One of the long-standing paradigms in computed tomography (CT) is that CT detectors should be large enough to cover the entire cross-section of the patient. The reason for this requirement is that, according to the classical theory, image reconstruction from truncated CT data (the so-called "interior problem") is non-unique. As a consequence, many aspects of truncated data inversion have not been studied in literature. In this project the PIs will theoretically analyze the problem of image reconstruction from truncated data with minimal prior knowledge. Both transmission CT and emission tomography (SPECT) with constant attenuation will be addressed. The PIs will study the questions of stability of the reconstructions in 2D and 3D, characterize the null-space of an integral transform arising in the interior problem, and analyze eigenfunctions of certain singular Sturm-Liouville problems. The PIs will also develop efficient algorithms for numerical solution of the interior problem. These algorithms will be implemented and tested using both simulated and real data.
At present, interior tomography is at the cusp of being directly applicable in general medical imaging diagnostics. The confluence of theoretical, algorithmic, computational, and technological advances gives the PIs hope that interior tomography can move from the "drawing boards" to practical medical imaging in the not so distant future. While there are still some open research problems, the results obtained recently show that the problems can be solved with a fairly high degree of confidence. If successful, this study will provide theoretically justified and practically applicable two and three-dimensional reconstruction algorithms for imaging a region of interest inside an object (e.g., the patient) from truncated data. The broader impact of the proposed research lies in the promise it holds for improvements in the practice of clinical medicine and biomedical science generally, as well as for industrial non-destructive testing. The most direct benefit is the reduction in the x-ray dose to the patient; other benefits include less expensive scanners, improved temporal resolution, increased scanner throughput, capability of imaging larger objects, reduced system cost, etc.
