Some Problems Of Integral Geometry In Advanced Imaging

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Title: Some Problems Of Integral Geometry In Advanced Imaging
Author: Gouia, Rim
Abstract: During the past decade , our society has become dependent on advanced mathematics for many of our daily needs . Mathematics is at the heart of the 21st century technologies and more specifically the emerging imaging technologies from thermoacoustic tomography (TAT ) and ultrasound computed tomography (UCT ) to non -destructive testing (NDT ) . All of these applications reconstruct the internal structure of an object from external measurements without damaging the entity under investigation . The basic mathematical idea common to such reconstruction problems is often based upon Radon integral transform .The Radon integral transform Rf puts into correspondence to agiven function f its integrals over certain subsets . In this work ,we will focus on the situation when the subsets are circles . Themajor problems related to this transform are the existence anduniqueness of its inversion , inversion formulas and the rangedescription of the transform . When Rf is known for circles of allpossible radii , there are well developed theories now addressingmost of the questions mentioned above . However , many of thesequestions are still open when Rf is available for only a part ofall possible radii , or when the support of f is outside the circle .The aim of my dissertation is to derive some new results about theexistence and uniqueness of the representation of a function by itscircular Radon transform with radially partial data for bothinterior and exterior problems . The presented new results open newfrontiers in the field of medical imaging such as intravascularultrasound (IVUS ) and transrectal ultrasound (TRUS ) .
URI: http : / /hdl .handle .net /10106 /5884
Date: 2011-07-14


Some Problems Of Integral Geometry In Advanced Imaging. Available electronically from http : / /hdl .handle .net /10106 /5884 .

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