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Pajevicsinisa - One of the best experts on this subject based on the ideXlab platform.
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Spectral Decomposition of a 4th-order covariance tensor
Signal Processing, 2007Co-Authors: J Basserpeter, PajevicsinisaAbstract:We propose a novel Spectral Decomposition of a 4th-order covariance tensor, @S. Just as the variability of vector (i.e., a 1st-order tensor)-valued random variable is characterized by a covariance ...
Sinisa Pajevic - One of the best experts on this subject based on the ideXlab platform.
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Spectral Decomposition of a 4th order covariance tensor applications to diffusion tensor mri
Signal Processing, 2007Co-Authors: Peter J Basser, Sinisa PajevicAbstract:We propose a novel Spectral Decomposition of a 4th-order covariance tensor, @S. Just as the variability of vector (i.e., a 1st-order tensor)-valued random variable is characterized by a covariance matrix (i.e., a 2nd-order tensor), S, the variability of a 2nd-order tensor-valued random variable, D, is characterized by a 4th-order covariance tensor, @S. Accordingly, just as the Spectral Decomposition of S is a linear combination of its eigenvalues and the outer product of its corresponding (1st-order tensors) eigenvectors, the Spectral Decomposition of @S is a linear combination of its eigenvalues and the outer product of its corresponding 2nd-order eigentensors. Analogously, these eigenvalues and 2nd-order eigentensors can be used as features with which to represent and visualize variability in tensor-valued data. Here we suggest a framework to visualize the angular structure of @S, and then use it to assess and characterize the variability of synthetic diffusion tensor magnetic resonance imaging (DTI) data. The Spectral Decomposition suggests a hierarchy of symmetries with which to classify the statistical anisotropy inherent in tensor data. We also present maximum likelihood estimates of the sample mean and covariance tensors associated with D, and derive formulae for the expected value of the mean and variance of the projection of D along a particular direction (i.e., the apparent diffusion coefficient or ADC). These findings would be difficult, if not impossible, to glean if we treated 2nd-order tensor random variables as vector-valued random variables, which is conventionally done in multi-variate statistical analysis.
Muneo Hori - One of the best experts on this subject based on the ideXlab platform.
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Application of Spectral Decomposition of Green's function to linear inverse problem
Engineering Analysis With Boundary Elements, 2004Co-Authors: Muneo HoriAbstract:An inverse analysis method using the Spectral Decomposition of Green's function is proposed for linear inverse problems of identifying inner sources from data of surface responses. It is assumed that Green's function of the corresponding physical problem is given. Applying the Spectral Decomposition, Green's function is discretized as a sum of eigen-values and eigen-functions. From the comparison of the measurement accuracy with the eigen-values, it is shown that responses which can be actually measured are given as a linear combination of eigen-functions corresponding to larger eigen-values. Such responses are found by determining coefficients of the eigen-functions from the measured data, and then sources which are predictable are determined just by calculating their coefficients for the eigen-functions. Without any ambiguity, the proposed method can determine the predictable inner sources from the data which are measured with the limited accuracy. A numerical simulation of solving a simple example problem is carried out to demonstrate the usefulness of the proposed inverse analysis method, and the results are discussed.
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Inversion Method Using Spectral Decomposition of Green’s Function
IUTAM Symposium on Field Analyses for Determination of Material Parameters — Experimental and Numerical Aspects, 2003Co-Authors: Muneo HoriAbstract:An inverse analysis method using the Spectral Decomposition of Green’s function is proposed. For linear inverse problems of identifying inner sources from surface responses, Green’s function, which is a solution of the corresponding physical problem, relates the sources to the responses and hence behaves as a compact operator. A suitable inverse operator that maps the responses to the sources can be determined by using the Spectral Decomposition of Green’s function. The proposed method numerically computes the Spectral Decomposition and determines the inverse operator. It is possible to regard a simple inverse analysis method, which uses a point-wise discretization of Green’s function and computes a generalized inverse matrix using the singular-value Decomposition, as an approximation to compute the inverse operator. The accuracy, however, could be much lower than the proposed method, because of the point-wise discretization and the less accurate computation of the Spectral Decomposition. Illustrative examples are solved to demonstrate the usefulness of the proposed inverse analysis method, as well as errors inherent to inverse analysis methods which do not compute the Spectral Decomposition of Green’s function properly.
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Inverse analysis method using Spectral Decomposition of Green's function
Geophysical Journal International, 2001Co-Authors: Muneo HoriAbstract:SUMMARY An inverse analysis method using the Spectral Decomposition of Green's function is proposed. For linear inverse problems of identifying inner sources from surface responses, Green's function, relates the sources to the responses. Since this function behaves as a compact operator, it admits the Spectral Decomposition and is expressed as a sum of distinct eigen values and eigen functions. A suitable inverse operator that maps the responses to the sources is then determined. The proposed method consists of the following three procedures: (1) numerically computing the Spectral Decomposition and determining the inverse operator; (2) estimating a response function using a set of measured data; and (3) predicting a source function from the response function using inverse operator. A simple inverse analysis method, which uses a pointwise discretization of Green's function and computes a generalized inverse matrix applying the singular-value Decomposition, is regarded as an approximation to compute the inverse operator. The accuracy, however, is much lower than the proposed method, because of the pointwise discretization and the less accurate computation of the Spectral Decomposition. Illustrative examples are solved to demonstrate the usefulness of the proposed inverse analysis method. Errors due to improper calculation of the inverse operator of Green’s function are shown.
John P. Castagna - One of the best experts on this subject based on the ideXlab platform.
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comparison of Spectral Decomposition methods
First Break, 2006Co-Authors: John P. CastagnaAbstract:John P. Castagna, University of Houston, and Shengjie Sun, Fusion Geophysical discuss a number of different methods for Spectral Decomposition before suggesting some improvements possible with their own variation of ‘matching pursuit’ Decomposition. In seismic exploration, Spectral Decomposition refers to any method that produces a continuous time-frequency analysis of a seismic trace. Thus a frequency spectrum is output for each time sample of the seismic trace. Spectral Decomposition has been used for a variety of applications including layer thickness determination (Partyka et al, 1999), stratigraphic visualization (Marfurt and Kirlin, 2001), and direct hydrocarbon detection (Castagna et al., 2003; Sinha et al., 2005). Spectral Decomposition is a non-unique process, thus a single seismic trace can produce various time-frequency analyses. There are a variety of Spectral Decomposition methods. These include the DFT (discrete Fourier Transform), MEM (maximum entropy method), CWT (continuous wavelet transform), and MPD (matching pursuit Decomposition). None of these methods are, strictly speaking, ‘right’ or ‘wrong’. Each method has its own advantages and disadvantages, and different applications require different methods. The DFT and MEM involve explicit use of windows, and the nature of the windowing has a profound effect on the temporal and Spectral resolution of the output. In general, the DFT is preferred for evaluating the Spectral characteristics of long windows containing many reflection events, with the spectra generally dominated by the spacing between events. The MEM is often difficult to parameterize and may produce unstable results. The CWT is equivalent to temporal narrow-band filtering of the seismic trace and has an advantage over the DFT for broad-band signals in that the window implicit in the wavelet dictionary is frequency dependent. The CWT has a great disadvantage, however, in that the wavelets utilized must be orthogonal. The commonly used Morlet wavelet, for example, has poor vertical resolution due to multiple side lobes. Furthermore, for typical seismic signals, the implicit frequency dependent windowing of the CWT is not particularly important, and experience has shown that a DFT with a Gaussian window of appropriate length produces almost the same result as a CWT with a Morlet wavelet. MPD (Mallat and Zhang, 1993) is a more computationally intensive process than the others, but, as will be shown in this paper, it has superior temporal and Spectral resolution if a compact mother wavelet is utilized. Matching pursuit Decomposition involves cross-correlation of a wavelet dictionary against the seismic trace. The projection of the best correlating wavelet on the seismic trace is then subtracted from that trace. The wavelet dictionary is then cross-correlated against the residual, and again the best correlating wavelet projection is subtracted. The process is repeated iteratively until the energy left in the residual falls below some acceptable threshold. As long as the wavelet dictionary meets simple admissibility conditions, the process will converge. Most importantly, the wavelets need not be orthogonal. The output of the process is a list of wavelets with their respective arrival times and amplitudes for each seismic trace. The inverse transform is accomplished simply by summing the wavelet list and the residual, thus reconstructing the original trace. The wavelet list is readily converted to a timefrequency analysis by superposition of the wavelet frequency spectra. Simple matching pursuit has difficulty in properly determining the precise arrival time of interfering wavelets – usually it will slightly misplace the wavelets which will also result in a slightly incorrect wavelet center frequency. Also, it can be seen that the process is path dependent: a slight change in the seismic trace may result in an entirely different order of subtraction. Thus, it may result in lateral instability of the non-uniqwue time-frequency analyses. Cross-correlation of the wavelet dictionary against the seismic trace is essentially a continuous wavelet transform, so it can be seen that the method involves iteratively performing hundreds, if not thousands, of wavelet transforms for each seismic trace. In this paper, we utilize a variation of matching pursuit called exponential pursuit Decomposition (EPD). The method treats complex interference patterns as containing ‘gravity wells’ at the correct wavelet locations, and the selected wavelet location is iteratively attracted to the correct location. The profound advantage of EPD over other methods is that there is no windowing, and corresponding Spectral smearing. The spectra for reflections from isolated interfaces that can be resolved by the method are the same as the seismic wavelet producing those reflections. The method can thus be used with confidence for direct hydrocarbon indication and stratigraphic visualization for thin beds. The classical Heisenberg Uncertainty Principle states that the product of temporal and frequency resolution is constant. One must normally pay the price of decreasing resolution in one domain, to increase resolution in the other. In EPD, there is no windowing and it is the bandwidth of the digital seismic data that limits resolution, not the windowing process. Thus, the Heisenberg Uncertainty Principle does not come into play. As a result, EPD provides better temporal AND Spectral resolution than the other methods. In comparing Spectral Decomposition methods, it is important to keep in mind what the goal of the analysis is.
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Inverse Spectral Decomposition
SEG Technical Program Expanded Abstracts 2004, 2004Co-Authors: Oleg Portniaguine, John P. CastagnaAbstract:This paper introduces a method which Spectrally decomposes a seismic trace by solving an inverse problem. In our technique, the reverse wavelet transform with a library of complex wavelets serves as a forward operator. The inversion reconstructs the wavelet coefficients that represent the seismic trace and satisfy an additional constraint. The constraint is needed as the inverse problem is non-unique. We show synthetic and real examples with three different types of constraints: 1) minimum L2 norm, 2) minimum L1 norm, and 3) sparse spike, or minimum support constraint. The sparse-spike constraint has the best temporal and frequency resolution. While the inverse approach to Spectral Decomposition is slow compared to other techniques, it produces solutions with better time and frequency resolution than popular existing methods.
J Basserpeter - One of the best experts on this subject based on the ideXlab platform.
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Spectral Decomposition of a 4th-order covariance tensor
Signal Processing, 2007Co-Authors: J Basserpeter, PajevicsinisaAbstract:We propose a novel Spectral Decomposition of a 4th-order covariance tensor, @S. Just as the variability of vector (i.e., a 1st-order tensor)-valued random variable is characterized by a covariance ...
