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Classification of remote earth resources sensing data according to normed exponential density statistics is reported. The use of density models appropriate for several physical situations provides an exact solution for the probabilities of classifications associated with the Bayes discriminant procedure even when the covariance matrices are unequal.
Two types of discrete adaptive procedures are proposed for updating statistical parameters such as the means and covariance matrices. These two types include nonweighted adaptive procedures and weighted adaptive procedures. It is believed that the use of the proposed procedures will enhance the performance accuracy of a pattern recognition system. The procedures discussed include an approach for a weighted mean vector adaptation.
A hierarchical structured control model was used to describe pilot performance in a multivariable control problem. The model is based on the assumption that a well trained operator optimized a quadratic optimization criterion. The model considers some special characteristics of the human operator as time delay, neuromuscular time lag, observation and motor noise and the ability to extract the first derivative of a displaced value. Numerical results of a simple human engineering example using the BBN model demonstrate the influence of the variation of different parameters on covariance matrices of the state and observation vectors.
The degree of dependence among the atmospheric optical thicknesses that are measured in nonselective absorption bands is studied. The observations were made previously in many spectral bands within the range 0.36-2.4 micron from near sea level in two continents where urban and industrial pollutions were weak. The sample covariance matrices and corresponding eigenvalues and eigenvectors are computed. The two highest eigenvalues account for 90% of the total variance in 10 spectral bands within the range 0.4-1.6 micron. The linear regression of the optical thickness on the total precipitable water vapor is computed to determine the attenuation coefficient that is associated with water vapor. This coefficient shows a rapid power-law decrease with wavelength in the visible spectrum and indicates that numerous water particles of radius 0.03-0.06 micron cause the attenuation.
A dispersion analysis considering 3 beta uncertainties in platform, vehicle, and environmental parameters is described. Powered explicit guidance used to develop closed loop steering commands, the nominal profile, and entry interface conditions are discussed. The groundrules and assumptions for the analysis are reviewed. The results presented include dispersion data at specific time slices from liftoff to entry interface, covariance matrices, summary data, and exchange ratios.
The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.
Coding is given for maximum likelihood and Bayesian estimation of the vector p of multinomial cell probabilities from incomplete data. Also included is coding to calculate and approximate elements of the posterior mean and covariance matrices. The program is written in FORTRAN 4 language for the Control Data CYBER 170 series digital computer system with network operating system (NOS) 1.1. The program requires approximately 44000 octal locations of core storage. A typical case requires from 72 seconds to 92 seconds on CYBER 175 depending on the value of the prior parameter.
A hill sliding technique was devised to extract Gaussian clusters from the multivariate probability density estimate of sample data for the first step of iterative unsupervised classification. Each cluster was assumed to posses a unimodal normal distribution. A clustering function proposed distinguished elements of a cluster under formation from the rest in the feature space. Initial clusters were extracted one by one according to the hill sliding tactics. A dimensionless cluster compactness parameter was proposed as a universal measure of cluster goodness and used satisfactorily in test runs with LANDSAT multispectral scanner data. The normalized divergence, defined by the cluster divergence divided by the entropy of the entire sample data, was utilized as a general separability measure between clusters. An overall clustering objective function was set forth in terms of cluster covariance matrices, from which the cluster compactness measure could be deduced. Minimal improvement of initial data partitioning was evaluated by this objective function in eliminating scattered sparse data points. The hill sliding clustering technique developed herein has the potential applicability to decomposition any multivariate mixture distribution into a number of unimodal distributions when an appropriate distribution function to the data set is employed.
A region of mixed pixels can be characterized through the probability density function of proportions of classes in the pixels. Using information from the spectral vectors of a given set of pixels from the mixed pixel region, expressions are developed for obtaining the maximum likelihood estimates of the parameters of probability density functions of proportions. The proportions of classes in the mixed pixels can then be estimated. If the mixed pixels contain objects of two classes, the computation can be reduced by transforming the spectral vectors using a transformation matrix that simultaneously diagonalizes the covariance matrices of the two classes. If the proportions of the classes of a set of mixed pixels from the region are given, then expressions are developed for obtaining the estmates of the parameters of the probability density function of the proportions of mixed pixels. Development of these expressions is based on the criterion of the minimum sum of squares of errors. Experimental results from the processing of remotely sensed agricultural multispectral imagery data are presented.
A least squares method proposed previously for solving a general class of problems is expanded in two ways. First, covariance matrices related to the solution are calculated and their interpretation is given. Second, improved methods of solving the normal equations related to those of Marquardt (1963) and Fletcher and Powell (1963) are developed for this approach. These methods may converge in cases where Newton's method diverges or converges slowly.
Analysis of the tapes of the Detroit, Michigan scene, which were received in fully processed format with geometric and radiometric correction, shows evidence of an along line data slip every sixteenth line in TM channel 2. Very large scale products were therefore generated in false color using channels 1, 3, and 4. Subjective evaluation of these enhanced scenes indicates that they are acceptable for interpretation at scales up to 1:50,000 and should be useful for change mapping probably up to 1:24,000 scale. The significant striping visible in water bodies for both the natural color and false color products indicates that the detector calibration is probably performing below the preflight specification. Variance-covariance matrices were computed and principal component analysis were performed for a set of 512 x 512 windows within the Arkansas scene. Initial analysis shows the shortwave infrared channels (TM 5 and 6) are a highly significant data source. The thermal channel (TM 7) shows negative correlation with TM 1 through 4.
Linear discriminant analysis for a two-class case is studied in the presence of misallocation in training samples. A general appraoch to modeling of mislocation is formulated, and the mean vectors and covariance matrices of the mixture distributions are derived. The asymptotic distribution of the discriminant boundary is obtained and the asymptotic first two moments of the two types of error rate given. Certain numerical results for the error rates are presented by considering the random and two non-random misallocation models. It is shown that when the allocation procedure for training samples is objectively formulated, the effect of misallocation on the error rates of the Bayes linear discriminant rule can almost be eliminated. If, however, this is not possible, the use of Fisher rule may be preferred over the Bayes rule.
Two fully processed test tapes were enhanced and evaluated at scales up to 1:10,000, using both hardcopy output and interactive screen display. A large scale, the Detroit, Michigan scene shows evidence of an along line data slip every sixteenth line in TM channel 2. Very large scale products generated in false color using channels 1,3, and 4 should be very acceptable for interpretation at scales up to 1:50,000 and useful for change mapping probably up to scale 1:24,000. Striping visible in water bodies for both natural and color products indicates that the detector calibration is probably performing below preflight specification. For a set of 512 x 512 windows within the NE Arkansas scene, the variance-covariance matrices were computed and principal component analyses performed. Initial analysis suggests that the shortwave infrared TM 5 and 6 channels are a highly significant data source. The thermal channel (TM 7) shows negative correlation with TM 1 and 4.
AESOP is a computer program for use in designing feedback controls and state estimators for linear multivariable systems. AESOP is meant to be used in an interactive manner. Each design task that the program performs is assigned a "function" number. The user accesses these functions either (1) by inputting a list of desired function numbers or (2) by inputting a single function number. In the latter case the choice of the function will in general depend on the results obtained by the previously executed function. The most important of the AESOP functions are those that design,linear quadratic regulators and Kalman filters. The user interacts with the program when using these design functions by inputting design weighting parameters and by viewing graphic displays of designed system responses. Supporting functions are provided that obtain system transient and frequency responses, transfer functions, and covariance matrices. The program can also compute open-loop system information such as stability (eigenvalues), eigenvectors, controllability, and observability. The program is written in ANSI-66 FORTRAN for use on an IBM 3033 using TSS 370. Descriptions of all subroutines and results of two test cases are included in the appendixes.
A detailed spectral analysis is conducted of thematic mapper and MSS data for an area near Des Moines, Iowa. Data are utilized from 7 blocks distributed throughout the area which included agricultural, forest, suburan, urban, and water scene types. The blocks are processed using a clustering algorithm to produce up to 18 cluster groupings for each block. Each cluster class is then identified with a ground-cover class using aerial photography and maps of the area. The clusters from each of the 7 blocks are inspected with regard to separability, mean, and variances. The separability measure used in the transformed divergence function or processor measures the statistical distance between classes based on class means and covariance matrices. The measure has a maximum value of 2,000 and the minimum of 0. Spectrally, very close classes will typically have values as low as 50 to 500.
Comparisons were made between subscenes from the first TM scene acquired of the Washington, D.C. area and a MSS scene acquired approximately one year earlier. Three types of analyses were conducted to compare TM and MSS data: a water body analysis, a principal components analysis and a spectral clustering analysis. The water body analysis compared the capability of the TM to the MSS for detecting small uniform targets. Of the 59 ponds located on aerial photographs 34 (58%) were detected by the TM with six commission errors (15%) and 13 (22%) were detected by the MSS with three commission errors (19%). The smallest water body detected by the TM was 16 meters; the smallest detected by the MSS was 40 meters. For the principal components analysis, means and covariance matrices were calculated for each subscene, and principal components images generated and characterized. In the spectral clustering comparison each scene was independently clustered and the clusters were assigned to informational classes. The preliminary comparison indicated that TM data provides enhancements over MSS in terms of (1) small target detection and (2) data dimensionality (even with 4-band data). The extra dimension, partially resultant from TM band 1, appears useful for built-up/non-built-up area separation.
The measurement of the earth's radiation budget has been chosen to illustrate the technique of objective system design. The measurement process is an approximately linear transformation of the original field of radiant exitances, so that linear statistical techniques may be employed. The combination of variability, measurement strategy, and error propagation is presently made with the help of information theory, as suggested by Kondratyev et al. (1975) and Peckham (1974). Covariance matrices furnish the quantitative statement of field variability.
A systematic approach for the identification of terrain media such as vegetation canopy, forest, and snow covered fields is developed using the optimum polarimetric classifier. The covariance matrices for the various terrain cover are computed from theoretical models of random medium by evaluating the full polarimetric scattering matrix elements. The optimal classification scheme makes use of a quadratic distance measure and is applied to classify a vegetation canopy consisting of both trees and grass. Experimentally measured data are used to validate the classification scheme. Theoretical probability of classification error using the full polarimetric matrix are compared with classification based on single features including the phase difference between the VV and HH polarization returns. It is shown that the full polarimetric results are optimal and provide better classification performance than single feature measurements.