Benno Schorr

A general method for three-dimensional filter computation (positron emission tomography application)

Application of the Fourier space deconvolution algorithm to three-dimensional (3D) reconstruction problems necessitates the computation of a frequency space filter; which requires taking the 3D Fourier transform of the system response function. In this paper, it is shown that for system response functions of the specific form d(theta, phi)/r2, with d(theta, phi) an angular function describing the imaging system, the filter computation can always be reduced to a single integration which, in many cases, may be performed analytically. Complete expressions are derived for the general 3D filter, and two examples are given to illustrate the use of such expressions.

Image Reconstruction for a Rotating Positron Tomograph

A high-resolution positron tomograph consisting of two high-density avalanche chambers mounted on a rotating gantry has been installed in the Nuclear Medicine Department of the Cantonal Hospital in Geneva. Positron annihilation data are collected from six detector positions and back-projected in real time to form a single three-dimensional image. Up to 16 transverse or longitudinal sections may be imaged simultaneously. The derivation of the appropriate deconvolution filter is described in this paper, and the first images taken in the transverse mode are presented. It is shown that quantitative information can be obtained from such images.

Filters for three-dimensional limited-angle tomography

One approach to the three-dimensional, tomographic reconstruction of data from a planar dual-detector stationary positron camera necessitates frequency-space filtering of the image obtained from back-projection. It is therefore useful to have a closed-form expression for the appropriate filter, but the derivation is complicated by the fact that the camera measures only a limited angular range of projections. Nevertheless, suitable expressions can be obtained and are here derived for detectors of rectangular and circular cross-section.

Inventory control based on advanced probability theory, an application

Whenever stock is placed as a buffer between consumption and supply the decision when to replenish the stock is based on uncertain values of future demand and supply variables. Uncertainty exists about the replenishment lead time, about the number of demands and the quantities demanded during this period. We develop a new analytical expression for the reorder point, which is based on the desired service level and three distributions: the distribution of the quantity of single demands during lead time, the distribution of the lengths of time intervals between successive demands, and the distribution of the lead time itself. The distribution of lead time demand is derived from the distributions of individual demand quantities and not from the demand per period. It is not surprising that the resulting formulae for the mean and variance are different from those currently used. The theory developed is also applicable to periodic review systems. The system has been implemented at CERN and enables a significant enhancement of the service level, while reducing the average stock.

On an iterative method for the unfolding of spectra

Abstract In ref. [5] an iterative method has been formally proposed for the numerical solution of a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of both the kernel and the right-hand side. Solving such an equation is also known as unfolding or deconvolution. In this paper, we report the main results of our study (ref. [8]) on a motivation for this iterative method and its convergence, taking into account a result (ref. [7]) on global convergence for a special discrete version of the iteration procedure. In presenting the results, priority has been given to the applicability of the method and not to its mathematical analysis, which may be found in ref. [8]. A numerical example from high-energy physics is presented.

On the choice of the class intervals in the application of the chi-square test

MANN and WALD [3] have given, for the first time, a formula for the choice of the optimal number k of equi-probable class intervals for the x 2-test.This formula leads to relatively high k for different numbers of the sample size. For the determination of k they used the supremum norm in the space of distribution functions as distance notion.BEIER-KUchler and NEUMANN [5] have improved these results and they found out that k can be reduced. In our paper a different norm in the space of distribution functions was used and the results are in good agreement with those of BEIER-KUchler and NEUMANN.

Volume measurement in vivo using positron tomography

The high density avalanche chamber (HIDAC) positron camera has been used for tomographic imaging of the human thyroid in vivo. Images are made seven and twenty-four hours after oral administration to the patient of the positron-emitting radionuclide124I. As a consequence of the high spatial resolution of the camera (2.5 mm fwhm), the functional volume of the thyroid may be estimated from the tomographic images to within about 12%. This paper discusses some of the studies performed to ensure reconstructed images free from artifacts and distortions, and, as illustration of the technique, two typical patient cases are presented.

Asymptotic expansions for the Landau density and distribution functions

Asymptotic expansions are given for the Landau function φ(λ)=12πi∫σ+i∞σ−i∞eμlnμ+λμdμ and for its integral, as λ → ± ∞. Further, it is shown that φ(λ) is indeed a probability density function, as is always assumed in applications.

Vertex fitting of several helices in space

Abstract A method is presented for the determination of the common interaction vertex of high-energy particle tracks for the case where the spectrometer target is situated in a strong magnetic field. A least-squares technique is used to fit the vertex position and provide improved estimates of the individual track parameters subject to several constraints. Examples of the method for various track multiplicities are discussed.