Larry A. Shepp

A Statistical Model for Positron Emission Tomography

Abstract Positron emission tomography (PET)—still in its research stages—is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner. Much as in X-ray transmission tomography and other modes of computerized tomography, the quality of the reconstructed image in PET is very sensitive to the mathematical algorithm to be used for reconstruction. In this article, we tailor a mathematical model to the physics of positron emissions, and we use the model to describe the basic image reconstruction problem of PET as a standard problem in statistical estimation from incomplete data. We describe various estimation procedures, such as the maximum likelihood (ML) method (using the EM algorithm), the method of moments, and the least squares method. A computer simulation of a PET experiment is then used to demonstrate the ML and the least squares reconstructions. The main purposes of this article are to report on what we believe is an...

Risk vs. profit potential: A model for corporate strategy

Abstract A firm whose net earnings are uncertain, and that is subject to the risk of bankruptcy, must choose between paying dividends and retaining earnings in a liquid reserve. Also, different operating strategies imply different combinations of expected return and variance. We model the firms cash reserve as the difference between the cumulative net earnings and the cumulative dividends. The first is a diffusion (additive), whose drift/volatility pair is chosen dynamically from a finite set, A . The second is an arbitrary nondecreasing process, chosen by the firm. The firms strategy must be nonclairvoyant . The firm is bankrupt at the first time, T , at which the cash reserve falls to zero ( T may be infinite), and the firms objective is to maximize the expected total discounted dividends from 0 to T , given an initial reserve, x ; denote this maximum by V ( x ). We calculate V explicitly, as a function of the set A and the discount rate. The optimal policy has the form: 1. (1) pay no dividends if the reserve is less than some critical level, a , and pay out all of the excess above a ; 2. (2) choose the drift/volatility pairs from the upper extreme points of the convex hull of A , between the pair that minimizes the ratio of volatility to drift and the pair that maximizes the drift; furthermore, the firm switches to successively higher volatility/drift ratios as the reserve increases to a . Finally, for the optimal policy, the firm is bankrupt in finite time, with probability one.

COMPUTERIZED TOMOGRAPHY: THE NEW MEDICAL X-RAY TECHNOLOGY

Computerized X-ray tomography is a completely new way of using X-rays for medical diagnosis. It gives physicians a more accurate way of seeing inside the human body and permits safe, convenient, and quantitative location of tumors, blood clots and other conditions which would be painful, dangerous, or even impossible to locate by other methods. Although each tomography machine costs hundreds of thousands of dollars, hundreds of tomography machines are already in use. A mathematical algorithm to convert X-ray attenuation measurements into a cross-sectional image plays a central role in tomography. Sophisticated mathematical analysis using Fourier transforms has led to algorithms which are much more accurate and efficient than the algorithm used in the first commercial tomography machines. We show how some of the algorithms in actual use have been developed. We also discuss some related mathematical theorems and open questions. 1. Introduction. In computerized tomography, X-ray transmission measurements are recorded on a computer memory device rather than on film, and a sophisticated mathematical algorithm is applied. This produces a numerical description of tissue density as a function of position within a thin slice through the body. The physician examines this function by use of visual displays. In the ordinary medical use of X-rays, the picture is something like a shadow; any feature in line with denser bone tissue tends to be blocked out. In other words, if we could make a great many pictures, each of a thin slice perpendicular to the beam of X-rays, the actual X-ray picture is formed by superposition of all these hypothetical pictures, i.e., it is a kind of multiple exposure. Com- puterized X-ray tomography provides a picture of a single thin slice through the body, without superposition. The word tomography is related to the Greek word tomos meaning cut or slice. Imagine a thin slice, say through the head, perpendicular to the main body axis. Several hundred parallel X-ray pencil beams are projected through the head in the plane of this slice, and the attenuation of X-rays in each beam is measured separately and recorded. (In earlier machines a single beam has been used by translating it parallel to itself within the plane; some of the later machines discussed in Section 3 use fan rather than parallel arrays of beams.) Another set of parallel beams is used within the same plane but at an angle of perhaps 10 or so with the first set, and measurements are taken again. The process is repeated until measurements have been taken for a grid covering all directions in the plane. An elaborate calculation then permits approximate reconstruction of the X-ray attenuation density as a function of position within the slice. In appropriate units, tissue density in the head varies roughly between 1.0 and 1.05 with the exception of bone which has a density of about 2. Some features of medical interest are indicated by variations of density as small as .005. Reconstructing tissue density with adequate accuracy at a sufficiently fine grid of points is thus a challenging project. Mathematically we may describe the problem as follows. Consider a fixed plane through the body. Let f(x,y) denote the density at the point (x,y), and let L be any line in the plane. Suppose we direct a thin beam of X-rays into the body along L, and measure how much the intensity is attenuated by going through the body. It is easy to see that the logarithm of the attenuation factor is given

Covering the circle with random ARCS

AbstractArcs of lengthsln, 0<ln+1<=ln<1,n=1,2,…, are thrown independently and uniformly on a circumferenceC of unit length. The union of the arcs coversC with probability one if and only ifn

The discrete Radon transform and its approximate inversion via linear programming

sumlimits_{n = 1}^infty {n^{ - 2} exp left( {l_1 + ... + 1l_n } right) = infty }

Two-dimensional remote air-pollution monitoring via tomography

n.

A Stochastic Model of Fragmentation in Dynamic Storage Allocation

Let ({eta}=left({eta_j}right)) be a sequence of independent Gaussian, means 0, Varriance 1, random variable in all of this paper. We shall then study the distribution of n n