ON A MODIFICATION OF THE HAMMING METHOD FOR SUMMING DISCRETE FOURIER SERIES AND ITS APPLICATION TO SOLVE THE PROBLEM OF CORRECTION OF THERMOGRAPHIC IMAGES Текст научной статьи по специальности «Математика»

Discrete & Continuous Models

#& Applied Computational Science 2022, 30 (4) 342-356

ISSN 2658-7149 (online), 2658-4670 (print) http^journ^s.rudn.ruA^ph

Research article

UDC 519.6

PACS 07.05.Tp, 02.60.Pn, 02.70.Bf DOI: 10.22363/2658-4670-2022-30-4-342-356

On a modification of the Hamming method for summing discrete Fourier series and its application to solve the problem of correction of thermographic images

Evgeniy B. Laneev, Obaida Baaj

Peoples' Friendship University of Russia (RUDN University), 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

(received: August 31, 2022; revised: December 12, 2022; accepted: December 19, 2022)

Abstract. The paper considers mathematical methods of correction of thermographic images (thermograms) in the form of temperature distribution on the surface of the object under study, obtained using a thermal imager. The thermogram reproduces the image of the heat-generating structures located inside the object under study. This image is transmitted with distortions, since the sources are usually removed from its surface and the temperature distribution on the surface of the object transmits the image as blurred due to the processes of thermal conductivity and heat exchange. In this paper, the continuation of the temperature function as a harmonic function from the surface deep into the object under study in order to obtain a temperature distribution function near sources is considered as a correction principle. This distribution is considered as an adjusted thermogram. The continuation is carried out on the basis of solving the Cauchy problem for the Laplace equation — an ill-posed problem. The solution is constructed using the Tikhonov regularization method. The main part of the constructed approximate solution is presented as a Fourier series by the eigenfunctions of the Laplace operator. Discretization of the problem leads to discrete Fourier series. A modification of the Hamming method for summing Fourier series and calculating their coefficients is proposed.

Key words and phrases: thermogram, ill-posed problem, Cauchy problem for the Laplace equation, Tikhonov regularization method, discrete Fourier series

1. Introduction

Thermal imaging methods are widely used in medicine as a means of early diagnostics [1-4]. Visualization (thermogram) of the temperature distribution on the surface of the patient's body contains information about sources of heat inside the body associated with the functioning of internal organs. In particular, it contains information about temperature anomalies associated with pathologies of internal organs. The image on the thermogram, as a rule,

© Laneev E.B., BaajO., 2022

This work is licensed under a Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by-nc/4.0/legalcode

is distorted due to the process of thermal conductivity, heat exchange and the relative remoteness of heat sources from the surface of the body.

Within the framework of the chosen mathematical model, it is possible to correct the image on the thermogram in order to increase the effectiveness of diagnostics. Since the evolution of the temperature distribution in the patient's body is relatively slow, it makes it possible to use stationary models, in particular, models of harmonic temperature distribution. As an adjusted thermogram, we will consider the temperature distribution near the sources obtained by the continuation of the harmonic function from the boundary (similar to the continuation of gravitational fields in geophysics problems [5]).

In [6], based on the method [7], one of the possible solutions to such a problem is proposed. The problem, as ill-posed, is solved using the Tikhonov regularization method [8]. When forming computational algorithms, discrete Fourier series [9, 10] are used, the coefficients of which are calculated from functions depending on the coefficient number [11]. To sum up such series, a modification of the Hamming method [9] is proposed here.

2. Mathematical model and inverse problem

As a mathematical model, we consider a homogeneous heat-conducting body in the form of a rectangular cylinder

D(F, x) = {(x, y,z)- 0 < x < lx ,0<y<ly, F(x, y) < z < x} C K3, (1) limited by the surface

The domain D(F, x) contains heat sources with a time-independent density function p, creating a stationary (harmonic) temperature distribution in the body. We associate the density function of heat sources with the anomalies under study. We assume that on side faces r of the cylindrical domain D(F, x) a temperature equal to zero is maintained, and on the surface S of the form (2) there is convective heat exchange with the external environment of temperature U0, described by Newton's law, according to which the density of the heat flux at the point of the surface S it is directly proportional to the temperature difference inside and outside.

It should be borne in mind from a physical point of view that despite the fact that the density of sources does not depend on time, the heat released by them is diverted across the boundary, the overall temperature distribution does not change over time, although the distribution gradient corresponds to stationary heat flows.

In the domain D(F, x) of the form (1), the temperature distribution is the solution of a mixed boundary value problem for the Laplace equation

S = {(x, y,z) ■ 0 < x <lx,0<y<ly,z = F(x, y)}.

(2)

We'll assume that we also know that

ax < F(x, y) <a2 < H, (x, y) e n, n = {(x, y) : 0 <x <lx, 0 < y < L}.

(3)

(4)

(Au(M) = p(M), M e D(F, œ),

^ =h(U0 -u) , on s s

Mlr = 0,

u is bounded at z ^ œ.

(5)

We assume that the function p is such that the solution of the problem (5) exists in C2(D(F, <))^C1 (D(F,<)). In particular, the solution of the problem (5) allows us to find m|5, i.e. the temperature distribution u on the surface of S, which we will call a thermogram.

Now let the thermogram be obtained as a result of measurements and the density of p is unknown. Let us now set the inverse problem. We set the problem of continuation of the temperature distribution from the surface towards the sources in order to obtain an adjusted thermogram as the temperature distribution ulz=H on the plane z = H, closer to the density carrier than the surface S. The plane z = H is related to the surface S by the condition (3).

We assume that the carrier of the function p is located in the domain z > H, then the solution of the problem (5) in the domain

D(F, H) = {(x,y,z)-. 0<x<lx ,0<y<ly, F(x, y) < z < H} (6)

satisfies the Laplace equation. The set of side faces of the domain D(F, H) is denoted by rH.

Inverse problem. Let the function be given within the framework of the model (5)

f = uls, (7)

and the density of p is unknown. It is required to find u\z=H. It is required to find u\z=H.

Since the value of H sufficiently arbitrarily defines the plane between the support of p and the surface S, then in fact the inverse problem consists in obtaining a solution u in the domain D(F, H) (6) of the boundary value problem

(Au(M) = 0, MeD(F,H),

^s = f,

du dn

= h(Uo - f)

(8)

m|t = 0.

We assume that the function f in (7), (8) is taken from the set of solutions to the direct problem (5), so the solution to the inverse problem exists in C 2(D(F,H))nC 1(D(F,H)).

Note that in the problem (8) on the surface S of the form (2), Cauchy conditions are set, that is, the boundary values f of the desired function u and the values of its normal derivative are set, so the problem (8) has a unique

solution. The boundary z = H of the domain D(F, H) is free and, thus, the problem (8) is unstable with respect to errors in the data, i.e. ill-posed.

The function ulz=H will be considered as an adjusted thermogram. Since the plane z = H is located closer to the support of density p, it should be expected that the corrected thermogram more accurately conveys information about the distribution of heat sources than the original thermogram.

3. Approximate solution of the inverse problem.

Let the function f in the problem (8) be given with an error, that is, instead of f, the function fs is given, so that

IfS -fha{n)

In [6], an approximate solution of the ill-posed problem (8) is constructed in the form

u5a (M)=v5a (M) + ¥ (M), MeD(F,H), (9)

where function (integral over a rectangle n of the form (4)))

(M) = J [h(U0 -f (xp,yp))^(M,P)\p€Sn1(xp,yp)-n

-fs(xP,yP)(ni,VPp(M,P))\pes]dxPdyP (10) is calculated using the problem (8) data, the Dirichlet problem source function

<P(M,P) = j^ £ —-x

J J Z-^i l-

KnxM kmyM nnxp Kmyp X Sin —--Sin —--Sin —-— Sin —-—

ix iy ix iy

in the cylinder

= {(x, y,z)- 0 < x <lx ,0<y<ly, —x <z<»)cR3, the normal to the surface S of the form (2)

n = grad (F(x,y)-z) = VxyF- k, nx = |nj.

The function vsa, which is an approximation to the density potential p [12] was obtained in [6] using the Tikhonov regularization method [5]

s (m\ = — ^ ^nm(a) exp{knm(zM — a)l • KUXm ^myM

Va (M)= nhi 1 + a exp{2knm (H-a)} sin lx sin ly , (11)

where a > 0,

/2 2\ 1/2 in2 m2X '

^nm 12 + 12

\ 'Jx 'Jy

^nm (a) — Fourier coefficients of the function (M) of the form (10)

, , 4 [' ? nnx . nmy , , , .

Q'nm(a) =—— I (x,y,a) sin ——sin—— dxdy, a < a1. (12)

^X^V J 'JX 'J'U

n

For the Fourier coefficients &sam (a) in [11] the expression

&nm (a) = H,nrn (a) + H,nrn (a) (13)

is obtained, where

4

j

n

n,nm(a) = —l h(U0 -fd(x,y))x

e k™(F(*,y) a) nnx nmy , A,

x-—-n1(x,y) sin——sin—— axay, (14)

2^nm J x 'J,y

4 [ ?5f knrrXF(x;y) a) ^ r ^ _ nnx . nmy

fd (x, y)-—-FI (x, y) cos —— sin —— dxdy+

lli ' ^ y 21 k " J J

n

4 i'x, .nme knrrXF(x;y) a) nnx nmy YY l f0(x,y)-Tfi-T-Fy{x,y) si^ —co^—— '

x y ^ Jy' Jnrn Jx Jy

+ -2- if S(x, y)e-knm(F(x'y)-a) sin ^ sin ^dxdy. (15)

'JX Jru J JX 'J'U

n

Thus, the Fourier coefficients &sam(a) are calculated as the sum of formally calculated Fourier coefficients in accordance with (12) over orthogonal systems

n œ r -s œ

nnx nmy I I nnx nmy sin —:- sin —:- } , { cos —:- sin

1 / / 1

(16)

f nnx nmy 1 { sin —— cos —-— } I ^x ^y J

n,m=1

of functions depending, apart from the arguments x and y, on the number nm of the Fourier coefficients.

4. Formation of an approximate solution based on discrete Fourier series

When discretizing the [13] problem (8) and performing numerical calculations using the formulas (11), (13), (14), (15) it is natural to pass to calculating the values of the approximate solution of the problem (8) on the grid of the values of the arguments x and y

U= ^a^ ,y3) : xz = ^, i = 0,l,...,Nx, y3 = ^, j = 0,l,...,Ny(17)

In this case, there is no need to use infinite Fourier series. One can pass to discrete Fourier series [9, 10], in this case two-dimensional.

The discrete Fourier series has an interpolation property, that is, the discrete Fourier series (by definition, representing a finite sum) with coefficients calculated by the corresponding formulas coincides on the grid with the values of the function. For example, if on the grid

xz = ijj, i = 0,l,...,N, (18)

the grid function f = (f0, f1,...,fN—1 ,fN) is given (when expanding into a discrete Fourier series in terms of sines, we assume that f0 = fN = 0). Then the function f can be represented by a discrete Fourier series in terms of sines [10]

N-1 N—1 ^

fi =Y,h sin ^ = Y,bk sin 1 = 0,1,...,N, (19)

i=i L i=i

where the coefficients bk are calculated by the formula (equivalent to the trapezoid formula for the corresponding integral in the theory of Fourier series):

h = 2n fi sin k = l,...,N-l. (20)

i=1

In other words, if the discrete series coefficients are calculated in accordance with the formula (20), then the discrete series (19) is exactly equal to the values of the function fi, i = 0,l,..., N.

Applying discrete Fourier series to the approximate solution (9) on the grid (17) for each fixed z, a2 < z < H, will lead to the formula for v5a:

Ny—1 N* — 1 / \ f! f M

(„s ) („) = -^ ^ Km(a) exp{fc.m(z-a)} . Km ( )ii h ¿1 l + «exp|2A;_(H - a)} sin Nx sin Ny , (21)

i = 0,...,Nx, j = 0,..., Ny.

In this case, the integrals in calculating &sam(a) by the formulas (13), (15), (14) it is natural to replace with formulas corresponding to the calculation of the coefficients of the discrete Fourier series of the form

4

Nx-lNy-1

v^ / n nm nmj (n,m) sin — sin

i=1 j=1 x y

=1 3=1 x

4 Nx-1Ny-1

+--^^ ^^ pij (n,m) sin

Nx-1Ny-1

(n = 1,2,..., Nx -1, m = l,2,..., Ny - 1) on the grid u of the form (17). To simplify the notation of integrands in integrals corresponding to systems (16), the notation f, g, p is introduced. A feature of calculating the coefficients of a discrete series in this case is that the functions f, g, p, in addition to the arguments xi and yrj, depend on the indices n and m of the Fourier coefficients.

5. Summation of a discrete Fourier series and calculation of its coefficients by the Hamming method

Here we give some modification of the Hamming method [9] and its proof, related to the representation of a function as a discrete Fourier series in terms of sines or cosines on the interval [0,1].

We now assume that the coefficients of the discrete series of some grid function w

k=1

are calculated formally in accordance with (20), where the values of the function f formally depend on the number k of the coefficient bk, i.e.

Let us show that the idea of Hamming algorithm [9] for calculating coefficients bk of a discrete Fourier series (23) is also applicable to this situation, that is, to calculating the sum (24).

We fix the number k of the Fourier coefficient. Let us denote for brevity tk = nk/N and consider the recurrent formulas

N-1

k = l,...,N-l (24)

(25)

Un = 0

o - 0>

U1 = fN (k) Um = (2 cos

(2 cos tk ) Um-1 - Um-2 + fN-m+1 (k), m = 2,3,..., N.

Our task is to show that the Fourier coefficients bk of the form (24) of the function w of the form (23) can be calculated by the formula

N

bk = (k) sin ^T = ^KTUnsintk,

i=0

(27)

where UN is calculated by recurrent formulas (26).

For simplicity of notation, the dependence of Um on k is not indicated. Note also that the recurrent formulas (26) use the values fN(k),..., fx(k), the value f0 (k) is not used when forming UN.

Algorithm (26), (27) obviously allows to avoid calculation of sines in (24) with argument nki/N when changing indices i and k.

Let us represent the function f(k) as a sum of functions f(k), each of which is a vector with zero coordinates, except for the coordinate with number i equal to fi(k), that generally speaking, not equal to zero:

N

№ = Y,f & (k),

i=0

(28)

f (0 (k) = (k), ...,$ (k)) = (0,0,...,0, fz(k), 0,...,0)

f«

Note that if the upper and lower indices do not coincide, the coordinate of the function f(k) is equal to zero. Note also that when the grid function is represented by a sine series, due to (25) f(0) (k) = f(N) (k) = (0,0,... ,0).

We apply the recursive formulas (26) to each function f(k), i = 0,1,2,..., N (for a fixed k), denoting result as U:

u(

U-U,

(1

Uo

ui1

u{1

uo

TT (i . i^m

= 0,

= f(0 (k),

= 2 cos tk uj°_i - U}(0U + f[N-m+i (k), rn = 2,3,..., N,

= 0, = f'î!)(k),

= 2 cos tk Ugh - + ftti^i (k), m = 2,3,..., N,

= 0, = $ (k),

m

(0)

= 2 cos tk uOh - U^_2 + f(;)_m+i (k), m = 2,3,

r«

(i)

uO»' = 0,

tc = iï'w,

Ui»< = 2 cos tt [f(l-i - r/("-2 + Wm+iW, m = 2,3, ...,N.

Summing up the corresponding parts of all equalities, due to the linearity of the recursive formulas, we obtain

N

r(N)

t(N)

=0,

i=0

N N

Y^"(k),

i=o i=o

N N N

= 2 cos tk £ U^-1 - £ uiS-2 + £ f.

N

(i)

N-m+1

(k), m = 2,3,...,N.

i=o

i=o

i=o

i=o

Considering (28) for the sum f (i)(k) and taking the notation

N

= Um, m = 0,

,N,

(29)

i=o

we obtain the recurrent formulas (26). Thus, to prove the formula (27), it

(i)

suffices to calculate U"N) for all i = 0,..., N.

Let us calculate U^, singling out the cases i = 0,1, N separately. Applying the recurrent formulas (26) to f(i) (k) for i = 0,1 gives

u0O) = 0,

u'f = f'ff(k) = 0,

u20° = fff-1(k) = 0, uiOh = /20) (k) = 0,

US' = f10)(k) = 0 = v00),

uO1' =0,

U^ = fW (k) = 0,

U21 = №1 (k) = 0,

u^h = fH) (k) = 0 = v(1), =f11] (k) = h (k) = v1V).

(30)

Applying the recurrent formulas (26) to f(i) (k) for i = N gives

U (N) =0 = y (N) u0 — 0 — vo ,

r(N) _ AN)

U[1 =f^) (k) = fN(k) = V1

(N)

u™ = (2 cos tk)V£-1 - VX-2, m = 2,3,..., N.

Note that for all values of m = 2,3,..., N in (31) the value +1 since the upper index is not equal to the lower one.

( )

( )

(31)

(k) = 0,

Applying the recurrent formulas (26) to f (t) (k) for i = 2,...,N — 1 gives

uC) = 0,

Un'-i = fN'-(N-i)+1 (k) = fi+1 (k) = 0 = Vq) ,

u

(i)

N-i+1

= f

u

«

N—i+m

(i)

N—(N—i+1)+1 (i)

(i) (h) = 0 = v(%),

(i)ru\ — f fb'\ — M(z)

(32)

(k) = f(t) (k) = f% (k) = V(),

= (2 cos t^Li -Vr

r(z)

m—2,

Here we took into account that /¡^)-(N-l+m)+1 (k) = fi+1-m(k) = 0, m = 2,3,... ,i, because the upper and lower indices do not match. In the formulas (30), (31), (32) we introduced the notation

m = 2,3,... ,i. f«

u

N—i+m

= y(%)

— * m )

i = 0,...,N, m = 0,1,2,... ,i.

(i)

It is easy to see that quantities Vm are calculated using the same formulas (26), "skipping" the first N — i — 1 zeros for Um . The introduction of the quantity Vm allows us to obtain an explicit formula for it:

(33)

v(i) = f (k) sin mtk Vm =h (k) sin tk

m = 0,1,... ,i,

(34)

for each i = 0,1,..., N.

Let us prove it using the method of induction for m. As follows from (30), (32), this equality holds for m = 0,1. Let's prove it for m = 0,1,... ,i. Let equality (34) hold for m — 2 and m — 1. Let's prove for m.

V® = (2 cos tk)V£_i -VZ—.2 =

(i)

Ai)

= fi (k) = m

2 cos t

sin(m — 1)tk sin(m — 2)th

k~

sin tk sin tk

sin mtk + sin(m — 2)tk — sin(m — 2)tk sin ti.

= f(k) sin mtk

Note that from (33), in particular, U^ = V-'Z) and from the formula (34) for m = i we obtain

r(i)

= v(:) = f< (k)sin ltk

sin t

k

Summing according to (29) with m = N, and using (24), we obtain

N N

rW _ V^tAi)

Un = Y,UN =

1

N

sin ti

i=0 i=0 i=0 From here we obtain the formula (27).

^fz(k')sinitk = bk

1 N sin tk 2 .

Let us now proceed to calculating the coefficients of the discrete Fourier series of the grid function w in cosine expansion. On the same grid (18) consider the discrete Fourier series in cosines expansion [10]

N-l u-

wi = Y + £ cos ^ + ^ (-1)i, i = 0,1,...,N, (36)

k=1

whose coefficients afe are calculated on the basis of the grid function f depending on the number by the formula

<* = I (^ + £ f-(»cos f + ^r-)"). (37)

We introduce the grid function

.f( k)=( ^2r, fi (k), f2 (k),..., fN-i (k), (38)

that differs from f in that the current and last coordinates are divided in half. Replacing f in (26) with f, we get formulas (33), (34).

Note that the recurrent formulas (30) for i = 0 imply U^ = 0, = 0. For i = 1,2,..., N from (33), (34) we obtain

Ulli = vBi = m sins-1)tt, t=1,2,...,N. (39)

sin ik

Consider the following construction (summation starts from i = 1, since

< = Upl-i = 0):

N N

cos tk UN - UN-i + f0(k) = cos tk -£ ufih + f0(k) =

i=i i=i N

(i) (i)

£[costku'§ - U(ff-i]+f0(k).

i=i

Replacing U1^ and Ul(i>-i according to formulas (35) and (39), we obtain

i=i

(i) (i)

N

costkUN - UN-i + f0(k) = £ [costkv}i) - V>i] + f0(k) =

i=i

.cos tk sini tk - sin(z - 1) tk

= £fi(k)——1 h ( ) ' + fo(v =

i=i K

? ,, , cos tk sinitk - cos tk sinitk + sintk cos itk ~ , ,

= £ f\(.k)-----sfei-"-"-~ + fo(k) =

i=i

N N—1

f + J N(K) cos !Mk

f fi (k) cos itk + fo (k) = fo (k) + f f% (k) cos itk + fN(k) cos Ntk

i=1

ff fz (k) cos uk+m—i)K

i=1 i=1

N—1

i=1

From here according to (37) we obtain a formula for the coefficients of the discrete Fourier series in the cosine expansion

22 ak = N [costkUN — Un—I + fo^ = N

cos tk UN — UN—i + MO-

, (40)

moreover, the quantities UN and UN-1 are calculated by the formulas (26),

in which the grid function f is replaced by f, which is related to f by the formula (38).

Note that the formulas (26), (27) can also be used to sum the Fourier series (23), since the formulas (24) and (23) differ only by a factor. Note that when summing the series (23), we, of course, do not obtain f(k), but we obtain some function w. Accordingly, the formulas (26), (40) can also be used to sum the Fourier series (36), since the formulas (37) and (36) differ only by the multiplier.

The formulas (26), (27), (40) can be used for two-dimensional discrete Fourier series both for calculating the coefficients and for summing the Fourier series (21). In this case, the formulas (26), (27) for each fixed pair nm are applied sequentially over each index i and j corresponding to the variables x and y.

The formulas (26), (40) for calculating the coefficients of the discrete Fourier series in cosine expansion, of course, are also valid in the case when f0 = fN = 0, which corresponds to the formulas (22).

6. Conclusion and discussion

Formulas (21), (22), (26), (27), (40) as a solution to the problem (8) can be used for mathematical processing of thermograms taken with a thermal imager in medicine [4] in order to correct the image on the thermogram. Note that taking into account the influence of blood flow leads to the need to use the metaharmonic equation [14], [15] in problem (8).

The thermogram, with one or another certainty, conveys an image of the structure of heat sources inside the body. However, within the framework of the task (8), the image on the thermogram can be refined. In this case, we consider the function fs as the original thermogram, and the function v5a \z=H as the corrected thermogram. Since the function v5a\Z=H is the temperature distribution on a plane closer to the investigated heat sources than the original surface S, we can expect a more accurate reproduction of the source image on the calculated thermogram v5a\Z=H.

The results of calculations performed on a model example show the effectiveness of the proposed method and algorithm based on the formulas (9), (10), (11), (13) that can be used for processing thermal images.

Note that the method of summation of discrete Fourier series, described in Section 5, can be used to solve other problems, the solutions of which can be obtained in the form of Fourier series in terms of eigenfunctions of the Laplace operator in a rectangle.

Acknowledgments

The research is supported by the Russian Foundation for Basic Research (grant №20-01-00610a).

References

[1] E. F. J. Ring, "Progress in the measurement of human body temperature," IEEE Engineering in Medicine and Biology Magazine, vol. 17, no. 4, pp. 19-24, 1998. DOI: 10.1109/51.687959.

[2] E. Y. K. Ng and N. M. Sudarshan, "Numerical computation as a tool to aid thermographic interpretation," Journal of Medical Engineering and Technology, vol. 25, no. 2, pp. 53-60, 2001. DOI: 10.1080/ 03091900110043621.

[3] B. F. Jones and P. Plassmann, "Digital infrared thermal imaging of human skin," IEEE Eng. in Med. Biol. Mag., vol. 21, no. 6, pp. 41-48, 2002. DOI: 10.1109/memb.2002.1175137.

[4] G. R. Ivanitskii, "Thermovision in medicine [Teplovideniye v meditsine]," Vestnik RAN, vol. 76, no. 1, pp. 44-53, 2006, in Russian.

[5] A. N. Tikhonov, V. B. Glasko, O. K. Litvinenko, and V. R. Melihov, "On the continuation of the potential towards disturbing masses based on the regularization method [O prodolzhenii potentsiala v storonu vozmushchayushchih mass na osnove metoda regulyarizatsii]," Izvestiya AN SSSR. Fizika Zemli, no. 1, pp. 30-48, 1968, in Russian.

[6] E. B. Laneev, N. Y. Chernikova, and O. Baaj, "Application of the minimum principle of a Tikhonov smoothing functional in the problem of processing thermographic data," Advances in Systems Science and Applications, vol. 1, pp. 139-149, 2021. DOI: 10.25728/assa.2021.21. 1.1055.

[7] E. B. Laneev, "Construction of a Carleman function based on the Tikhonov regularization method in an ill-posed problem for the Laplace equation," Differential Equations, vol. 54, no. 4, pp. 476-485, 2018. DOI: 10.1134/S0012266118040055.

[8] A. N. Tikhonov and V. J. Arsenin, Methods for solving ill-posed problems [Metody resheniya nekorrektnyh zadach]. Moscow: Nauka, 1979, in Russian.

[9] R. W. Hamming, Numerical methods for scientists and engineers. New York: McGraw-Hill Book Company, 1962.

[10] E. B. Laneev, Numerical methods [Chislennye metody]. Moscow: RUDN, 2005, in Russian.

[11] O. Baaj, "On the application of the Fourier method to solve the problem of correction of thermographic images," Discrete and Continuous Models and Applied Computational Science, vol. 30, no. 3, pp. 205-216, 2022. DOI: 10.22363/2658-4670-2022-30-3-205-216.

[12] E. B. Laneev, Ill-posed problems of continuation of harmonic functions and potential fields and methods for their solution [Nekorrektnye zadachi prodolzheniya garmonicheskih funkcij i potencialnyh polej i metody ih resheniya]. Moscow: RUDN, 2006, in Russian.

[13] E. B. Laneev, M. N. Mouratov, and E. P. Zhidkov, "Discretization and its proof for numerical solution of a Cauchy problem for Laplace equation with inaccurately given Cauchy conditions on an inaccurately defined arbitrary surface," Physics of Particles and Nuclei Letters, vol. 5, no. 3, pp. 164-167, 2002. DOI: 10.1134/S1547477108030059.

[14] H. Pennes, "Analysis of tissue and arterial blood temperature in the resting human forearm," J. Appl. Physiol., no. 1, pp. 93-122, 1948.

[15] J. P. Agnelli, A. A. Barrea, and C. V. Turner, "Tumor location and parameter estimation by thermography," Mathematical and Computer Modelling, vol. 53, no. 7-8, pp. 1527-1534, 2011. DOI: 10.1016/j.mcm. 2010.04.003.

For citation:

E. B. Laneev, O. Baaj, On a modification of the Hamming method for summing discrete Fourier series and its application to solve the problem of correction of thermographic images, Discrete and Continuous Models and Applied Computational Science 30 (4) (2022) 342-356. DOI: 10.22363/2658-46702022-30-4-342-356.

Information about the authors:

Laneev, Evgeniy B. — Doctor of Physical and Mathematical Sciences, Professor of Mathematical Department of Peoples' Friendship University of Russia (RUDN University) (e-mail: [email protected], phone: +7(903)1333622, ORCID: https://orcid.org/0000-0002-4255-9393, ResearcherlD: G-7887-2016, Scopus Author ID: 24366681900)

Baaj, Obaida — Post-Graduate Student of Mathematical Department of Peoples' Friendship University of Russia (RUDN University) (e-mail: [email protected], phone: +7(916)6890863, ORCID: https://orcid.org/0000-0003-4813-7981, ResearcherlD: ADK-3603-2022, Scopus Author ID: 57222870471)

УДК 519.6

PACS 07.05.Tp, 02.60.Pn, 02.70.Bf DOI: 10.22363/2658-4670-2022-30-4-342-356

Об одной модификации метода Хемминга суммирования дискретных рядов Фурье и её применение для решения задачи коррекции термографических изображений

Е. Б. Ланеев, Обаида Бааж

Российский университет дружбы народов, ул. Миклухо-Маклая, д. 6, Москва, 117198, Россия

Аннотация. В работе рассматриваются математические методы коррекции термографических изображений (термограмм), полученных с помощью тепловизора, в виде распределения температуры на поверхности исследуемого объекта. Термограмма воспроизводит изображение тепловыделяющих структур, расположенных внутри исследуемого объекта. Это изображение передаётся с искажениями, так как источники, как правило, удалены от его поверхности и распределение температуры на поверхности объекта передаёт изображение как размытое за счёт процессов теплопроводности и теплопереноса. В работе в качестве принципа коррекции рассматривается продолжение функции температуры как гармонической функции с поверхности вглубь исследуемого объекта с целью получения функции распределения температуры вблизи источников. Такое распределение рассматривается как скорректированная термограмма. Продолжение функции температуры осуществляется на основе решения задачи Коши для уравнения Лапласа — некорректно поставленной задачи. Построение решения проводится с использованием метода регуляризации Тихонова. Основная часть построенного приближённого решения представлена в виде ряда Фурье по собственным функциям оператора Лапласа. Дискретизация задачи приводит к дискретным рядам Фурье. Для суммирования рядов Фурье и вычисления коэффициентов в работе предложена модификация метода Хемминга.

Ключевые слова: термограмма, некорректная задача, задача Коши для уравнения Лапласа, метод регуляризации Тихонова, дискретный ряд Фурье