Application of Multi-Dimensional Fourier Transforms to Tomography, pp. 321-370
Authors: (Natalie Baddour, Dept. of Mechanical Engineering, Univ. of Ottawa, Ottawa, Ontario, Canada)
Abstract: The Fourier transform is one of the most useful tools in the scientific and engineering repertoire. By examining a function in the frequency domain, additional information and insights may be obtained. The Fourier transform may be applied to transform from temporal to frequency domains and/or from spatial to spatial-frequency domains. Furthermore, it may be extended into multi-dimensions, thereby expanding its scope to combined time and space problems. One particular result that has spawned great applications in the field of imaging is the classical Fourier diffraction theorem of ultrasound tomography, which itself may be considered to be an analog of the Fourier slice theorem of straight-ray tomography. These theorems relate the Fourier transform of measured data to the Fourier transform of the object being imaged. Image reconstruction algorithms are subsequently based on the fact that the object may be reconstructed if enough knowledge about its Fourier transform can be obtained through measurements. In fact, this theorem permits far greater generalization and can be applied to other imaging modalities using different physical mechanisms such as thermography, photoacoustics or diffuse photon density waves. In this chapter, generalizations of the classical Fourier diffraction theorem are developed and applied to other imaging modalities via consistent application of multi-dimensional Fourier transforms. These generalizations naturally lead to considerations of multi-dimensional Fourier transforms in curvilinear coordinates.