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General classes of nonlinear and linear transformations were investigated for the reduction of the dimensionality of the classification (feature) space so that, for a prescribed dimension m of this space, the increase of the misclassification risk is minimized.
The computational procedure and numerical results are presented for a new method to solve Kuessner's integral equation in the case of subsonic compressible flow about harmonically oscillating planar surfaces with controls. Kuessner's equation is a linear transformation from pressure to normalwash. The unknown pressure is expanded in terms of prescribed basis functions and the unknown basis function coefficients are determined in the usual manner by satisfying the given normalwash distribution either collocationally or in the complex least squares sense. The present method of solution differs from previous ones in that the basis functions are defined in a continuous fashion over a relatively small portion of the aerodynamic surface and are zero elsewhere. This method, termed the local basis function method, combines the smoothness and accuracy of distribution methods with the simplicity and versatility of panel methods. Predictions by the local basis function method for unsteady flow are shown to be in excellent agreement with other methods. Also, potential improvements to the present method and extensions to more general classes of solutions are discussed.
The authors review some of the properties of the pseudoinverse and oblique pseudoinverse of a linear transformation T from one finite-dimensional inner-product space into another, and then use these properties and a theorem of Milne (1968), which states that the oblique pseudoinverse can be expressed in terms of a weak generalized inverse and two projection operators, in order to compute a mean-axis influence coefficient matrix for the dynamic analysis of an elastic body. Some eigenvalue invariance properties of the mean-axis structural dynamics equations are demonstrated, and on the simple example of a uniform beam, it is shown that the finite frequencies and mode shapes of the mean axis structural system are identical to the nonzero frequencies and mode shapes of the free structure.
A mathematical analysis technique called characteristic vector analysis, reported by Simonds (1963), is used to determine spectral dye densities in multiemulsion film such as color or color-IR imagery. The technique involves examining a number of sets of multivariate data and determining linear transformations of these data to a smaller number of parameters which contain essentially all of the information contained in the original set of data. The steps involved in the actual procedure are outlined. It is shown that integral spectral density measurements of a large number of different color samples can be accurately reconstructed from the calculated spectral dye densities.
The author has identified the following significant results. Both LANDSAT imagery and digital data were studied for usefulness in surveying water conditions of Minnesota lakes. Initial consideration was given to analysis of LANDSAT image densities because of the low technologic and cost requirements. The techniques employed, however, yield inconsistent and unreliable results. A set of criteria are given for using LANDSAT data in identification of three categories of particulate contaminants in Lake Superior. A linear transformation giving the relationship between the residual LANDSAT intensities and concentrations of three contaminants was obtained from correlation of remote sensing data with insitu measurements. LANDSAT imagery was found useful in placing peat bogs and fens in their respective geologic settings. Artificial disturbances and drainageways in peatlands could be recognized and classified.
The effects of nonlinear diffusive heat transport in North's (1975) model is reexamined. By inserting the nonlinearity into the original energy balance equation rather than the spectrally transformed linear algebraic equations, and including one more mode in the spectral expansion, it is found that nonlinear diffusion does give a significantly different sensitivity for states near the current climate. For linear diffusion, an increase of 0.1% of the solar constant from the present value is sufficient to melt all the ice on the earth, whereas an increase of about five times as large is required for nonlinear diffusion. This difference in sensitivity shows the importance of dynamical modeling in simple climate models.
A simple physiological model of mortality kinetics is used to assess the intuitive concept that the aging rates of populations are proportional to their mortality rates. It is assumed that the vitality of an individual can be expressed as a simple summation of the weighted functional capacities of its organs and homeostatic systems that are indispensable for survival. It is shown that the mortality kinetics of a population can be derived by a linear transformation of the frequency distribution of vitality, assuming a uniform constant rate of decline of the physiological functions. A simple comparison of two populations is not possible when they have different vitality frequency distributions. Analysis of the data using the model suggests that the differences in decline of survivorship with age between the military pilot population, a medically insured population, and the control population can be accounted for by the effect of physical selection on the vitality frequency distribution of the screened populations.
A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach.
To facilitate geologic interpretation of satellite elevation potential field data, analysis techniques are developed and verified in the spherical domain that are commensurate with conventional flat earth methods of potential field interpretation. A powerful approach to the spherical earth problem relates potential field anomalies to a distribution of equivalent point sources by least squares matrix inversion. Linear transformations of the equivalent source field lead to corresponding geoidal anomalies, pseudo-anomalies, vector anomaly components, spatial derivatives, continuations, and differential magnetic pole reductions. A number of examples using 1 deg-averaged surface free-air gravity anomalies of POGO satellite magnetometer data for the United States, Mexico, and Central America illustrate the capabilities of the method.
If a machine is to perform a given subtask autonomously, it will require an internal model which, combined with operator and environmental inputs, can be used to generate the manipulator functions necessary to complete the task. This paper will advance a technique based on linear transformations by which short, supervised periods of manipulation can be accomplished. To achieve this end a distinction will be made between tasks which can be completely defined during the training period, and tasks which can be only partially defined prior to the moment of execution. A further distinction will be made between tasks which have a fixed relationship to the manipulator base throughout the execution period, and tasks which have a continuously changing task/base relationship during execution. Finally, through a rudimentary analysis of the methods developed in this paper, some of the practical aspects of implementing a supervisory system will be illustrated
Effective micro-structure properties of two-dimensional large repetitive frame-like structures are derived using straightforward combinations of energy methods of linear transformations. Once the actual structure is identified symmetry considerations are used in order to identify its independent property constants. The actual values of these constants are constructed according to a building block format which is carried out in the three consecutive steps: (1) all basic planar lattices are identified, (2) effective continuum properties are derived for each of these planar basic grids using energy methods, and (3) orthogonal transformations are finally used to determine the contribution of each basic set to the overall effective continuum properties of the structure.
Results are reported from an experiment designed to control the vibratory motion of a beam at NASA, Langley Research Center. The experimental setup consists of a free-free uniform beam acted upon by four electromagnetic force actuators, with the motion being measured by nine displacement sensors. The entire assembly is linked to a CDC Cyber 175 computer which permits on-line real-time computation of the control forces. The control scheme is based on Independent Modal-Space Control (IMSC), whereby the modal forces are computed using a nonlinear, on-off control law. The actuator forces are then synthesized using a linear transformation, resulting in quantized forces. The sensors data is processed by modal filters. It is observed that the controls designed on the basis of the IMSC method are very effective in suppressing the vibratory motion of the beam, even though there is about a 50 percent differene between the actual and the analytically computed frequencies.
The radiometric data quality of the LANDSAT 4 multispectral scanner (MSS) was examined using several LANDSAT 4 frames. It was found that LANDSAT 4 MSS produces high-quality data of the caliber experienced with previous LANDSATS. For example, the detector equalization procedure worked well, leaving a residual banding effect of about 0.3 digital counts RMS, close to the theoretical minimum value of quantization error. Nevertheless, artifacts of the data were found, two of which were not experienced in previous MSS data. A low-level coherent noise effect was observed in all bands, with a magnitude of about 0.5 digital counts and a frequency of approximately 28 KHz (representing a wavelength of about 3.6 pixels); a substantial increase in processing complexity would be required to reduce this artifact in the data. Also, a substantial scan-length variation (of up to six pixels) was noted in MSS data when the TM sensor was operating; the LANDSAT 4 correction algorithms being applied routinely by the EROS Data Center to produce a p-type data should remove most of this variation. Between-satellite calibrations were examined in paired LANDSAT 3 and LANDSAT 4 MSS data sets, which were closely matched in acquisition time and place. Radiometric comparisons showed that all bands were highly linear in digital counts, and a well-determined linear transformation between the MSS's was established.
Automatic pre-processing technique called Principal Components (PRINCO) in analyzing LANDSAT digitized data, for land use and vegetation cover, on the Brazilian cerrados was evaluated. The chosen pilot area, 223/67 of MSS/LANDSAT 3, was classified on a GE Image-100 System, through a maximum-likehood algorithm (MAXVER). The same procedure was applied to the PRINCO treated image. PRINCO consists of a linear transformation performed on the original bands, in order to eliminate the information redundancy of the LANDSAT channels. After PRINCO only two channels were used thus reducing computer effort. The original channels and the PRINCO channels grey levels for the five identified classes (grassland, "cerrado", burned areas, anthropic areas, and gallery forest) were obtained through the MAXVER algorithm. This algorithm also presented the average performance for both cases. In order to evaluate the results, the Jeffreys-Matusita distance (JM-distance) between classes was computed. The classification matrix, obtained through MAXVER, after a PRINCO pre-processing, showed approximately the same average performance in the classes separability.
A comparison was made of two photogrammetric measurement techniques which can be used for the determination of the spatial loctions of targets. One technique is known as the Direct Linear Transformation (DLT), and the other is referred to as the Bundle method. Each technique utilizes triangulation of image data from two or more cameras, but they differ in their method of determining camera parameters. While the study was performed with simulated data, the geometries, image data accuracies and other parameters were set to simulate conditions for a two-camera, photogrammetric system which will be used at the National Transonic Facility at the NASA's Langley Research Center. For the conditions studied the Bundle technique was found to have smaller errors. Both methods were sensitive to the spatial distribution of control-point targets, but this effected the accuracy of the Bundle algorithm less than that of the DLT algorithm.
A novel calibration approach was used for Landsat 4 TM simulator data acquired at the Patrick Draw test site, Wyoming, where the 'true' reflectance spectrum for any pixel area cannot be practically determined because of mixing of spectral components. An empirically derived function was first applied to the scanner data to remove the brightness gradient due to directional scattering and surface geometry. Based on the observation that the distribution of scanner brightness values (DN) were Gaussian, the minimum, mean, and maximum DN for each band were extracted and plotted versus the similar percent reflectance values determined from a large representative collection of field reflectance data for the area. These plots provided linear transformations relating DN value to reflectance in each band.
A Landsat-4 Thematic Mapper (TM) image of the Wind River Basin area in Wyoming is currently under analysis for stratigraphic and structural mapping and for assessment of spectral and spatial characteristics using visible, near infrared, and short wavelength infrared bands. To estimate the equivalent Lambertian surface reflectance, TM radiance data were calibrated to remove atmospheric and instrumental effects. Reflectance measurements for homogeneous natural and cultural targets were acquired about one year after data acquisition. Calibration data obtained during the analysis were used to calculate new gains and offsets to improve scanner response for earth science applications. It is shown that the principal component images calculated from the TM data were the result of linear transformations of ground reflectance. In images prepared from this transform, the separation of spectral classes was independent of systematic atmospheric and instrumental factors. Several examples of the processed images are provided.
In connection with the current interest in agile spacecraft maneuvers, it has become necessary to consider the nonlinear coupling effects of multiaxial rotation in the treatment of command generation and tracking problems. Multiaxial maneuvers will be required in military missions involving a fast acquisition of moving targets in space. In addition, such maneuvers are also needed for the efficient operation of robot manipulators. Attention is given to details regarding the direct nonlinear command generation and tracking, an approach which has been successfully applied to the design of control systems for V/STOL aircraft, linearizing transformations for spacecraft controlled with external thrusters, the case of flexible spacecraft dynamics, examples from robot dynamics, and problems of implementation and testing.