=Paper=
{{Paper
|id=Vol-2711/paper51
|storemode=property
|title=The Method of Reconstructing Discontinuous Functions Using Projections Data and Finite Fourier Sums
|pdfUrl=https://ceur-ws.org/Vol-2711/paper51.pdf
|volume=Vol-2711
|authors=Oleh M. Lytvyn,Oleksandra Lytvyn,Oleh O. Lytvyn,Vitaliy Mezhuyev
|dblpUrl=https://dblp.org/rec/conf/icst2/LytvynLLM20
}}
==The Method of Reconstructing Discontinuous Functions Using Projections Data and Finite Fourier Sums==
https://ceur-ws.org/Vol-2711/paper51.pdf
The Method of Reconstructing Discontinuous Functions
Using Projections Data and Finite Fourier Sums
Oleh M. Lytvyn1[0000-0003-2775-8471], Oleksandra Lytvyn2[0000-0001-5224-2221],
Oleh O. Lytvyn3[0000-0003-2357-3816], Vitaliy Mezhuyev4[0000-0002-9335-6131]
1 Ukrainian engenering-pedagogical academy, Universitetska str. 16, Kharkiv, 61003, Ukraine
academ_mail@ukr.net
2 Kharkiv National University of Radio Electronics, Nauki avenue 14, Kharkiv, 61166, Ukraine
litvinog@ukr.net
3 Ukrainian engenering-pedagogical academy, Universitska str. 16, Kharkiv, 61003, Ukraine
olegolitvin55@gmail.com
4 FH JOANNEUM University of Applied Sciences, Kapfenberg, Austria,
vitaliy.mezhuyev@fh-joanneum.at
Abstract. Product quality control is one of the important areas of the produc-
tion management process. In particular, when designing and manufacturing
some particularly important objects, there is a need to check their internal struc-
ture for defects in them.
The methods of computed tomography are by far the leading among the meth-
ods of flaw detection. Therefore, developing and researching methods for veri-
fying 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 pro-
jections coming from a computer tomograph. It is proposed to use discontinu-
ous splines for the automatic representation of these functions, with known
lines of discontinuity of a special form, in the form of a single analytical ex-
pression. 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 pro-
posed method does not require the decomposition of a discontinuous compo-
nent into a Fourier series. This allows the approximation to be obtained using
tomography data without the Gibbs phenomenon.
Keywords: computer tomography, reconstruction, image, discontinuous func-
tion, discontinuous spline, sum Fourier.
1 Introduction
As is known [1, 2, 3], the approximation of discontinuous functions of one and
many variables by finite Fourier sums leads to the Gibbs phenomenon [4]. This phe-
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0). ICST-2020
nomenon also occurs in computer tomography [5-7]. For this case, in the works of
Gottlieb and Gustafsson [8, 9], various methods of constructing finite Fourier sums
are given; Fourier coefficients are multiplied by the factors determined in a proper
way in order to reduce the influence of the Gibbs phenomenon on the final result.
In [10], the method of an approximate representation of the function of two varia-
bles by finite Fourier sums was investigated, in which the Fourier coefficients are
found helping projections using the direct and inverse Radon transformation This
method is known as the Direct Fourier Method (DFM).
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 p and inverse
discrete function transformation f using a fast Fourier transform.
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 interpola-
tion, constructed using Fourier series and the central section theorem (Sampling The-
orem).
The second problem is that in practice contours of images have sharpening, angu-
lar 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 phenome-
non 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 coeffi-
cients.
The second way is using the fact that the amplitude of the oscillation is propor-
tional to the jump of the function f . Jumps have different values for typical applica-
tions.
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 [1, 2] are used, which has a smoothing effect, but not
an increase in the accuracy of the result.
In this paper, it is proposed to generalize the method introduced in [11, 12] to the
case of approximating discontinuous functions of two variables using projections and
finite Fourier sums for one important class of discontinuous functions. This corre-
sponds to a new approach to the problem of research, which does not lead to the
Gibbs phenomenon.
2 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 = 1, M .
Lk
k (1)
In the future we will assume, that the object of research belongs to the square
D = 0,1 . This problem can be interpreted as a problem of studying the densi-
2
ty 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 [12] was used.
According to this method, the solution of the task was sought in the form of
a Fourier sum.
N N
f ( x, y ) S N , N ( x, y ) = Fk ,l ei 2 ( kx + ly ) , (2)
k=−N l=−N
where the Fourier coefficients are calculated by the formula
Fk , l = f ( x, y ) e − i 2 ( 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 [12].
3 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.
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 contin-
uously 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 ). (3)
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 bounda-
ries k : wk ( x, y ) = 0 domains Dk : wk ( x, y ) 0, wk ( x, y ) C r ( R 2 ), k = 0, M − 1,
r = 0,1 ; D0 D1 DM .
Theorem 1. If
f 0 ( x, y ), w0 ( x, y ) 0,
f ( x, y ), w ( x, y ) 0, w ( x, y ) 0,
1 0 1
f ( x, y ) = (4)
f k ( x, y ), wk −1 ( x, y ) 0, wk ( x, y ) 0,
f M ( x, y ), wM −1 ( x, y ) 0,
then we can represent f ( x, y ) in the following analytical form:
1 M −1 wk ( x, y )
f ( x, y ) = f 0 ( x, y ) + f M ( x, y ) + ( f k +1 ( x, y ) − f k ( x, y ) ) . (5)
2 k =0 wk ( x, y )
Proving. Let it M = 1. Then the function [13]:
1 w0 ( x, y )
f ( x, y ) = f 0 ( x, y ) + f1 ( x, y ) + ( f1 ( x, y ) − f 0 ( x, y ) )
2 w0 ( x, y )
and has properties:
if w0 ( x, y ) 0 , then
( f0 ( x, y ) + f1 ( x, y) − ( f1 ( x, y) − f 0 ( x, y))) = f 0 ( x, y),
1
f ( x, y ) =
2
if w0 ( x, y ) 0 , then
f ( x, y ) = ( f 0 ( x, y ) + f 1 ( x, y) + ( f 1 ( x, y) − f 0 ( x, y))) = f 1 ( x, y).
1
2
That is, the assertion of Theorem 1 for M = 1 is fulfilled.
Let it M = 2. Then the function:
1 1 wk ( x, y )
f ( x, y ) = f 0 ( x, y ) + f 2 ( x, y ) + ( f k +1 ( x, y ) − f k ( x, y ) )
2 k =0 wk ( x, y )
takes a look
1 w0 ( x, y )
f ( x, y ) = f 0 ( x, y ) + f1 ( x, y ) + ( f1 ( x, y ) − f 0 ( x, y ) ) +
2 w0 ( x, y )
w1 ( x, y )
+ ( f 2 ( x, y ) − f1 ( x, y ) ) .
w1 ( x, y )
This feature has the following properties:
if w0 ( x, y ) 0 , w1 ( x, y ) 0 , then
f ( x, y ) =
1
2
(
f 0 ( x, y ) + f 2 ( x, y ) − ( f1 ( x, y ) − f 0 ( x, y ) ) −
)
− ( f 2 ( x, y ) − f1 ( x, y ) ) = f 0 ( x, y ) ,
if w0 ( x, y ) 0 , w1 ( x, y ) 0 , then
f ( x, y ) =
1
2
(
f 0 ( x, y ) + f 2 ( x, y ) + ( f1 ( x, y ) − f 0 ( x, y ) ) −
)
− ( f 2 ( x, y ) − f1 ( x, y ) ) = f1 ( x, y ) ,
if w0 ( x, y ) 0 , w1 ( x, y ) 0 , then
f ( x, y ) =
1
2
(
f 0 ( x, y ) + f 2 ( x, y ) + ( f1 ( x, y ) − f 0 ( x, y ) ) +
)
+ ( f 2 ( x, y ) − f1 ( x, y ) ) = f 2 ( 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
1 M −1
f ( x, y ) =
2
f 0 ( x , y ) + f M ( x , y ) + ( f k +1 ( x, y ) − f k ( x, y ) )( −1) = f 0 ( x, y ).
k =0
Let 1 p M − 2 and
w0 ( x, y ) 0,..., wp −1 ( x, y) 0, w p ( x, y) 0,..., wM −1 ( x, y) 0.
Then formula (5) gives equality
1 p −1
f ( x, y ) =
2
f 0 ( x , y ) + f M ( x , y ) + ( f k +1 ( x, y ) − f k ( x, y )) (1) +
k =0
M −1
+ ( f k +1 ( x, y ) − f k ( x, y ) )( −1) = f p ( x, y ) .
k= p
The latter is obtained after the disclosure of the brackets and obvious transfor-
mations. 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
Let’s introduce the functions:
f k− ( x, y ) = lim f ( u, v ), f k+ ( x, y ) = lim f ( u, v ), k = 0, M − 1.
(u ,v )→( x , y ) (u ,v )→( x , y )
wk ( u , v ) 0, wk ( u , v ) 0,
wk ( x , y ) = 0 wk ( x , y ) = 0
Construct auxiliary functions:
− f k+−1 ( x, y ) wk ( x, y ) + f k− ( x, y ) wk −1 ( x, y )
h0 ( x, y ) = f 0− ( x, y ), hk ( x, y ) = ,
wk −1 ( x, y ) − wk ( x, y )
k = 1, M − 1, hM = f M+ −1 ( x, y ).
Denote:
h0 ( x, y ), w0 ( x, y ) 0,
h ( x, y ), w ( x, y ) 0, w ( x, y ) 0,
k k −1
sp( x, y ) =
k
(6)
k = 1, M − 1
h ( x, y ), w ( x, y ) 0.
M M − `1
Theorem 2. Function sp ( x, y ) is a discontinued spline with properties:
lim sp ( u , v ) = lim f ( u , v ) = f p− ( x, y ) ,
(u ,v )→ ( x , y ) (u ,v )→ ( x , y )
w p ( u , v ) 0, w p ( u , v ) 0,
wp ( x , y ) = 0 wp ( x , y ) = 0
lim sp ( u , v ) = lim f ( u , v ) = f p+ ( x, y ) , p = 0, M − 1.
(u ,v )→ ( x , y ) (u ,v )→( x, y )
w p ( u , v ) 0, w p ( u , v ) 0,
wp ( x , y ) = 0 wp ( x , y ) = 0
Proving. Let 1 p M − 2 , then
− f p+−1 (u , v) w p (u , v ) + f p− (u , v )w p −1 (u , v )
lim sp ( u, v ) = lim hp ( u, v ) = lim =
(u ,v )→ ( x , y ) (u ,v )→( x , y )
(u ,v )→( x , y )
w p ( u , v ) 0,
wp −1 (u , v) − wp (u , v)
w p ( u , v ) 0, w p ( u , v ) 0,
wp ( x , y ) = 0
wp ( x , y ) = 0 wp ( x , y ) = 0
− f p+−1 (u , v) wp (u , v) f p− (u , v ) w p −1 (u , v )
= lim + lim =
(u ,v )→ ( x , y )
w p ( u , v ) 0,
wp −1 (u , v) − wp (u , v) (u ,v )→ ( x , y )
w p ( u , v ) 0,
wp −1 (u , v ) − w p (u , v )
wp ( x , y ) = 0 wp ( x , y ) = 0
= 0 + f p− ( x, y ) = f p− ( x, y ).
Similarly
− f p+ (u , v) w p +1 (u , v) + f p−+1 (u , v) w p (u , v)
lim sp (u , v ) = lim h p +1 (u , v ) = lim =
(u ,v )→( x , y ) (u ,v )→( x , y )
(u ,v )→( x , y )
w p ( u , v ) 0,
w p (u , v) − w p +1 (u , v)
w p ( u , v ) 0, w p (u ,v ) 0,
w p ( x, y )=0 w p ( x, y )=0
w p ( x, y )=0
− f p+ (u , v) w p +1 (u , v) f p−+1 (u , v) w p (u , v)
= lim + lim =
(u ,v )→( x , y )
w p (u ,v ) 0,
w p (u , v) − w p +1 (u, v) (u ,v )→( x , y )
w p ( u , v ) 0,
w p (u , v) − w p +1 (u , v)
w p ( x, y )=0 w p ( x, y )=0
= f p+ ( x, y ) + 0 = f p+ ( x, y ).
Thus, the function sp ( x, y ) has the same unilateral boundaries as the func-
tion f ( x, y ) on the lines p : wp ( x, y ) = 0, p = 1, M − 2 . Let p = 0. Then
lim sp ( u, v ) = lim h0 ( u, v ) = lim f 0− (u, v) = f 0− ( x, y ),
(u , v )→( x , y )
(u ,v )→( x , y ) (u ,v )→( x , y ) w0 ( u , v ) 0,
w0 ( u , v ) 0, w0 ( u , v ) 0,
w0 ( x , y ) = 0
w0 ( x , y ) = 0 w0 ( x , y ) = 0
− f 0+ (u , v) w1 (u , v) + f 1− (u , v) w0 (u , v)
lim sp (u , v ) = lim h1 (u , v ) = lim =
(u ,v )→( x , y ) (u ,v )→( x , y ) (u ,v )→( x, y )
w (u ,v ) 0,
w 0 (u , v ) − w1 (u , v )
w 0 (u ,v ) 0, w 0 (u ,v ) 0, 0
w 0 ( x, y )=0 w 0 ( x, y )=0 w 0 ( x, y )=0
− f 0+ (u , v) w1 (u , v) f 1− (u , v) w0 (u , v)
= lim + lim =
( u , v ) → ( x , y ) w (u , v ) − w (u , v ) ( u , v ) → ( x , y ) w (u , v ) − w (u , v )
w (u ,v ) 0,
0
0 1 w (u ,v ) 0, 0
0 1
w 0 ( x, y )=0 w 0 ( x, y )=0
= f 0+ ( x, y ) + 0 = f 0+ ( x, y ).
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 follow-
ing chain of equalities.
lim sp ( u, v ) = lim hM −1 ( u , v ) =
(u ,v )→ ( x , y ) (u ,v )→( x , y )
wM −1 ( u , v ) 0, wM −1 ( u , v ) 0,
wM −1 ( x , y ) = 0 wM −1 ( x , y ) = 0
− f M+ − 2 (u , v) wM −1 (u , v) + f M− −1 (u , v ) wM − 2 (u , v )
= lim =
(u ,v )→ ( x , y )
w ( u , v ) 0,
wM −2 (u , v) − wM −1 (u , v)
M −1
wM −1 ( x , y ) = 0
− f M+ − 2 (u , v) wM −1 (u , v) f M− −1 (u , v) wM − 2 (u , v )
= lim + lim =
M − 2 (u , v ) − wM −1 (u , v ) M − 2 (u , v ) − wM −1 (u , v )
(u ,v )→( x , y ) w (u ,v )→( x , y ) w
w M −1 ( u , v ) 0, w ( u , v ) 0,
M −1
wM −1 ( x , y ) = 0 wM −1 ( x , y ) = 0
= 0 + f M− −1 ( x, y ) = f M− −1 ( x, y ).
lim sp (u , v ) = lim hM (u , v ) = lim f M+ −1 (u , v) = f M+ −1 ( x, y ).
( u ,v ) →( x , y ) ( u ,v )→( x , y ) ( u ,v ) →( x , y )
w M −1 ( u ,v ) 0 , w M −1 ( u ,v ) 0 , w M −1 ( u ,v ) 0 ,
w M −1 ( x , y ) = 0 w M −1 ( x , y ) = 0 w M −1 ( x , y ) = 0
Theorem 2 is proved.
In the case if the function f k++1 ( x, y ) − f k− ( x, y ) 0 is in some parts of at least one
line wk ( x, y ) = 0, then its decomposition in the finite Fourier sum is accompanied by
the Gibbs phenomenon. For this case, in the writings of Gottlieb and Gustafsson [8, 9]
various methods of constructing finite Fourier sums are given; Fourier coefficients are
multiplied by the corresponding factors.
5 Implementation of the method
In this article we propose a solution to the problem of approximating discontinu-
ous 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 bounda-
ries 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 discontinu-
ous functions using of the finite Fourier sums as follows
( x, y ) = f ( x, y ) − sp( x, y ),
( x, y ) T N ( x, y ).
For further use the following statement.
Theorem 3. The function ( x, y ) belongs to the class of continuous periodic func-
tions, i. e.
( x, y ) C [0,1] 2 : ( x, y ) C [0,1] 2 : (0, y ) = (1, y ), ( x,0) = ( x,1).
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 [0,1] 2.
Theorem 4. If a function f ( x, y ) is nonperiodic and has first-kind breaks only
on lines inside the domain [0,1]2, then function:
f ( x, y ) = f ( x, y ) − ( x, y ),
( x, y ) = f (0, y )(1 − x) + f (1, y ) x + f ( x,0)(1 − y ) +
+ f ( x,1) y − f (0,0)(1 − y ) + f (0,1) y (1 − x) − f (1,0)(1 − y ) + f (1,1) y x
is periodic with period 1 for both variables, if it periodically extends to the entire
plane Oxy.
Proving. Find traces [11] of the function ( x, y ) on all four sides of the
square [0,1]2. As a result we will receive:
f (0, y ) = f (0, y ) − (0, y ) = f (0, y ) − f (0, y ) = 0,
f (1, y ) = f (1, y ) − (1, y ) = f (1, y ) − f (1, y ) = 0,
f ( x,0) = f ( x,0) − ( x,0) = f ( x,0) − f ( x,0) = 0,
f ( x,1) = f ( x,1) − ( x,1) = f ( x,1) − f ( x,1) = 0.
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.
Theorem 4 is proved.
Remark. In this way, we can represent each nonperiodic function f ( x, y ) in
the form:
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 func-
tion 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 al-
so known as boundaries f k+ ( x, y ) and f k− ( x, y ) in the form:
w ( x, y ) w ( x, y )
f k+ ( x, y) = f k +1 x − wk ( x, y ) k , y − wk ( x, y) k ,
x y
w ( x, y ) w ( x, y )
f k− ( x, y ) = f k x − wk ( x, y ) k , y − wk ( x, y ) k ,
x y
wk ( x, y )
wk ( x, y ) = .
w ( x, y ) wk ( x, y )
2 2
− k +
x y
Step 2. Find the difference:
( x, y ) = f ( x, y ) − sp ( x, y ).
Step 3. For the function ( x, y ) we find:
( 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.
Step 4. Submit the function f ( x, y ) as a sum:
f ( x, y ) = sp ( x, y ) + ( x, y ) + ( x, y ). (7)
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 com-
ponent 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 [12].
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 = t p
( x, y )ds =
kx + ly = t p
f ( x, y )ds − sp( x, y) + ( x, y) ds.
kx + ly = t p
(8)
7 Example
Denote:
w( x, y ) = ( x − 0.5) 2 + ( y − 0.5) 2 .
If the lines wk ( x, y ) = 0, k = 0, M − 1 are circles wk ( x, y ) w ( x, y ) − rk = 0,
k = 0, M − 1 , that is:
D0 : w( x, y ) r0 , , Dk : w( x, y ) rk , k = 1, M − 1, DM = [0,1]2
then the functions f k ( x, y ) will have the following boundary properties:
f k− ( x, y ) = f k ( rk cos , rk sin ) , k = 0, M − 1,
f k+ ( x, y ) = f k +1 ( rk cos , rk sin ) , k = 0, M − 1,
r 0, 0 2.
Here r and coordinates of the point ( x, y ) in the polar coordinate system with
the center at the point ( 0.5, 0.5) .
In this case, the formula (6) for sp ( x, y ) the functions hk ( x, y ) can be written as:
h0 ( x, y ) = f 0 ( r0 cos , r0 sin ) , , hk ( x, y ) =
w( x, y ) − rk w( x, y ) − rk −1
= f k ( rk −1 cos , rk −1 sin ) + f k ( rk cos , rk sin ) , k = 1, M − 1,
rk −1 − rk rk − rk −1
hM ( x, y ) = f M ( 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 un-
known function f ( x, y ) by means of projections, coming from a computer tomo-
graph, for the case, when known line break function f ( x, y ) and known unilateral
boundaries functions for f k ( 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.
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