Hall, Eric B. | Search NASA

NASA NTRS1996

A Velocity Distribution Model for Steady State Heat Transfer

Consider a box that is filled with an ideal gas and that is aligned along Cartesian coordinates (x, y, z) having until length in the 'y' direction and unspecified length in the 'x' and 'z' directions. Heat is applied uniformly over the 'hot' end of the box (y = 1) and is removed uniformly over the 'cold' end (y = O) at a constant rate such that the ends of the box are maintained at temperatures T(sub 0) at y = O and T(sub 1) at y = 1. Let U, V, and W denote the respective velocity components of a molecule inside the box selected at some random time and at some location (x, y, z). If T(sub 0) = T(sub 1), then U, Y, and W are mutually independent and Gaussian, each with mean zero and variance RT(sub 0), where R is the gas constant. When T(sub 0) does not equal T(sub 1) the velocity components are not independent and are not Gaussian. Our objective is to characterize the joint distribution of the velocity components U, Y, and W as a function of y, and, in particular, to characterize the distribution of V given y. It is hoped that this research will lead to an increased physical understanding of the nature of turbulence.

Hall, Eric B.↗

NASA NTRS1995

The composite classification problem in optical information processing

Optical pattern recognition allows objects to be recognized from their images and permits their positional parameters to be estimated accurately in real time. The guiding principle behind optical pattern recognition is that a lens focusing a beam of coherent light modulated with an image produces the two-dimensinal Fourier transform of that image. When the resulting output is further transformed by the matched filter corresponding to the original image, one obtains the autocorrelation function of the original image, which has a peak at the origin. Such a device is called an optical correlator and may be used to recognize the locate the image for which it is designed. (From a practical perspective, an approximation to the matched filter must be used since the spatial light modulator (SLM) on which the filter is implemented usually does not allow one to independently control both the magnitude and phase of the filter.) Generally, one is not just concerned with recognizing a single image but is interested in recognizing a variety of rotated and scaled views of a particular image. In order to recognize these different views using an optical correlator, one may select a subset of these views (whose elements are called training images) and then use a composite filter that is designed to produce a correlation peak for each training image. Presumably, these peaks should be sharp and easily distinguishable from the surrounding correlation plane values. In this report we consider two areas of research regarding composite optical correlators. First, we consider the question of how best to choose the training images that are used to design the composite filter. With regard to quantity, the number of training images should be large enough to adequately represent all possible views of the targeted object yet small enough to ensure that the resolution of the filter is not exhausted. As for the images themselves, they should be distinct enough to avoid numerical difficulties yet similar enough to avoid gaps in which certain views of the target will be unrecognized. One method that we introduce to study this problem is called probing and involves the creation of the artificial imagery. The second problem we consider involves the clasification of the composite filter's correlation plane data. In particular, we would like to determine not only whether or not we are viewing a training image, but, in the former case, we would like to determine which training image is being viewed. This second problem is investigated using traditional M-ary hypothesis testing techniques.

Hall, Eric B.↗