Interferometric 2D and 3D tomography of photoelastic media

The experimental study of mechanical stress distribution in biological tissues in-vivo is of interest for some biomedical applications. This work considers the problem of light and stressed tissue interaction and the inverse problem of stress field reconstruction in 2D and 3D cases. Optical tomography is one of the most promising methods of solving these problems. This technique involves the reconstruction of the refractive index field using the measurements of waveform distortion. The reconstruction of stress field requires establishing the relation of the stress tensor to the variation of refraction index. A simple photoelastic model is a reasonable first approximation due to normal functioning of biological tissue. The propagation equation that describes the light propagation through the optically active elastic media obtained in the solving of the forward problem in terms of geometric optics approach. Interferometric, shlieren and depolarization methods of experimental data acquisition are considered. In general, 3D state of a stressed tissue should be described by six components of the stress tensor, but only three propagation equations appear to be independent. To close the system of equations, we have used three partial differential equilibrium equations with appropriate boundary conditions. The system of equations of interferometric tomography is studied in detail. In this case, the separation of stress tensor components results from analytical solving in the Radon domain. For special case of 2D deformation we need only one propagation equation and two equilibrium equations. It is shown that 3D problem can not be reduced to 2D problem in the general case of tensor field tomography. This sends us in search of special 3D algorithms. The use of wavelets is one of perspective ways of tomographic reconstruction under strong noise. 2D and 3D algorithms of the inverse Radon transform through inverse wavelet transform in noisy conditions have been developed.