Tomography and Microlocal Analysis

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Tomography is the mathematics and engineering used to image the internal structure of an object from indirect data. Tomography algorithms produce images (reconstructions) of the inside of objects from this indirect data. Important types of tomography include X-ray, sonar, radar, and seismic imaging. Standard tomographic algorithms work well with complete data--enough data to image the object accurately and stably (for example, X-ray data from views all around the body provides complete X-ray tomography data). However, many tomographic problems have limited data--some data are missing and standard reconstruction algorithms typically do not image all features of the object and might produce artifacts in the reconstruction. Added streaks and other artifacts are common in limited data reconstructions, and the results of this project will characterize the artifacts, analyze why they occur, and determine how to suppress them. This characterization of artifacts will allow researchers to distinguish features of the object from artifacts in the reconstruction. The research to suppress artifacts will provide algorithms that produce clearer reconstructions in which artifacts are less visible. Direct applications of this research include X-ray tomography, synchrotron imaging, seismic imaging, and radar. This project focuses on inverse problems in tomography, which allows researchers to image the internal structure of an object from indirect data. There are many effective reconstruction algorithms that utilize complete tomographic data (for example, X-ray data from views all around the body); however, many tomographic problems involve limited data, in which some data are missing. The goal of this project is to develop new mathematical methods in micro local analysis to understand and solve limited data tomography problems. Currently, this paradigm is valid only for limited data sets with smooth boundaries that have specific orientations. The resulting reconstruction methods will be tested on industrial synchrotron data. The investigator will also analyze models for seismic imaging and develop the microlocal analysis of the normal operator and generalizations, including calculating its symbol. This calculation will be used to refine the operator so that the reconstruction highlights object boundaries and is uniform throughout the object. The symbol indicates which object singularities will be visible and which will be invisible in the reconstruction. The operator will be implemented using the approximate inverse and tested on data generated from the wave equation and real data. These methods will also be extended to bistatic radar applications, characterized by a transmitter and receiver flying independently in opposite directions.