Optical Imaging: 3D Approximation and Perturbation Approaches for Time-domain Data

The reconstruction method presented here is based on the diffusion approximation for the light propagation in turbid media and on a minimization strategy for the output-least-squares problem. A perturbation approach is introduced for the optical properties. Here, the number of free variables of the inverse problem can be strongly reduced by exploiting a priori information such as the search for single inhomogeneities within a relatively homogeneous object, a typical situation for breast cancer detection. Higher accuracy and a considerable reduction of the computational effort are achieved by solving a parabolic differential equation for a perturbation density, i.e. the difference between the photon density in an inhomogeneous object and the density in the homogeneous case being given by an analytic expression. The calculations are performed by a 2D FEM algorithm, however, as a time-dependent correction factor is applied, the 3D situation is well approximated. The method was successfully tested by the University of Pennsylvania standard data set. Data noise was generated and taken into account in a modified data set. The influence of different noise on the reconstruction results is discussed

Optical imaging: three-dimensional approximation and perturbation approaches for time-domain data

The reconstruction method presented here is based on diffusion approximation for light propagation in turbid media and on a minimization strategy for the output-least-squares problem. A perturbation approach is introduced for the optical properties. Here we can strongly reduce the number of free variables of the inverse problem by exploiting a priori information such as the search for single inhomogeneities within a relatively homogeneous object, a typical situation for breast cancer detection. We achieve higher accuracy and a considerable reduction in computational effort by solving a parabolic differential equation for a perturbation density, i.e., the difference between the photon density in an inhomogeneous object and the density in the homogeneous case being given by an analytic expression. The calculations are performed by a two-dimensional finite-element-method algorithm. However, as a time-dependent correction factor is applied, the three-dimensional situation is well approximated. The method was successfully tested by use of the University of Pennsylvania standard data set. Data noise was generated and taken into account in a modified data set. The influence of different noise on the reconstruction results is discussed

Mathematical modelling of indirect measurements in scatterometry

In this work, we illustrate the benefits and problems of mathematical modelling and effective numerical algorithms to determine the diffraction of light by periodic grating structures. Such models are required for reconstruction of the grating structure from the light diffraction patterns. With decreasing structure dimensions on lithography masks, increasing demands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical method offer access to the geometrical parameters of periodic structures including pitch, side-wall angles, line heights, top and bottom widths. The mathematical model for scatterometry is based on the Helmholtz equation derived as a time-harmonic solution of the Maxwell equations. It determines the incident and scattered electric and magnetic fields, which fully specify the light propagation in a periodic two-dimensional grating structure. For numerical simulations of the diffraction patterns, a standard finite element method (FEM) or a generalized finite element method (GFEM) is used for solving the elliptic Helmholtz equation. In a first step, we performed systematic forward calculations for different varying structure parameters to evaluate the applicability and sensitivity of different scatterometric measurement methods. Furthermore our programs include several iterative optimization methods for reconstructing the geometric parameters of the grating structure by the minimization of a functional. First reconstruction results for different test data sets are presented

Parameter sensitivity in near-infrared imaging

To model the photon migration in highly scattering media, we use an approximation of the Boltzmann equation, the diffusion equation. A prerequisite for handling the inverse problem consists in solving the forward problem under realistic conditions. We discuss the influence of boundary conditions on the light propagation. The boundary conditions at the walls surrounding the object highly sensitively influence the photon flux at the boundary which means that the time-resolved transmittance is affected. An algorithm for the determination of boundary parameters is introduced and demonstrated by an instructive example. We use the finite element method for the time-resolved case as a basic method in combination with a minimization strategy. The boundary conditions are determined as conditions of the third kind, i.e. the photon density is proportional to the outward photon flux at the boundary

Mathematical modelling of indirect measurements in periodic diffractive optics and scatterometry

In this work, we illustrate the benefits and problems of mathematical modelling and effective numerical algorithms to determine the diffraction of light by periodic grating structures. Such models are required for reconstruction of the grating structure from the light diffraction patterns. With decreasing structure dimensions on lithography masks, increasing demands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical method offer access to the geometrical parameters of periodic structures including pitch, side-wall angles, line heights, top and bottom widths. The mathematical model for scatterometry is based on the Helmholtz equation derived as a time-harmonic solution of Maxwell's equations. It determines the incident and scattered electric and magnetic fields, which fully specify the light propagation in a periodic two-dimensional grating structure. For numerical simulations of the diffraction patterns, a standard finite element method (FEM) or a generalized finite element method (GFEM) is used for solving the elliptic Helmholtz equation. In a first step, we performed systematic forward calculations for different varying structure parameters to evaluate the applicability and sensitivity of different scatterometric measurement methods ..

Transillumination imaging performance using time domain data

Light propagation in highly scattering media can be numerically simulated by solving the diffusion equation by the finite element method (FEM). Employing an iterative algorithm, the FEM solution of the forward problem is applied to the inverse imaging problem. Good test results were previously achieved when absorbers were searched in different objects. Now the reconstruction of scattering is also taken into account. Simulated measurement data are used to test and evaluate the method at various objects with tissue-like properties. Resulting problems are very ill posed. The algorithm is specially adapted to the illposedness of the problem. Improvements in reconstruction results can be achieved in two ways, first by adapting the detector arrangements and, secondly, by using a regularization strategy. The effectiveness of these methods is demonstrated by instructive examples