The inverse thermography problem as the inverse potential problem for bodies of constant thickness

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1. Introduction Digital technologies have penetrated into all branches of human activity, and one of the actual problems is to improve the quality and informativeness of the presentation of research results, in particular the quality of images obtained based on measurement data, by their mathematical (digital) processing. This applies, for example, to images obtained using thermal imaging methods with the help of a thermal imager, which records thermal electromagnetic radiation from the surface of the object under study in the infrared range. In particular, in medicine, thermal imaging has become an effective tool for early diagnosis [1-4]. The image on the thermogram, which is a map of the temperature distribution on the surface of the patient’s body, makes it possible to assess functional anomalies in the state of their internal organs. At the same time, the image on the thermogram is sometimes somewhat distorted due to thermal conductivity and heat exchange processes. In the works [5, 6], a method © 2026 E. B. Laneev, A. M. Gaipova This work is licensed under a Creative Commons “Attribution-NonCommercial 4.0 International” license. for correcting the image on a thermogram based on harmonic continuation (similar to problems in geophysics [7]) is proposed. In this paper, within the framework of the chosen mathematical model, the image of the density distribution function of the temperature field sources is considered as the corrected thermogram. It is proposed to obtain this function as the result of solving the corresponding inverse problem of the metaharmonic potential. The inverse potential problem is constructed based on the inverse problem for a certain mixed boundary value problem for the metaharmonic equation. The metaharmonic equation allows one to take into account heat transfer due to blood flow [8, 9]. The inverse problem under consideration is ill-posed, since small errors in the initial data (the initial thermogram) may correspond to significant errors in the solution of the inverse problem. To construct a stable approximate solution, the Tikhonov regularization method is used [10]. 2. Problem statement Let’s consider a physical and then a mathematical model, within which we will set up an inverse problem. The physical model is a homogeneous heat-conducting body in the form of a cylinder of rectangular cross-section, bounded by the surface S and containing heat sources with a density function that does not depend on time, creating a stationary temperature distribution in the body. We associate the density function of heat sources with the object under study. We assume that the lateral faces of the cylinder are maintained at zero temperature, and that there is convective heat exchange with a zero-temperature medium on the surface S, described by Newton’s law, according to which the heat flux density at a point on the surface is directly proportional to the temperature difference between the inside and the outside of the body. A metaharmonic equation is used to account for blood flow. Let us move on to the mathematical model. In an infinite cylinder with a rectangular cross-section D∞ = {(x, y, z) ∶ 0 < x < lx, 0 < y < ly, -∞ < z < ∞} ⊂ ℝ3 consider a cylindrical region D(F, ∞) = {(x, y, z) ∶ 0 < x < lx, 0 < y < ly, F(x, y) < z < ∞}, bounded by a surface S = {(x, y, z) ∶ 0 < x < lx, 0 < y < ly, z = F(x, y) < H}. (1) Let be the set of side faces of the domain D(F, ∞). In the domain D(F, ∞), consider the following mixed boundary value problem for the metaharmonic equation u(M) = (M), M ∈ D(F, ∞), u | n |S = h(U0 - u)| , | S (2) u| = 0, u is bounded when z → ∞. Problem (2) corresponds to a steady-state temperature distribution created by heat sources with a distribution function , on the surface S there is a third boundary condition corresponding to convective heat exchange with a medium at temperature U0 = 0 with a coefficient h, and the boundary has a zero temperature. We assume that the function is such that the solution of problem (2) exists in C2(D(F, ∞)) ⋂ C1(D(F, ∞)). In particular, the solution of problem (2) allows us to find the temperature distribution u|S on the surface S, which we call a thermogram. Now let’s set the inverse problem. Inverse problem Let the function be unknown in the model (2), but let the function f = u|S, (3) be given. We need to find the function. We note that the density recovery is associated with the same difficulties as the solution of the inverse potential problem [11], for which significant restrictions on the classes of uniqueness are known. Here we will consider density functions for bodies of constant thickness (x, y, z) = (x, y) (z - H) (H + s - z), (4) belonging to the Sretensky uniqueness class [12]. In this case, we assume that the depth H and the thickness s are known, and the function (x, y) is the unknown quantity in the inverse problem. In the model (2), the inverse problem is reduced to the following problem for the functions u and : u(M) - k2u(M) = (M), M ∈ D(F, ∞), u|S = f , (5) u | n |S = -hf | , | S u| = 0. We assume that the function f in (3), (5) is taken from the set of solutions of the problem (2) for the densities (4), so the solution of the inverse problem with respect to the function exists in L2. We note that in the problem (5), the Cauchy conditions are specified on the surface S of the form (1), i.e., the boundary values f of the unknown function u and the values of its normal derivative are given, therefore problem (5) has a unique solution [13] with respect to u. As will be shown below, the problem with respect to is reduced to an inverse potential problem for the densities (4) belonging to Sretensky’s uniqueness class. Problem (5), as a Cauchy problem for an elliptic equation, is not stable with respect to errors in the function f , i.e., it is ill-posed. Methods for regularizing Cauchy problems for elliptic equations are relevant even for simple domains [14, 15]. We will consider the function (x, y) as an adjusted thermogram. Let’s construct an explicit representation of the exact solution to problem (5). 3. Exact solution to the inverse problem We construct the exact solution of the problem (5) following the scheme of [16-18]. Let us consider the source function (M, P) of the following Dirichlet problem for the metaharmonic equation in the cylinder D∞: P (M, P) - k2 (M, P) = - MP, M, P ∈ D∞, |xP=0,lx = 0, |yP=0,ly = 0, → 0 when |zP| → ∞. The function has the form (M, P) = exp(-krMP) + W (M, P), 4 rMP where rMP is the distance between points M and P, and W (M, P) is the metaharmonic function of P. The source function can be obtained by the reflection method as a sum of point source functions with a period of 2lx on x and 2ly on y. 4 (M, P) = 1 ∞ ∑ n,m=-∞ exp{-kr1,nm} ( r1,nm - exp{-kr2,nm} r2,nm - exp{-kr3,nm} r3,nm + exp{-kr4,nm} r4,nm ), where so that r1,00 = rMP. r1,nm = [(xM - xP + 2lxn)2 + (yM - yP + 2lym)2 + (zM - zP)2]1/2, r2,nm = [(xM + xP + 2lxn)2 + (yM - yP + 2lym)2 + (zM - zP)2]1/2, r3,nm = [(xM - xP + 2lxn)2 + (yM + yP + 2lym)2 + (zM - zP)2]1/2, r4,nm = [(xM + xP + 2lxn)2 + (yM + yP + 2lym)2 + (zM - zP)2]1/2, Let M ∈ D(-∞, F), where D(-∞, F) = {(x, y, z) ∶ 0 < x < lx, 0 < y < ly, -∞ < z < F(x, y)}. Applying the second Green formula in the domain D(F, ∞) to the function u(P) - the solution of the problem (5) and the function (M, P), we obtain ∫ [ (M, P) u(P) - u(P) P (M, P)]dvP = D(F,∞) = ∫ [ u (P) (M, P) - u(P) (M, P)]d P, M ∈ D(-∞, F) or, for the metaharmonic operator D(F,∞) n nP ∫ [ (M, P)( - k2)u(P) - u(P)( P - k2) (M, P)]dvP = D(F,∞) = ∫ [ u (P) (M, P) - u(P) (M, P)]d P, M ∈ D(-∞, F). D(F,∞) n nP Since (M, P) is a metaharmonic function of P, and the function u satisfies the inhomogeneous metaharmonic equation: ( P - k2)u(P) = (P), we obtain ∫ (M, P) (P)dvP = ∫ [ u (P) (M, P) - u(P) (M, P)]d P, M ∈ D(-∞, F). D(F,∞) D(F,∞) n nP Given the homogeneous boundary conditions for and u on the lateral faces of the cylindrical domain D(F, ∞), we obtain ∫ D(F,∞) (M, P) (P)dvP = ∫ S [ - hf (P) (M, P) - f (P) (M, P)]d P, M ∈ D(-∞, F). (6) n P Let us introduce the notation (M) = ∫ S [ - hf (P) (M, P) - f (P) (M, P)]d P, M ∈ D(-∞, F). (7) n P The function is calculated from the known function f and can be considered as a known function. Then (6) can be written as ∫ (M, P) (P)dvP = (M), M ∈ D(-∞, F). (8) D(F,∞) Given that according to (4) (x, y, z) = (x, y) (z - H) (H + s - z), we can write the potential in (8) as ∫ D(F,∞) lx (M, P) (P)dvP = ∫ 0 ly dxP ∫ 0 H+s dyP (xP, yP) ∫ H dzP (M, P), M ∈ D(-∞, F). Let a < min F(x, y) and the point M is located on the rectangle (x,y) (a) = {(x, y, z) ∶ 0 < x < lx, 0 < y < ly, z = a}. (9) Using the representation for the source function when zP > a we obtain (M, P) = 2 lxly ∞ ∑ n,m=1 e-knm(zP-a) sin knm nxM sin lx myM sin ly nxP sin lx myP , ly where ∫ D(F,∞) lx (M, P) (P)dvP = ∫ 0 ∞ ly dxP ∫ 0 dyP (xP, yP)K(M, P), M ∈ M ∈ (a). (10) K(M, P) = 4 ∑ Knm sin nxM sin myM sin nxP sin myP . (11) lxly n,m=1 s lx ly lx sinh s n2 m2 ly 1/2 Knm = exp { - knm(H + 2 - a)} 2 2 , knm = ( 2 + k l nm x 2 + k2) l y . (12) Then, from (10) and (8), we obtain an integral equation of the first kind ∫ K(M, P) (P)dxPdyP = (M), M ∈ (a). (13) (H) From equation (13), taking into account the expansion (11), when zM = a, we obtain a relation between the Fourier coefficients of the unique solution and the Fourier coefficients of the right-hand side ̃nmKnm = ̃nm(a), (14) where ̃nm(a) are the Fourier coefficients of the function (M)|M∈ (a): nx ̃nm(a) = 4 ∫ (x, y, a) sin sin my dxdy. lxly (a) lx ly Let us note that formula (14) characterizes the decrease of the Fourier coefficients ̃nm(a) with increasing n and m if the function f is such that it ensures the existence of a solution to the problem (5) and, consequently, of the function . Expressing the Fourier coefficients ̃nm from (14), we obtain (x, y) = ∞ ∑ n,m=1 ̃nm(a)K-1 sin nx lx sin my ly . (15) nm The series (15) converges in L2 if the solution of problem (5) with respect to exists for the given function f . Formula (15), where the function is of the form (7), gives an explicit expression for solving the inverse problem. 4. Approximate solution to the problem Let the function f in problem (5) be given with an error, that is, instead of the function f , the function f is given, such that ‖f - f ‖L2(S) ⩽ . We construct an approximate solution of the ill-posed problem (5) that converges to the exact solution when → 0. The function of the form (7) in this case can be obtained approximately: (M) = ∫ S [ - hf (P) (M, P) - f (P) (M, P)]d P. (16) n P Applying the Cauchy-Bunyakovsky inequality to the difference of the functions (16) and (7) for M ∈ ( a), a < min F(x, y), we obtain an estimate for the right-hand side of the integral equation (13) (x,y) | (M) - (M)|M ⩽ h max ( ∫ 2(M, P)d )1/2‖f - f ‖ + ∈ (a) M∈ (a) S P L2(S) + max ( ∫ [ (M, P)]2d P)1/2‖f - f ‖L S ⩽ C . (17) M∈ (a) S nP 2( ) As an approximate solution to equation (13), we will consider the extremal of the Tikhonov functional [10] with a zero-order stabilizer M [w] = ‖Kw - ‖2 + ‖w‖2 , > 0, (18) ‖ ‖L2( (a)) L2( (H)) where (a) and (H) are the domains of the form (9). The extremal can be obtained as a solution of the Euler equation for the functional (18) K∗Kw + w = K∗ (a), which in the Fourier coefficients of the function w has the form K ̃ nm nm 2 w̃nm + w̃nm = Knm (a), where Knm has the form (12) and ̃ (a) = 4 ∫ (x, y, a) sin nx sin my dxdy nm lxly (a) lx ly - Fourier coefficients of the function (M)|M∈ (a). Solving the equation for the Fourier coefficients of the extremal, we find the approximation w to the function in the domain D(-∞, H): ∞ ̃ (a)K-1 nx my nm w (x, y) = ∑ nm nm sin sin . (19) n,m=1 1 + K-2 lx ly nm We note that the terms of the series (19) differ from the terms of the series (15) by the regularizing factor (1 + K-2 })-1, which ensures the convergence of the series. Theorem 1. Let the solution of problem (5) exist. Then for any = ( ) > 0 such that ( ) → 0 and ( ) /√ ( ) → 0 as → 0, the function w of the form (19) converges as → 0 to the exact solution in L2. Proof. Let us estimate the difference ‖w - ‖ ⩽ ‖w - w ‖ + ‖w - ‖, (20) where w is a function of the form (19) with an exact function f w (M) = ∞ ∑ n,m=1 ̃nm(a)K-1 sin nm nm 1 + K-2 nxM lx sin myM ly . Let us estimate the difference w - w in (20) using the estimate (17) ∞ | 4 K 2 -1 nm | ̃ 1/2 2 ̃ | | | ‖w (M) - w (M)‖ = | lxly ∑ ( 1 + K-2 ) | nm(a) - nm(a)| | ⩽ n,m=1 nm | ex x ⩽ C1 max [ 1 + e2x ]‖ (a) - (a)‖ ⩽ C1 √2 . (21) Let us estimate the difference w - in (20) | 4 ∞ | -2 K 2 nm | 1/2 2 ̃ | | | 4 ∞ -2 K 2 nm 2 1/2 | ‖w - ‖ ⩽ | lxly ∑ ( 1 + K-2 ) |Knm nm(a)| | = | lxly ∑ ( 1 + K-2 ) ̃nm| . n,m=1 nm | | n,m=1 nm | Since the series depending on the parameter is majorized by a convergent numerical series with nm coefficients ̃2 , a limit transition in is possible, and thus ‖w - ‖ → 0 при → 0. (22) From (20), (21), and (22), and the conditions of the theorem, it follows that ‖w ( ) - ‖ → 0 при → 0. 5. Results and Discussion The proven theorem allows us to consider the series (19) as an explicit formula for calculating an approximate solution to problem (5), robust to an error in specifying the thermogram f . Note that for a flat surface S, calculating the integral (16) is significantly simplified. When specifying a non-flat surface S approximately, the results of works [19, 20] on calculating the normal to an approximately specified surface can be used to perform calculations using formula (16). Conclusion A stable solution to the inverse problem (5) can be used for mathematical processing of thermograms, particularly in medicine [1], for the purpose of thermal image correction. A thermogram obtained with a thermal imager conveys with varying accuracy the structure of heat sources within the body. Refinement of the thermogram image can be performed within the framework of problem (5). In this case, the function f is compared to the original thermogram, and the function can be considered as the result of thermogram processing. Since the function represents the intensity distribution of heat sources on the rectangle (H), one can expect a more accurate reproduction of their image in the calculated thermogram compared to the original thermogram f .

About the authors

Evgeniy B. Laneev

RUDN University

Email: elaneev@yandex.ru
ORCID iD: 0000-0002-4255-9393
Scopus Author ID: 24366681900
ResearcherId: G-7887-2016

Doctor of Physical and Mathematical Sciences, Professor of the Mathematical Institute named after S. M. Nikolsky of RUDN University

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Aigul M. Gaipova

RUDN University

Author for correspondence.
Email: 1032169239@rudn.ru
ORCID iD: 0009-0008-1345-6265

Post-Graduate student of the Mathematical Institute named after S. M. Nikolsky of RUDN University

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

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