) = 2^{2l} , N_{l,4,i}^{\prime \prime } \left( {\frac{i - 2}{{2^{l} }}} \right) = - 2^{2l + 1} , N_{l,4,i}^{\prime \prime } \left( {\frac{i - 1}{{2^{l} }}} \right) = 2^{2l} . \hfill \\ \end{gathered} $$

Let’s introduce the following bundle kernels (values on the \([ - 1;1] \times [ - 1;1] \cap Z^{2}\) media) [16]:

$$ \begin{gathered} D_{0,0} = \frac{1}{36}\left( {\begin{array}{*{20}c} 1 & 4 & 1 \\ 4 & {16} & 4 \\ 1 & 4 & 1 \\ \end{array} } \right), D_{1,0} = \frac{{2^{l - 1} }}{6}\left( {\begin{array}{*{20}c} { - 1} & { - 4} & { - 1} \\ 0 & 0 & 0 \\ 1 & 4 & 1 \\ \end{array} } \right), D_{0,1} = \frac{{2^{l - 1} }}{6}\left( {\begin{array}{*{20}c} { - 1} & 0 & 1 \\ { - 4} & 0 & 4 \\ { - 1} & 0 & 1 \\ \end{array} } \right),\, \hfill \\ D_{1,1} = 2^{2l - 2} \left( {\begin{array}{*{20}c} 1 & 0 & { - 1} \\ 0 & 0 & 0 \\ { - 1} & 0 & 1 \\ \end{array} } \right), D_{2,0} = \frac{{2^{2l} }}{6}\left( {\begin{array}{*{20}c} 1 & 4 & 1 \\ { - 2} & { - 8} & { - 2} \\ 1 & 4 & 1 \\ \end{array} } \right), D_{0,2} = \frac{{2^{2l} }}{6}\left( {\begin{array}{*{20}c} 1 & { - 2} & 1 \\ 4 & { - 8} & 4 \\ 1 & { - 2} & 1 \\ \end{array} } \right). \hfill \\ \end{gathered} $$

Then the \(\tilde{f}\) following equalities are fair for the values of the locally approximately splined and its private derivatives in the nodes \(\left( {\frac{\lambda }{{2^{j + l} }},\frac{\rho }{{2^{j + l} }}} \right)\) grids [12, 13]. From the Cox - de Boer relation (9) we find

$$ \begin{gathered} D^{{\left( {d_{1} ,d_{2} } \right)}} \tilde{f}\left( {2^{ - j - l} \lambda ,2^{ - j - l} \rho } \right) = D_{{d_{1} ,d_{2} }}^{ - } *R_{j + l} (\lambda + 2,\rho + 2), \hfill \\ d_{1} ,d_{2} = 0,1,2, \,\,\,\,d_{1} + d_{2} \le 2;\,\,\,\,\lambda = k_{j + l,1,0} , \ldots ,k_{j + l,1,1} , \,\,\,\,\rho = k_{j + l,2,0} , \ldots ,k_{j + l,2,1} . \hfill \\ \end{gathered} $$
(17)

4 Algorithms App

One of the methods for producing structures from composite materials is the automated calculation method, in which tapes are placed on the surface of the technological mandrel using pressure rollers [17, 18].

The surface of the fan blade is defined by some sets of points in several sections. First, we model this surface using a locally approximating spline of two variables [19]. Then we find the values of this spline at the nodes of the finer mesh and shift the surface points corresponding to these nodes normal to the surface by a height equal to the thickness of the tape. Next, we carry out the wavelet decomposition with zeroing the wavelet coefficients. As a result, surface smoothing occurs [20].

5 Conclusion

The article presents computational algorithms for locally approximating splines based on the use of the inverse discrete wavelet transform and convolution transform. The algorithms are applied to the development of a part of the CAD system for the manufacture of structures from composite materials by automated calculation.