Mathematical Sciences: Computed Tomography and Sampling

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9404436 Faridani Tomography produces the reconstruction of a generalized density function f from a large number of its line integrals. Beyond its initial use in medical imaging, it has found a growing number of scientific and industrial applications. Ordinary tomography is not local; the standard formulas for reconstruction at a single point require measurements at all lines within some plane containing the point. Local tomography produces the reconstruction of a linear combination of functions related to f. This reconstruction is local; for reconstruction at a point, attenuation measurements are needed only along lines very close to that point. This makes it possible to reconstruct a region of interest within a large object without having to scan the whole object. The term computed tomography describes a class of methods for imaging the interior of opaque objects. Characteristic of these methods is that the image is computed from a large number of indirect measurements. Mathematically, the image represents some density function, and the measurements are line integrals of this function. A well-known example is CAT scans. Here the density function is the x-ray absorption coefficient, and the measurements are obtained by recording the attenuation of thin x-ray beams traversing the patient. Beyond its original application in medical imaging, computed tomography has found a growing number of scientific and industrial applications. The research involves applications of tomography in areas such as medical imaging (imaging the coronary arterial tree), biological research (imaging the fine structure of tissue using the emerging technology of high-resolution Micro-CT scanners), and quantum optics. Part of this research project concerns the technique of `local tomography'. While ordinary tomography requires scanning the whole object, local tomography allows for scanning only a region of interest. Local tomography is of practical interest sin ce it requires less expensive scanning equipment, reduces x-ray exposure, and makes it possible to image objects too large to be scanned in their entirety. At present, local tomography allows one to identify the shapes of different features, but does not produce the correct density differences. The goal is to develop a method which allows reconstruction of density differences. This will further increase the usefulness of local tomography for many applications. Furthermore, questions of optimal sampling, resolution, and reconstruction algorithms in three-dimensional local tomography will be investigated in collaboration with researchers at the Mayo Clinic, who are building a high-resolution Micro-CT scanner. The second part of this research continues work in Shannon sampling theory and its application to computed tomography. Shannon sampling theory is of fundamental importance in signal processing. The aliasing error for sampling non-bandlimited functions on non-equidistant but periodic sampling sets will be investigated, and more general classes of sampling sets will be studied. The results will be applied to computed tomography, resulting in error estimates and new efficient sampling schemes. The deeper understanding of optimal sampling and reconstruction algorithms thus obtained will allow to fully exploit the capabilities of current tomographic scanners and have implications for the design of future equipment.