Product quality control is one of the important areas of the production management process. In particular, when designing and manufacturing some particularly important objects, there is a need to check their internal structure for defects in them. The methods of computed tomography are by far the leading among the methods of flaw detection. Therefore, developing and researching methods for verifying the internal structure of multilayered objects using tomographic methods is an urgent task. This work is dedicated to this task. This paper investigates the method of approximation of discontinuous functions of two variables by discontinuous splines. These functions describe the internal structure of the 2D body. The unknown parameters are found in them using projections coming from a computer tomograph. It is proposed to use discontinuous splines for the automatic representation of these functions, with known lines of discontinuity of a special form, in the form of a single analytical expression. It is also proposed to use the O. M. Lytvyn method for calculating the Fourier coefficients of two variables using periodic discontinuous splines of one variable and projections. This allows you to submit discontinuous functions in the form of a discontinuous spline sum and a finite Fourier sum. Thus, the proposed method does not require the decomposition of a discontinuous component into a Fourier series. This allows the approximation to be obtained using tomography data without the Gibbs phenomenon.
As is known [
nomenon also occurs in computer tomography [
In [
To obtain experimental data in the DFM method, a parallel scan scheme is used. The main idea of the DFM method is to find Fourier transforms of projections p and use them to find the Fourier transform of a function f that describes the investigated image.
Numerical implementation includes a discrete Fourier transform discrete function transformation f using a fast Fourier transform. p and inverse
Therefore, to restore the image N N the method DFM falls into a number of very fast methods, with the number of arithmetic operations is proportional N 2 log N . But there are two problems that arise with its numerical implementation. The first problem is the need to perform an interpolation procedure in Fourier's space, which is a complicated procedure.
The authors of the DFM method assert that all polynomial interpolation methods are not suitable for this purpose. Therefore, they propose another method of interpolation, constructed using Fourier series and the central section theorem (Sampling Theorem).
The second problem is that in practice contours of images have sharpening, angular points, which leads to an increase in the breaks in the function f . It is well known that in the approximation of this function by finite Fourier sums, the Gibbs phenomenon arises. It generates nonphysical oscillations in the form, that is oscillations, which the original does not have.
The authors of the DFM method claim that they have investigated several methods of combating the Gibbs phenomenon and as a result proposed two different ways to eliminate most of the oscillations.
The first one is a simple application of an exponential filter for Fourier coefficients.
The second way is using the fact that the amplitude of the oscillation is proportional to the jump of the function f . Jumps have different values for typical applications.
For example, a person's skull has a greater density than the brain inside, which leads to the largest jump of function near the surface of the skull.
Also the finite sum of Fejer [
In this paper, it is proposed to generalize the method introduced in [
Method of finite Fourier sums
The problem of image reconstruction is to restore the function f (x, y) based on known projection data – the values of integrals k along the lines Lk that cross the object of the study: f (x, y) dl = k , k = 1, M .
Lk
In the future we will assume, that the object of research belongs to the square D = 0,12 . This problem can be interpreted as a problem of studying the density f (x, y) inside a body on the plane Oxy by methods of X-ray computerized tomography.
To solve the problem, the method proposed by O. M. Lytvyn in [
k =−N l=−N Fk,l = f ( x, y) e−i2(kx+ly)dxdy.
D
The peculiarity and advantage of the developed method is that it proposes explicit formulas for the approximate calculation of the Fourier coefficients of the function of two variables by means the values of projections coming from a computer tomograph to a processor.
This led to the solution of the problem to the calculation of integrals. The choice of a system of straight lines by which projective data is given, and hence the form of integrals, and the form of formulas for their calculation, is determined by the values of the indices k and l in the Fourier sum.
To calculate the Fourier coefficients by means of projections, it is necessary to
consider separately the cases concerning the signs k , l and their mutual position on
the numerical axis ( k l, k l, k = l ). A detailed description of the method is given
in [
Formulation of the problem and idea of the method
In this paper, we propose explicit formulas for the construction of discontinuous splines of two variables with first kind discontinuities on the boundary of a system of embedded one to two two-dimensional domains. (1) (2) ries k : wk (x, y) = 0 r = 0,1 ; D0 D1
Theorem 1. If domains DM .
A method of their use is proposed for approximating the discontinuous functions of two variables by finite Fourier sums, in which Fourier coefficients are calculated only for that component of an approximate function, which is a continuous or continuously differentiable function.
The basic idea of the method is as follows: the discontinuous function f (x, y) is replaced by the sum of the discontinuous spline sp(x, y) and the continuous function F (x, y) .
f (x, y) = sp(x, y) + F (x, y).
In this paper, we assume that the function is periodic with period 1 by variable x and period 1 by variable y and has known breaks of the first kind at the boundaDk : wk (x, y) 0, wk (x, y) C r (R2 ), k = 0, M −1, f0 (x, y), w0 (x, y) 0, f1(x, y), w0 (x, y) 0, w1(x, y) 0, f (x, y) = fk (x, y), wk −1(x, y) 0, wk (x, y) 0, fM (x, y), wM −1(x, y) 0, then we can represent f (x, y) in the following analytical form:
1 M −1 f ( x, y) = 2 f0 (x, y) + fM ( x, y) + ( fk +1 ( x, y) − fk ( x, y)) k =0 wk ( x, y) wk ( x, y) .
Proving. Let it M = 1. Then the function [
1 f ( x, y) = 2 f0 (x, y) + f1 ( x, y) + ( f1(x, y) − f0 ( x, y)) w0 ( x, y) w0 ( x, y) and has properties: if w0 (x, y) 0 , then if w0 (x, y) 0 , then
1 f ( x, y) = ( f0 (x, y) + f1(x, y) − ( f1(x, y) − f0 (x, y))) = f0 (x, y), 2 (3) (4) (5) f (x, y) = 12 ( f0 (x, y)+ f1(x, y) + ( f1(x, y) − f0 (x, y))) = f1(x, y). That is, the assertion of Theorem 1 for M =1 is fulfilled.
Let it M = 2. Then the function: f (x, y) = 12 f0(x, y)+ f2 (x, y)+ ( fk+1(x, y) − fk (x, y)) 1 k=0 wk (x, y) wk (x, y) takes a look f (x, y) = 12 f0(x, y)+ f1(x, y)+( f1(x, y) − f0 (x, y)) w0 (x, y) w0 (x, y) + + ( f2(x, y) − f1(x, y)) w1(x, y) w1(x, y) .
This feature has the following properties: if w0(x, y) 0 , w1(x, y) 0 , then if w0(x, y) 0 , w1(x, y) 0 , then if w0(x, y) 0 , w1(x, y) 0 , then f (x, y) = 12( f0(x, y)+ f2 (x, y) −( f1(x, y) − f0 (x, y)) −
−( f2(x, y) − f1(x, y))) = f0 (x, y), f (x, y) = 12( f0(x, y)+ f2 (x, y)+( f1(x, y) − f0 (x, y)) −
−( f2(x, y) − f1(x, y))) = f1(x, y), f (x, y) = 12( f0(x, y)+ f2 (x, y)+( f1(x, y) − f0 (x, y))+
+( f2(x, y) − f1(x, y))) = f2 (x, y).
That is, the assertion of Theorem 1 for M = 2 is fulfilled.
Let M 2, then formula (5) with w0(x, y) 0, w1(x, y) 0 , ..., wM−1(x, y) 0 gives equality
M−1 f (x, y) = 12 f0(x, y)+ fM(x, y)+ ( fk+1(x, y) − fk(x, y))(−1) = f0(x, y).
k=0
w0 (x, y) 0,..., wp−1(x, y) 0, wp (x, y) 0,..., wM −1( x, y) 0.
The latter is obtained after the disclosure of the brackets and obvious transformations. Let w0 (x, y) 0, w1(x, y) 0, …, wM −1 (x, y) 0. Then formula (5) gives: 1 M −1 f (x, y) = f 0 (x, y) + f M (x, y) + ( f k +1 (x, y) − f k (x, y))(1) = f M (x, y).
2 k =0 Thus, all the assertions of Theorem 1 are fulfilled. Theorem 1 is proved. 4
Construction of a split spline
fk− ( x, y) = lim f (u, v), fk+ ( x, y) = lim f (u, v), k = 0, M −1. (wwukk,((vux)→,,vy))(=x00,,y) (wwukk,((vux)→,,vy))(=x00,,y) − fk+−1(x, y)wk (x, y) + fk− (x, y)wk −1(x, y) wk −1(x, y) − wk (x, y)
h0 (x, y) = f0− (x, y), hk (x, y) = Denote: k = 1, M −1, hM = fM+ −1(x, y).
h0 (x, y), w0 (x, y) 0, hk (x, y), wk −1(x, y) 0, wk (x, y) 0, sp(x, y) = k = 1, M −1 hM (x, y), wM −`1(x, y) 0. (6) Theorem 2. Function sp( x, y) is a discontinued spline with properties: lim sp ( u, v) = lim f ( u, v) = f − ( x, y ) ,
p (u,v)→( x, y ) wp (u,v)0, wp ( x, y )=0 (u,v)→( x, y ) wp (u ,v )0, wp ( x, y)=0 lim sp ( u, v) = lim f ( u, v) = f + ( x, y ) , p p = 0, M − 1. (u,v)→( x, y) wp (u,v)0, wp ( x, y )= 0 (u,v)→( x, y) wp (u ,v)0, wp ( x, y)=0 Proving. Let 1 p M − 2 , then lim sp ( u, v) = lim h (u, v) =
p (u,v)→( x, y) wp (u,v)0, wp ( x, y)=0 (u,v)→( x, y) wp (u,v)0, wp ( x, y)=0
lim (u,v)→( x, y) wp (u,v)0, wp ( x, y)=0 =
lim (u,v)→( x, y) w wp (u,v)0, wp ( x, y)=0 − f + p −1 (u, v)w (u, v)
p p −1 (u, v) − w (u, v) p +
lim (u,v)→( x, y) w wp (u,v)0, wp ( x, y)=0 − p lim sp(u, v ) = lim h
(u, v ) = p +1 (u ,v)→( x, y ) w p (u ,v)0, w p ( x, y )=0
lim (u,v)→( x, y ) w p (u,v)0, w p ( x, y )=0 − f + p (u, v)w p +1 (u, v) + f − p +1 (u, v)w
lim (u ,v)→( x, y ) w w p (u ,v)0, w p ( x, y )=0
p − f + p
(u, v)w lim sp (u, v) = lim h (u, v) =
0 (u,v)→( x, y) (u,v)→( x, y) w0 (u,v)0, w0 (u,v)0, w0 ( x, y)=0 w0 ( x, y)=0
lim (u,v)→( x, y) w0 (u,v)0, w0 ( x, y)=0 0 − f (u, v) = f − 0 ( x, y), lim sp(u, v) = lim h (u, v) =
1 (u,v)→( x, y) (u,v)→( x, y) ww 00 ((ux,,vy))=00, ww 00 ((ux,,vy))=00,
lim (u,v)→( x, y) w 0 (u,v)0, w 0 ( x, y)=0 =
lim (u,v)→( x, y) w (u, v) − w (u, v) w 0 (u,v)0, 0 1 w 0 ( x, y)=0 (u,v)→( x, y) w (u, v) − w (u, v) w 0 (u,v)0, 0 1 w 0 ( x, y)=0 − f + (u, v)w (u, v) 0 1 + lim f − (u, v)w (u, v) 1 0 = = f + ( x, y) + 0 = f + ( x, y).
0 0 Thus, the function sp(x, y)
has the same unilateral boundaries on the lines p : wp (x, y) = 0, p = 0, M − 2 as the function f ( x, y) and the same breaks of the first kind. To prove the assertions of the theorem for p = M − 1 , we give the following chain of equalities.
− f 0+ (u, v)w1 (u, v) + f1− (u, v)w0 (u, v) w0 (u, v) − w1 (u, v) = =
lim (u,v)→(x, y) wwMM −−11 ((ux,,vy))=00, lim sp (u, v) = lim h
(u, v) = (u,v)→(x, y) wM −1 (u,v)0, wM −1 (x, y)=0
M −1 (u,v)→(x, y) wM −1 (u,v)0, wM −1 (x, y)=0 − fM+ −2 (u, v)wM −1 (u, v) + fM− −1 (u, v)wM −2 (u, v) w
M −2
lim + lim (wwuMM,v−−11) →((ux(,,vyx)),=y0)0, wM −2 (u, v) − wM −1 (u, v) (wwuMM,v−−11) →((ux(,,vyx)),=y0)0, wM −2 (u, v) − wM −1 (u, v) = − fM+ −2 (u, v)wM −1 (u, v) fM− −1 (u, v)wM −2 (u, v) lim sp(u, v)= lim h (u, v)= f + M −1 (u, v) = f + ( x, y).
M −1 (u,v)→( x, y) wM −1 (u,v)0, wM −1 ( x, y)=0
In the case if the function fk++1 (x, y) − fk− ( x, y) 0 is in some parts of at least one
line w (x, y) = 0, then its decomposition in the finite Fourier sum is accompanied by
k
the Gibbs phenomenon. For this case, in the writings of Gottlieb and Gustafsson [
= 0 + fM− −1 ( x, y) = fM− −1 ( x, y).
M (u,v)→( x, y) wM −1 (u,v)0, wM −1 ( x, y)=0
lim (u,v)→( x, y) wM −1 (u,v)0, wM −1 ( x, y)=0
Implementation of the method
In this article we propose a solution to the problem of approximating discontinuous functions using of the finite Fourier sums as follows. We give f (x, y) in the form of the sum of a discontinuous spline sp(x, y), which has the same unilateral boundaries on the lines of the discontinuity as the approximate function f (x, y), and the finite Fourier sum TN (x, y), which approximates the difference between the function f (x, y) and the indicated spline.
In this article we propose a solution to the problem of approximating discontinuous functions using of the finite Fourier sums as follows (x, y) = f (x, y) − sp(x, y),
(x, y) TN (x, y).
Theorem 3. The function (x, y) belongs to the class of continuous periodic functions, i. e.
(x, y) C [
The proof follows from the fact that, according to Theorem 2, the function (x, y)
on the lines of the discontinuity will have unilateral boundaries equal to zero. That is,
they are continuous throughout the region [
Theorem 4. If a function f ( x, y) is nonperiodic and has first-kind breaks only
on lines inside the domain [
Proving. Find traces [
Thus, the function f ( x, y) , that we consider to be extended periodically to the entire plane Oxy , will satisfy the conditions: f (x + T , y) = f (x, y), f (x, y + T ) = f (x, y), T = 1.
This means that the function f ( x, y) is periodic.
f (x, y) = ( x, y) + f (x, y)
and approximate the Fourier sums only f ( x, y), which allows a periodic extension to the entire plane Oxy . 6
A general algorithm for the approximation of a discontinuous function with the help of discontinuous splines and projective data
We now formulate a general algorithm for approximation a discontinuous function f (x, y) with the help of discontinuous splines and projections coming from a computer tomograph, considering the lines of discontinuity and one-sided boundaries known.
Step 1. We construct a discontinuous spline as a function sp(x, y), considering also known as boundaries fk+ (x, y) and fk− (x, y) in the form:
f k+ (x, y) = f k +1 x − wk (x, y) x wk (x, y) y , wk (x, y) =
wk (x, y) wk (x, y) 2 wk (x, y) 2 − x + y .
Step 3. For the function (x, y) we find:
(x, y) = f (x, y) − sp(x, y). (x, y) = (x, y) − (x, y), where
(x, y) = (0, y)(1 − x) + (1, y)x + (x, 0)(1 − y) + + (x,1) y − (0, 0)(1 − y) + (0,1) y (1 − x) − (1, 0)(1 − y) + (1,1) y x.
f (x, y) = sp(x, y) + (x, y) + (x, y).
Remark. In formula (7), the sum of first two terms is a no periodic discontinuous component of a function f (x, y) on a given system of lines; the third term is a component of a function that allows a periodic extension to the entire plane Oxy.
Since (x, y) is a continuous, periodic, and also if derivatives (1.0) (x, y),
(0,1) (x, y) are continuous, it can be approximately represented as a finite Fourier
sum, the Fourier coefficients of which are found by means of projections by the
method of O. M. Lytvyn [
Note, that from a computer tomograph projections come from an unknown function f (x, y) , therefore projections of the function (x, y) will be based on the formula:
kx+ly=tp (x, y)ds =
kx+ly=tp f (x, y)ds − sp(x, y) + (x, y) ds.
kx+ly=tp (7) (8)
Example
k = 0, M −1 , that is: w(x, y) = (x − 0.5) 2 + ( y − 0.5) 2 .
wk (x, y) = 0, k = 0, M −1 are circles wk (x, y) w( x, y) − rk = 0,
D0 : w(x, y) r0 , , Dk : w(x, y) rk , k = 1, M −1, DM = [
r 0, 0 2. the center at the point (0.5, 0.5).
Here r and coordinates of the point ( x, y) in the polar coordinate system with In this case, the formula (6) for sp( x, y) the functions hk (x, y) can be written as: h0 (x, y) = f0 (r0 cos , r0 sin ) , , hk (x, y) = = fk (rk −1 cos , rk −1 sin ) w(x, y) − rk + fk (rk cos , rk sin ) w(x, y) − rk −1 , k = 1, M −1, rk −1 − rk rk − rk −1 hM (x, y) = fM (rM −1 cos , rM −1 sin ).
The results of the computational experiment and their analysis are planned to be presented in further developments. 8
Conclusions 1. In this paper, we propose a general method for the approximation of an unknown function f ( x, y) by means of projections, coming from a computer tomograph, for the case, when known line break function f ( x, y) and known unilateral boundaries functions for fk (x, y).
2. To introduce the proposed method to practice, the authors plan to develop and study the method of finding the lines of discontinuity and unilateral boundaries of an unknown function f ( x, y) on the specified lines.