Computed Tomography and Sampling

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AbstractThe research proposed in this project involves Computed Tomography, Advanced Shannon Sampling Theory, applications of sampling theory in tomography, and combination of image reconstruction with image processing. Computed x-ray tomography produces images of opaque objects. The underlying mathematical problem consists of reconstructing a density function f(x) from finitely many measurements of its line integrals. Much of the present focus of the field is on fully three-dimensional imaging, stimulated by major algorithmic advances in recent years. Shannon sampling theory provides exact interpolation formulas for band-limited functions and plays a fundamental role in signal processing. It is used in tomography to identify efficient sampling schemes that allow for maximum resolution in the reconstructed images, as well as for error analysis of reconstruction algorithms. The research proposed in this project lies in the intersection of these areas and is motivated by applications. Its interconnected themes include the derivation of new sampling theorems; optimal data acquisition in 3D tomography; mathematically rigorous and practically useful analysis of recently developed 3D tomographic algorithms; local tomography in two and three dimensions; high-resolution reconstructions; and the close interaction of tomographic reconstruction with image processing, motivated by the experience that high-resolution reconstructions often carry more noise and can be further improved by state of the art image processing. In pursuing these themes the project will place an emphasis on interdisciplinary collaboration.Computed x-ray tomography (CT) is an increasingly popular technique for two- and three-dimensional (3D) imaging of opaque objects. An object is irradiated and the image is computed from measurements of the transmitted radiation. Beyond its famous use in medical imaging, tomography has found an increasing number of applications in science, engineering, and technology. The research undertaken in this project will contribute to make this technique even more efficient. It will develop and use Shannon type sampling theorems to identify optimal data acquisition schemes for 3D tomography and to analyze and improve algorithms for image reconstruction and image processing. This will contribute to the development of fully three-dimensional image reconstruction methods that require a minimal number of measurements and have high accuracy, optimal resolution, and minimal noise. The general approach taken in this project aims to unite research on fundamentals with interdisciplinary efforts driven directly by practical challenges presented by users of tomography. The project will also involve efforts to provide graduate students with an interdisciplinary environment centered around multi-dimensional imaging.