Some Problems Of Integral Geometry In Advanced Imaging

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).