The mathematical algorithms for tomographic reconstructions described in Chapter 3 are based on projection data. These projections can represent, for example, the attenuation of x-rays through an object as in conventional x-ray tomography, the decay of radioactive nucleoids in the body as in emission tomography, or the refractive index variations as in ultrasonic tomography. This chapter will discuss the measurement of projection data with energy that travels in straight lines through objects. This is always the case when a human body is illuminated with x-rays and is a close approximation to what happens when ultrasonic tomography is used for the imaging of soft biological tissues (e.g., the female breast). Projection data, by their very nature, are a result of interaction between the radiation used for imaging and the substance of which the object is composed. To a first approximation, such interactions can be modeled as measuring integrals of some characteristic of the object. A simple example of this is the attenuation a beam of x-rays undergoes as it travels through an object. A line

ii Acknowledgements Let me express my genuine appreciation to my thesis advisor, Professor Stanislav Rokhlin. Without his guidance and persistence this document could not have been brought to fruition. Also my thanks to my dissertation committee, Professors Laslo Adler, Carolyn Merry, and Roni Yagel; I sincerely appreciate their time, effort, and enthusiasm. I am also grateful to my fellow graduate students. I cannot mention them all, but allow me to make special acknowledgements to Wei Huang, Ligou Wang, and Dan Applegate. The camaraderie shared by graduate students is an important aspect of the graduate school experience that should not be discounted. My thanks also, of course, to my family and friends, for their steadfast support and devotion throughout the many years of my education. In particular, let me express my deepest appreciation to my mother, who more than anyone else has provided support when I needed it most.