In 1997 the PI and Kazantsev obtained the first explicit inversion formula for the emission tomography problem using the intimate connection between this problem and elliptic systems with operator coefficients.In this project the PI plans to generalize this method to the following cases: (1) cases with partial or incomplete data; (2) cases where the unknown function is replaced by a tensor field; (3) cases where a general scattering term is also present in a transport equation; (4) cases where straight line rays are replaced by geodesic lines which may be known or unknown; (5) cases where the attenuation coefficient is also unknown; (6) cases with reflection on the boundary. We also plan to consider inverse problems with multiple boundary measurements for elliptic and hyperbolic operators with partial Cauchy data and develop for them the method of Carleman estimates together with the geometric optics construction.
In 1895 Rontgen discovered the so-called X-rays and made the first radiograph (of his wife's hand).In 1917 Radon considered the problem of reconstructing a function on the plane from the integrals of this function over all possible straight lines and obtained an explicit inversion formula. In the mid seventies these two discoveries joined in the X-ray tomograph. In 1979 an English engineer, Hounsfield, and an American mathematician, Cormack, received the Nobel prize in medicine for its development. Using X-ray tomography, one can find the attenuation of a body. If we replace now the X-rays by more "soft" rays we obtain a much more difficult tomography problems. For example, in a seismic tomography one generates elastic waves in the earth using acoustical sources at the earth's surface.The waves that return to the surface of the earth are observed. The problem is to reconstruct the elastic properties of the earth from this data. In this case rays will be unknown curves that depends on the elastic properties. In the emission tomography problem it is required to find unknown radioactive source distribution in a body with a known attenuation. It was only in 1997 that the PI and Kazantsev obtained the first inversion formula for this problem with arbitrary smooth attenuation. In this project, the PI plans to generalize this result to the following cases: (1) cases with incomplete data and curved rays (this is important for oil exploration for instance); (2) cases where the attenuation coefficient is also unknown (this is important for medical imaging, since we reduce the need to use the X-ray tomography for determination of the attenuation); (3) cases where the unknown scalar function is replaced by a vector or a tensor field (non-destructive evaluation of vector and tensor fields in aerodynamics and material sciences respectively). The project will integrate research and education since the PI plans to use new facts from the project in advanced "Math and Engineering courses ".
