Equivalent Rectangle Research Articles

Transient phenomena analysis using dynamic speckle patterns

The full width at half maximum (FWHM) is sometimes used to characterize the autocorrelation function of the time history of a speckle pattern. We propose to include more autocorrelation points to diminish the variability of the measurement. The width of the equivalent rectangle (WER) and the X*LOG X measurements are defined and some simulations and experimental results obtained are shown.

Target Acquisition Criteria for Human Observers

A primary target acquisition threat to ground vehicle systems is the human observer whose psycho-visual capabilities are a critical element in the simulation of these phenomena. This paper discusses the Army's performance assessment methodology for determining signature countermeasures tradeoffs and the human factors elements essential for this type of analysis. Current Army acquisition models use a very simplistic model of the human visual perception system. The minimum resolvable contrast (MRC) models, for example, consider the target as an equivalent rectangle in a uniform background scene. The criteria for detection, recognition and identification are defined in terms of the number of line pairs across a critical target dimension. Although this simple model is convenient for many battlefield effectiveness simulations; the level of detail is insufficient for most CM applications. Several examples will illustrate the limitations of current target acquisition methodology for visual signature assessment and evaluation. A detailed description of the target/background scene and the observer psychovisual characteristics is essential for a robust simulation of these processes. In particular color, motion, shape, texture, search and clutter are essential parameters in any CM performance assessment methodology. The inclusion of these elements into future target acquisition models is the focal point for this discussion.

Accuracy of using point targets for SAR calibration

The peak and integral methods for radiometric calibration of a synthetic aperture radar (SAR) using reference point targets are analyzed. Both calibration methods are shown to be unbiased, but the peak method requires knowledge of the equivalent rectangle system resolution which is sensitive to system focus. Exact expressions for the RMS errors of both methods are derived. It is shown that the RMS error resulting from the peak method is always smaller than or equal to that from the integral method for a well-focused system. However, for robust radiometric calibration of SAR, or when nonlinear phase errors are present, the integral method is recommended, because it does not require detailed knowledge of the impulse response and the resulting RMS error is not dependent on system focus.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A theoretical analysis of interflow of water through surface soil horizons with implications for movement of chemicals in field runoff

Steady interflow of water in the near‐surface saturated soil horizons, which are the important nonpoint sources of applied agricultural chemicals appearing in field runoff, is investigated theoretically. Analytical solutions are obtained for some relevant cases of interflow in a sloping layered soil having a subsoil of lower permeability. Analysis is made for the effect of the relative permeability of the subsoil and of the depth of the subsoil above an impermeable base varying from zero to infinity, in relation to different ratios of slope length to soil profile depth. Solutions and analyses are also extended to a case where flow through the topsoil is saturated and steady, but where the subsoil is still accepting a constant downward flux of water uniformly over all its area. The solutions are for a rectangularly bounded, tilted flow medium; but for long slopes the details of the flow medium boundaries at the top and bottom ends should have negligible effects on overall flow. The results show that the interflow through each horizon of a layered soil profile can be approximated by a one‐dimensional Darcian flow parallel to the slope, if the ratio of slope length to soil profile depth is greater than about 6 to 10. This extends the finding reported earlier in the literature for a uniform soil. For an extreme case of a sloping soil with a semi‐infinitely deep subsoil on an impermeable base, the interflow through the topsoil, if it is highly conductive compared to the subsoil, can still be treated as a one‐dimensional flow. For such cases, the movement of soil chemicals with interflow can be described as a one‐dimensional miscible displacement process. A uniformly constant downward flux of water into the subsoil decreases interflow through the topsoil. The extent of the decrease depends upon the land slope and the relative conductivity of the subsoil. Of even greater significance for chemical transport is the finding that the cross section of the topsoil through which the interflow traverses and picks up its chemical load is drastically reduced by downward leakage, even when the subsoil conductivity is two orders of magnitude smaller than that of the topsoil. A simplified prediction of this region of interflow is deduced from the results. By representing this wedge‐shaped area by an equivalent rectangle, the chemical movement, for a first‐order estimation in field application, may still be approximated as a one‐dimensional displacement.

Focal spots: III. Field characteristics.

Experimental results for focal spot field characteristics are presented. These results do not agree completely with results published by Moores and Roeck but are in agreement with Doi's theory as applied to separable focal spots. A new image plane origin called the principal ray is proposed. The variation of the star test pattern image for separable, symmetric focal spots is described. The equation for the field variation of the RMS equivalent rectangle is presented.

Exit of Ground Waters on Drainage Boundaries

A known analytical solution provides the location of the exit point of the free surface of ground waters as a function of the slope angle and the reduced discharge in a form of a finite integral. This integral was evaluated numerically and a substitute polynomial equation provided to determine the function. Based on the exact position of the exit point, a new discharge formula free from any preliminary assumptions is derived. It covers the range of slope angles from near 0° to 180° for infinitely long aquifers. It was found that the same formula is applicable to certain finite lengths. Applicability of the function for the location of the exit point and the discharge formula for earth dams is considered. It is explained that a hydraulically equivalent rectangle, which substitutes the upstream prism, must have the width of the base substantially smaller than that used in older theories.

An optical information procedure for characterizing the shape of fine particle images

In a search for a method for characterizing the shape of particle images using optical information processing procedures, the technique of Redman and Reid for optical character recognition has been modified to explore the structure of several particle images. In the Redman and Reid method the Fourier transform of the image is interrogated by a rotating disc from which a sector has been removed. The waveforms generated by the interrogation of the diffraction pattern is subjected to Fourier analysis. The spectrum of the waves is characteristic of the shape of the character to be studied. Using this system we have explored the diffraction pattern of rectangles and ellipses of different shape. We then measured the diffraction patterns for several irregular particle profiles and compared them to the frequency spectrum generated for the rectangles and ellipses. It is demonstrated that using this procedure it is possible to obtain a shape characteristic spectrum by direct optical processing procedures which enable a profile to be described in terms of an equivalent ellipse or rectangle.

Resolution Limits with Propagation Phase Errors

Resolution limits and corresponding optimum linear apertures are determined in the presence of phase errors. Let α(t) be the phase aberration at position t across the aperture; it is assumed that the random process α has a power law structure function, E{[(α(t)-α(τ)]2}= c|t-τ|n. Beam tilting caused by the phase error is "removed" (for each sample of α), then resolution formulas are developed. An approximate analysis is obtained in closed form and yields an optimum resolution proportional to c1/n for O < n < 2. The exact analysis is given for Gaussian α, and again the optimum resolution is proportional to c1/n. In applications n= 5/3 is of interest, and in the Gaussian case the best obtainable equivalent rectangle resolution is ρ λ)/2π (0.975)c3/5 radians with a corresponding optimum linear aperture of 14c-3/5. When long exposures are considered, imaging without removing beam tilting is of interest, and resolution is degraded by a factor of about 2.5 for a linear aperture. Alternatively, in some applications optimum focus as well as beam tilt should be considered, in this case resolution is improved by a factor of about 1.4 (again for n= 5/3). Finally, joint (tilt corrected) optimization over aperture length and taper is treated; however, as one might expect, the use of taper offers negligible resolution improvement.

A note on the estimation of the optimum successive overrelaxation parameter for Laplace's equation

The optimum parameter for any given region is estimated by finding an equivalent rectangle intuitively. Experimental evidence is produced in support of the method for two dimensions and for three dimensions with axial symmetry.

On the maximal dilation of quasiconformal mappings

1. Let G and G' be two plane open sets and w(z) a topological mapping of G onto G'. By Q we denote any quadrilateral in G, i.e. the topological image of a closed square with a distinguished pair of opposite sides. The conformal modulus m of Q is the ratio m = a/b of the sides of a conformally equivalent rectangle R, the distinguished sides of Q corresponding to the sides of length b. We call this essentially unique conformal mapping of Q onto R the canonical mapping of Q. The modulus m is equal to the extremal distance of the two distinguished sides of Q with respect to Q. The maximal dilation of the mapping w(z) on G is the number

The Equivalent Rectangle in Prestressed Concrete Design

A simple procedure is presented for selecting cross-sectional shapes and dimensions for prestressed concrete beams in a manner similar to that used for selecting structural steel shapes. The requir...