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

Laneev, E. B.; Mouratov, M. N.; Zhidkov, E.P.

doi:10.1134/s1547477108030059

Cited by 4 publications

(8 citation statements)

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“…Stable solution of the inverse problem (9) can be used for mathematical processing of thermograms taken with a thermal imager, in particular, in medicine [4], in order to correct the image on the thermogram. Note that taking into account the blood flow leads to the need to use the metaharmonic equation [11,12] in problem (9). As already mentioned, a thermogram, with one or another reliability, convey an image of the structure of heat sources inside the body.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…As already mentioned, a thermogram, with one or another reliability, convey an image of the structure of heat sources inside the body. Refinement of the image on the thermogram can be carried out within the framework of the problem (9). In this case, the function 𝑓 is associated with the original thermogram, and the function 𝑢 𝐻 is considered the result of processing the thermogram.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…In [8], the following representation of the solution of the problem is obtained (9) 𝑢(𝑀 ) = 𝑣(𝑀 ) + Φ(𝑀 ), 𝑀 ∈ 𝐷(𝐹 , 𝐻), (12) where the function Φ is calculated on the known functions 𝑓 and 𝑓 1 , and the function 𝑣 has the form:…”

Section: Exact Solution Of the Inverse Problemmentioning

confidence: 99%

“…As follows from the formulas (15), ( 23), (20), when calculating the Fourier coefficient for each pair of indices 𝑛 and 𝑚, a superposition of the following calculations is required: summation of the series for 𝜑, integration on the surface 𝑆, integration on the rectangle Π(𝑎). Thus, when discretizing [9] the problem (𝑁 𝑥 points on the variable 𝑥, 𝑁 𝑦 points on the variable 𝑦) when calculating Fourier coefficients, about 𝑂(𝑁 𝑥 𝑁 𝑦 ) 4 operations are required. This is the largest volume of operations when constructing a solution to the problem (9), during which, in addition to time, there is a loss of accuracy and an additional error is formed in calculating the Fourier coefficients and solving the problem as a whole.…”

Section: Calculation Of Fourier Coefficientsmentioning

confidence: 99%

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On the application of the Fourier method to solve the problem of correction of thermographic images

Baaj

2022

Discrete and Continuous Models

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The work is devoted to the construction of computational algorithms implementing the method of correction of thermographic images. The correction is carried out on the basis of solving some ill-posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. This problem corresponds to the problem of the analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study towards the heat sources. The cylindrical region is bounded by an arbitrary surface and plane. On an arbitrary surface, a temperature distribution is measured (and thus is known). It is called a thermogram and reproduces an image of the internal heat-generating structure. On this surface, which is the boundary of the object under study, convective heat exchange with the external environment of a given temperature takes place, which is described by Newton’s law. This is the third boundary condition, which together with the first boundary condition corresponds to the Cauchy conditions - the boundary values of the desired function and its normal derivative. The problem is ill-posed. In this paper, using the Tikhonov regularization method, an approximate solution of the problem was obtained, stable with respect to the error in the Cauchy data, and which can be used to build effective computational algorithms. The paper considers algorithms that can significantly reduce the amount of calculations.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Exact Solution Of the Inverse Problemmentioning

confidence: 99%

Section: Calculation Of Fourier Coefficientsmentioning

confidence: 99%

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On the application of the Fourier method to solve the problem of correction of thermographic images

Baaj

2022

Discrete and Continuous Models

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“…when z M = a, z P = 0. The Fourier coefficients in the formula (46) are calculated without calculating the function Φ, similarly to [12]. When using the formula (46), integration is performed under the sign of the integral in (37) and under the sign of the sum in (55).…”

Section: Numerical Solution To the Optimization Problemmentioning

confidence: 99%

Correction of thermographic images based on the minimization method of Tikhonov functional

Baaj

Chernikova

Laneev

2022

YUGOSLAV J OPERATION

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The paper considers the method of correction of thermographic images (thermograms) obtained by recording in the infrared range of radiation from the surface of the object under study using a thermal imager. A thermogram with a certain degree of reliability transmits an image of the heat-generating structure inside the body. In this paper, the mathematical correction of images on a thermogram is performed based on an analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study towards the heat sources. The continuation is carried out by solving an ill-posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. To construct a stable solution to the problem, the principle of the minimum of the Tikhonov smoothing functional we used.

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

Laneev

Baaj²

2022

Discrete and Continuous Models

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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.

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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

Cited by 4 publications

References 0 publications

On the application of the Fourier method to solve the problem of correction of thermographic images

On the application of the Fourier method to solve the problem of correction of thermographic images

Correction of thermographic images based on the minimization method of Tikhonov functional

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

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