Loeve Expansion Research Articles

Feature extraction based on eigenvector analysis applied to the monostatic scattered field of a ribbed finite cylinder

Feature extraction based on eigenvector analysis is applied to the monostatic response of a ribbed finite cylindrical shell with hemispherical end caps. The method, which is based on the Karhunen–Loeve expansion, is applied in the frequency domain to extract features for optimal representation of the data. An orthonormal set of eigenvectors that form a set of basis functions are computed by diagonalizing the correlation matrix. The expansion of the monostatic scattered field with the resultant set of basis functions is optimal because a small number of the functions is required to approximate the scattered field at each orientation of the scatterer. Such a representation reduces the dimensionality of the problem by more than an order of magnitude. The method is applied to two frequency ranges. In the first case, enhancements in target strength are present due to the phase matching of the elastic waves to the acoustic waves in the exterior fluid. In the second case, Bloch wave resonances are present due to the periodicity of the ribs. It is shown that as larger variations are present in target strength as a function of frequency, a larger number of eigenvectors are necessary to approximate the scattering response.

A time domain binaural model based on spatial feature extraction for the head-related transfer function.

A complex-valued head-related transfer function (HRTF) can be represented as a real-valued head-related impulse response (HRIR). The interaural time and level cues of HRIRs are extracted to derive the binaural model and also to normalize each measured HRIR. Using the Karhunen-Loeve expansion, normalized HRIRs are modeled as a weighted combination of a set of basis functions in a low-dimensional subspace. The basis functions and the space samples of the weights are obtained from the measured HRIR. A simple linear interpolation algorithm is employed to obtain the modeled binaural HRIRs. The modeled HRIRs are nearly identical to the measured HRIRs from an anesthetized live cat. Typical mean-square errors and cross-correlation coefficients between the 1816 measured and modeled HRIRs are 1% and 0.99, respectively. The real-valued operations and linear interpolating in the model are very effective for speeding up the model computation in real-time implementation. This approach has made it possible to simulate real free-field signals at the two eardrums of a cat via earphones and to study the neuronal responses to such a virtual acoustic space (VAR).

Unsteady Two-Dimensional Flows in Complex Geometries: Comparative Bifurcation Studies with Global Eigenfunction Expansions

We present a bifurcation study of the incompressible Navier--Stokes equations in a model complex geometry: a spatially periodic array of cylinders in a channel. The dynamics of the flow include a Hopf bifurcation from steady to oscillatory flow at an approximate Reynolds number R of 350 and the appearance of a second frequency at approximately $R\simeq 890$. The multiple frequency dynamics include a substantial increase in spatial and temporal scales with Reynolds number as compared with the simple limit cycle oscillation present close to R = 350. Numerical bifurcation studies of the dynamics are performed using three forms of global eigenfunction expansions. The first basis set is derived through principal factor analysis (Karhunen--Loeve expansion) of snapshots from accurate direct spectral element numerical solutions of the Navier--Stokes equations. The second set is obtained from the eigenfunctions of the Stokes operator for this geometry. Finally eigenfunctions are derived from a singular Stokes operator, i.e., the Stokes operator modified to include a variable coefficient which vanishes at the domain boundaries. Truncated systems of ($\sim 100$) ODEs are obtained through projection of the Navier--Stokes equations onto the basis sets, and a comparative study of the resulting dynamical models is performed.

The 4 YearCOBENormalization and Large‐Scale Structure

We present an analysis of the 4 year data from the COBE DMR experiment. We use a Karhunen- Loeve expansion of the pixel data to calculate the normalization and goodness of -t of a range of models of structure formation. This technique produces unbiased normalization estimates and is capable of determining the normalization of any particular model with a statistical uncertainty of 7%. We present a parameterization of the normalization and likelihood function which summarizes our results for a wide range of models. We use the COBE normalization to compute small-scale Nuctuation ampli- tudes for a variety of models and discuss the implications of these results for theories of large-scale structure. Subject headings: cosmic microwave background E cosmology: theory E large-scale structure of universe E methods: statistical

A reduced dynamical model of convective flows in tall laterally heated cavities

Proper orthogonal decomposition (the Karhunen–Loeve expansion) is applied to convective flows in a tall differentially heated cavity. Empirical spatial eigenfunctions are computed from a multicellular solution at supercritical conditions beyond the first Hopf bifurcation. No assumption of periodicity is made, and the computed velocity and temperature eigenfunctions are found to be centro‐symmetric. A low-dimensional model for the dynamical behaviour is then constructed using Galerkin projection. The reduced model successfully predicts the first Hopf bifurcation of the multicellular flow. Results determined from the low-order model are found to be in qualitative agreement with known properties of the full system even at conditions far from criticality.

A novel kinetic model discrimination method: eigenfunction distribution via forced concentration oscillation

Forced feed concentration oscillation, a special transient method, is developed for rival kinetic model discrimination. Time-dependent concentration samples are taken from four axial positions of an isothermal tubular reactor packed with catalyst pellets. Karhunen—Loeve expansion is used to find eigenfunctions of the reacting components for different kinetic models, on the basis of a biorthogonal function expansion of spatial concentration correlation. Oxidation of benzene to maleic anhydride over V 2O 5 is employed as a working system for which six possible mechanisms are postulated. The inlet benzene concentration is imparted a square-wave oscillation to study the response at various axial positions. Model discrimination has shown that two out of the six possible models merit further discrimination. Measured concentration snapshots of a time series are applied to determine the test eigenfunctions, which are then compared with those of the rival models. Agreement between calculated and experimentally determined eigenfunctions testifies to the validity of the present method of model discrimination.

Micromagnetic modeling and experimental study of transition noise correlation in thin-film media

Micromagnetic modeling and time-domain experimental measurements are combined to study spatial noise correlations between two closely recorded transitions in oriented longitudinal thin-film media. The Karhunen–Loeve expansion method is utilized to characterize the noise correlations. The analysis shows that the noise correlation of the dipulses exhibits three principle modes: amplitude variation, shift in unison, and breathing. The supralinear increase of noise at small bit intervals in well-oriented longitudinal film media is dominated by the amplitude variation, which arises from large variation of the degree of percolation between adjacent transition boundaries. The results from the micromagnetic modeling shows excellent agreement with the results obtained from the experimental time-domain noise measurements.

Recognition of rotated images by invariant Karhunen–Loeve expansion

Given an ensemble of images, we consider a method for the extraction of optimal features that are invariant under two-dimensional rotations. Our method consists of first identifying some characteristics of the optimal basis functions and then applying the Karhunen–Loeve algorithm to determine them completely. We perform a few classification experiments with the invariant features, using a neural network as a classifier. The new method is compared with the Zernike moments method.

Auditory space modeling and simulation via orthogonal expansion and generalized spline model

A two-stage model that establishes a mathematical representation of auditory space is developed. The first stage of the model consists of a low-dimensional subspace representation for the free-field-to-eardrum transfer functions (FETF’s). The bases of this subspace are complex-valued eigentransfer functions (EF’s) obtained from the Karhunen–Loeve expansion of the measured FETF’s covariance matrix. Each FETF is represented as a weighted sum of the EF’s. The second stage of the model is a functional representation for the weights, termed spatial transformation characteristic functions (STCF’s), applied to the EF’s. The STCF’s are functions of azimuth and elevation. A generalized spline model is fitted to each STCF derived from measurements. The spline model filters out noise and permits interpolation of the STCF between measured directions. The FETF’s for an arbitrary direction is synthesized by weighting the EF’s with the smoothed and interpolated STCF’s. Using FETF’s sampled uniformly over the upper 3/4 sphere for one KEMAR ear, it is shown that 99.9% of the energy in the measured FETF’s is contained in a 16-dimensional subspace. The relative average mean-square error between 2320 measured and simulated FETF’s is found to be less than 0.25%. [Raw data provided by Dept. of Neurophysiology, University of Wisconsin−Madison.]

Convergence of Galerkin solutions using Karhunen–Loève expansions of inhomogeneous 1-D turbulence

The rate of convergence of the Karhunen–Loève expansion of an inhomogeneous, instantaneous random field is compared with that of Fourier expansion in relation to the Reynolds number. The model turbulence is generated by solving the Burgers’ equation with random forcing. The coefficients of the Fourier expansion are determined by a Galerkin solution scheme. The results show obvious superiority of the Karhunen–Loève expansion, especially for high Reynolds number flows.

Shear flow coherent structures via Karhunen–Loève expansion

The method of snapshots for Karhunen–Loève expansion has been applied to two-dimensional, two-component hot-wire data from the first pairing region of a weakly perturbed free shear layer. Almost 90% of the fluctuation energy is captured by the first four modes. The first and the second modes contain almost equal amounts of energy, as do the third and the fourth, and the time dependence for each pair is very nearly sinusoidal with the odd modes leading in phase by π/2.

Increased potential for tissue discrimination in quantitative magnetic resonance imaging

A method has been developed that substantially increases quantitative tissue discrimination of pathology in in-vivo magnetic resonance (MR) imaging. Results from a study of carcinoma and benign hypertrophy of the prostate show the potential for classification of tissues from individual patients and demonstrate increased class separation. This appears to be the firstapplication of pattern recognition techniques to relaxation-time data from in-vivo MR images. Multiple-point relaxation-time maps were classified using both statistical measures and texture descriptors derived from spatial grey level dependency matrices. These parameters were combined in a new pattern-recognition method based upon the Karhunen- Loeve expansion.

Orthogonal decomposition and transmission of nonstationary random processes

A relatively simple and straightforward procedure is given for representing analytically defined or data-based covariance kernels of arbitrary random processes in a compact form that allows its convenient use in later analytical random vibration response studies. The method is based on the spectral decomposition of the random process by the orthogonal Karhuen Loeve expansion and the subsequent use of least-squares approaches to develop an approximating analytical fit for the eigenvectors of the underlying random process. The resulting compact analytical representation of the random process is then used to derive a closed-form solution for the nonstationary response of a damped SDOF harmonic oscillator. The utility of the method for representing the excitation and calculating the mean square response is illustrated by the use of an analytically-defined covariance kernel widely employed in random vibration studies. It is shown that the method offers the potential of being a useful tool for feature extraction of experimentally measured covariance kernels of nonstationary random processes.

Use of dominant eigenvalues in evaluating the probability density function of the intensity for a polychromatic speckle pattern

A method has been described to determine a priori the optimum number of samples required to estimate the eigenvalues in the Karhunen–Loeve expansion of a polychromatic speckle pattern without solving the relevant integral equation. It is found that choosing samples less than this number leads to an incorrect estimate of the eigenvalues, while choosing a larger number of samples does not change the eigenvalues significantly. Thus the method allows the use of the Karhunen–Loeve expansion with minimum effort and leads to an accurate estimate of the probability density function of the intensity. Owing to the general nature of this method, it is also applicable to a large class of problems where the Karhunen–Loeve expansion is used.

A STUDY OF TIME‐SHIFTED SIGNALS AND THEIR EXPANSIONS

A simple extension of the concept of the inner product leads to orthonormal expansions of time‐shifted signals with coefficients dependent on the shift variable. It is shown that such expansions have their counterparts of Parseval's identity and Bessel inequality. The Projection Theorem holds, and a version of Mercer's theorem and Karhumen—Loeve's expansion are also shown to hold, in a non‐stochastic regime. The approach leads to new interpretations of time correlation functions and Fourier Series expansions.