This talk concerns the source reconstruction problem in a transport problem through an absorbing and scattering medium from boundary measurement data on an arc of the boundary. The method, specific to two dimensional domains, relies on Bukgheim’s theory of A-analytic maps and it is joint work with A. Tamasan (UCF) and H. Fujiwara (Kyoto U).

In this work we present a two-step method for the numerical solution of parabolic and hyperbolic Cauchy problems. Both problems are formulated in 2D and the proposed method is considered for the direct and the corresponding inverse problem. The main idea is to combine a semi-discretization with respect to the time variable with THE boundary integral equation method for the spatial variables. The time discretization reduces the problem to a sequence of elliptic stationary problems. The solution is represented using a single-layer ansatz and then we end up solving iteratively for the unknown boundary density functions. We solve the discretized problem on the boundary of the medium with the collocation method. Classical quadrature rules are applied for handling the kernel singularities. We present numerical results for different linear PDEs.

This is a joint work with R. Chapko (Ivan Franko University of Lviv, Ukraine) and B. T. Johansson (Linköping University, Sweden).

Optical Coherence Tomography (OCT), an imaging modality based on the interferometric measurement of back-scattered light, is known for its high-resolution images of biological tissues and its versatility in medical imaging. Especially in its main field of application, ophthalmology, the continuously increasing interest in OCT, aside from improving image quality, has driven the need for quantitative information, like optical properties, for a better medical diagnosis. In this talk, we discuss the quantification of the refractive index, an optical property which describes the change of wavelength between different materials, from OCT data. The presented method is based on a Gaussian beam forward model, resembling the strongly focused laser light typically used within an OCT setup. Samples with layered structure are considered, meaning that the refractive index as a function of depth is well approximated by a piece-wise constant function. For the reconstruction, a layer-by-layer method is presented where in every step the refractive index is obtained via a discretized $L^2$−minimization. The applicability of the proposed method is then verified by reconstructing refractive indices of layered media from both simulated and experimental OCT data.

Total variation ($TV$) regularization has been extensively employed in inverse problems in imaging. In this talk, we propose a new method of $TV$ regularization for medical image reconstruction, which extends standard regularization approaches. Our novel method may be conceived as an augmented version of typical $TV$ regularization. Within this approach, a new monitoring variable, $\omega(x)$, is introduced via an additional term in the minimization functional. The integration in this term is performed with respect to the $TV$ measure, corresponding to the deviation of the image, $u(x)$. The dual function $\omega(x)$ is the integrand of the additional term, and its smoothing nature compensates, when necessary, for the abruptness of the $TV$ measure of the image. It is within this dual variable that the regularity is imposed via the minimization process itself. The main purpose of the dual variable is to control the behavior of $u(x)$, especially regarding its discontinuity properties. Our preliminary results indicate the fast convergence rate of our method, thus highlighting its promising potential. This research is partially supported by the Horizon Europe project SEPTON, under grant agreement 101094901.