Source Imaging Problem Research Articles

Fast acoustic source imaging using multi-frequency sparse data

We consider the acoustic source imaging problems in using multiple frequency data at sparse directions/points. Using at most n observation directions/points, we show that the size and location of the source can be recovered. A non-iterative method is then proposed to image the source. The method is simple to implement and extremely fast since it only computes an indicator function on the domain of interest using matrix vector multiplications. Numerical examples are presented to validate the effectiveness of the method.

Use of group theory in the selection and description of regularization methods for functional source imaging

Commonly, functional source imaging problems are "partial" rather than "ordinary" inverse problems--wherein the defining operator consists of component operators that individually do not address all variables of the unknown. When this ordinary-to-partial transition is minimally constrained, algebraic principles can be used to derive a favored methodology--which we do here. The resulting Isotropy method is compared to two other regularization methods proposed for functional source imaging (Kalman and Joint Regularization). This theoretical support for the favored status of the Isotropy method is consistent with its favorable computational performance in low prior information settings, as indicated in recent publications.

The temporal prior in bioelectromagnetic source imaging problems.

The multiplicity of temporal priors proposed for regularization of the bioelectromagnetic source imaging problems [e.g., the inverse electrocardiogram (ECG) and inverse electroencephalogram (EEG) problems], is discordant with the fact that fundamental statistical principles sharply limit the choice. Thus, our objective is to derive the form of the prior consistent with the general unavailability of temporal constraints. Writing linear formulations of the inverse ECG and inverse EEG problems as H = FG + N (where the ith columns of matrices H, G, and N, are data, signal, and noise vectors at time step i, and F is the transfer matrix), and using the noninformative principle that features of the spatiotemporal prior not supplied a posteriori should be invariant under temporal transformations, we show that the implied spatiotemporal signal autocovariance matrix (of the vector formed by the entries of G) is given in block matrix form [equation in text] where Cg is a matrix of unit trace proportional to the autocovariance matrix of any column of G (representing supplied information regarding the spatial prior), epsilon[.] denotes expectation, superscript ' indicates transpose, [symbol in text] is the Kronecker product, [symbol in text] is Frobenius norm, and the "matrix scalar product" [symbol in text] indicates the inner product of the two vectors formed by the entries of the two adjacent matrices (i.e., A [symbol in text] B [triple bond] trace[A'B]). This result eliminates some uncertainties and ambiguities that have characterized spatiotemporal regularization methods--including eight methods previously introduced in this transactions. Ultimately, the result derives from an implied symmetry principle under which the form of a nontrivial noninformative temporal component of the prior can be identified. Among other things, separability of the spatiotemporal prior in terms of the above Kronecker product can be thought of as the expression of the lack of "entanglement" of the spatial and temporal contributions (a consequence of noninformativity). The approach is generalized to the important cases of non-Gaussian spatial priors, and signal and noise that are not independent (transfer matrix noise). We also demonstrate a means for computational complexity reduction, related to the application of a particular orthogonal transformation, having features dependent on whether or not the transfer matrix represents a surjective mapping.

Analysis of the compatibility conditions for the radon projections, with an application to seismology

AbstractIn the context of the Radon approach to tomographic imaging, we face the problem of ascertaining whether or not a set of functions can be considered a set of reliable projections of an unknown function. We first review some results about the singular system of the Radon operator. Then, we define a set of parameters that allows us to evaluate the degree of compatibility of a set of projections. We apply our analysis to the seismic source imaging problem. In particular, we show that the compatibility conditions can be used for the empirical Green function retrieval. © 2002 Wiley Periodicals, Inc. Int J Imaging Syst Technol 12, 112–116, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ima.10018

Second-Order Approximation of the Pseudoinverse for Operator Deconvolutions and Families of Ill-Posed Problems

We identify a more accurate means of deconvolving compact nondegenerate operators than that provided by independent application of standard regularization techniques to individual members of the implied continuum of first kind integral equations. For the problem defined by $$ h(x,t) = \int_Y f(x,y)g(y,t)dY, $$ where it is required to find g(y,t) given noise-corrupted versions of square-integrable f(x,y) and h(x,t), it is shown that greater accuracy results from optimizing solutions for each member of the sequence of integral equations derived from individual components of the singular value expansion of h(x,t), than from optimizing solutions to each of the equations determined by different fixed values of t. This result has particular implications for treatment of the bioelectric/biomagnetic source imaging problems, where the characteristic need to perform such a deconvolution has been previously handled by producing an individually regularized solution to the above equation for each t in a collection of ...