Research On Imaging Problems Based On Parabolic Equations

Optical tomography is a typical non-invasive medical imaging technique.It uses nearinfrared light as an excitation source to illuminate a biological tissue,which makes photons scattered and absorbed in the biological tissue.Then the light intensity distribution information on the surface can be measured by high-precision instruments.Finally,a stable reconstruction algorithm is used to reconstruct the optical parameters of the biological tissue.The high contrast of the optical parameters indicates that there are some anomalies inside the biological tissue.Optical tomography is usually formulated as an initial boundary value problem for partial differential equations.The governing equations include Maxwell's equations,radiation transport equations and diffusion equations,according to the spatial scale.This paper studies the forward and inverse problems of the diffuse optical tomography,based on the time-dependent diffusion equation model.First,using the boundary integral equation method,the initial boundary value problem for the diffusion equation is converted equivalently into a boundary integral equation system.In order to establish the numerical discretization scheme of the boundary integral equation system,the single-layer potential of the diffusion equation and its normal derivative are expressed as generalized Abel integrals for the time variable.In addition,their kernels depend on both the time and space variables.A stable discretization scheme for singular boundary integrals is established,based on the asymptotic behavior of the kernel functions.Second,an inverse boundary value problem for the diffusion equation is studied,which aims to reconstruct the geometric information(location,size,shape)on unknown inclusions inside the biological tissues from boundary measurement data.Then a non-iterative numerical method is proposed,with rigorous theoretical analysis.Finally,numerical experiments of the forward and inverse problems are carried out,and the numerical results greatly verify the feasibility and effectiveness of the proposed numerical methods.