Moon, Sunghwan (2013-12). Properties of Some Integral Transforms Arising in Tomography. Doctoral Dissertation. Thesis uri icon

abstract

  • This dissertation deals with several types of imaging: radio tomography, single scattering optical tomography, photoacoustic tomography, and Compton camera imaging. Each of these tomographic techniques leads to a Radon-type transform: radio tomography brings about an elliptical Radon transform, single scattering optical tomography reduces to the V-line Radon transform, and photoacoustic tomography with line detectors boils down to a cylindrical Radon transform. We also introduce a different Radon-type transform arising in photo acoustic tomography with circular detectors, and study mathematically similar object, a toroidal Radon transform. We also consider the cone transform arising in Compton camera imaging as well as the windowed ray transform. We provide inversion formulas for all these transforms. When given some Radon- type transform, we are interested not only in inversion formulas, but also in range conditions, and stability. We thus address range conditions, a stability estimate for some of the Radon-type transforms above.
  • This dissertation deals with several types of imaging: radio tomography, single scattering optical tomography, photoacoustic tomography, and Compton camera imaging. Each of these tomographic techniques leads to a Radon-type transform: radio tomography brings about an elliptical Radon transform, single scattering optical tomography reduces to the V-line Radon transform, and photoacoustic tomography with line detectors boils down to a cylindrical Radon transform. We also introduce a different Radon-type transform arising in photo acoustic tomography with circular detectors, and study mathematically similar object, a toroidal Radon transform. We also consider the cone transform arising in Compton camera imaging as well as the windowed ray transform.

    We provide inversion formulas for all these transforms. When given some Radon- type transform, we are interested not only in inversion formulas, but also in range conditions, and stability. We thus address range conditions, a stability estimate for some of the Radon-type transforms above.

publication date

  • December 2013