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Tradeoffs between measurement residual and reconstruction error in inverse problems with prior information

Description: In many inverse problems with prior information, the measurement residual and the reconstruction error are two natural metrics for reconstruction quality, where the measurement residual is defined as the weighted sum of the squared differences between the data actually measured and the data predicted by the reconstructed model, and the reconstruction error is defined as the sum of the squared differences between the reconstruction and the truth, averaged over some a priori probability space of possible solutions. A reconstruction method that minimizes only one of these cost functions may produce unacceptable results on the other. This paper develops reconstruction methods that control both residual and error, achieving the minimum residual for any fixed error or vice versa. These jointly optimal estimators can be obtained by minimizing a weighted sum of the residual and the error; the weights are determined by the slope of the tradeoff curve at the desired point and may be determined iteratively. These results generalize to other cost functions, provided that the cost functions are quadratic and have unique minimizers; some results are obtained under the weaker assumption that the cost functions are convex. This paper applies these results to a model problem from biomagnetic source imaging and exhibits the tradeoff curve for this problem.
Date: June 1, 1995
Creator: Hughett, P.
Partner: UNT Libraries Government Documents Department

An optimal constrained linear inverse method for magnetic source imaging

Description: Magnetic source imaging is the reconstruction of the current distribution inside an inaccessible volume from magnetic field measurements made outside the volume. If the unknown current distribution is expressed as a linear combination of elementary current distributions in fixed positions, then the magnetic field measurements are linear in the unknown source amplitudes and both the least square and minimum mean square reconstructions are linear problems. This offers several advantages: The problem is well understood theoretically and there is only a single, global minimum. Efficient and reliable software for numerical linear algebra is readily available. If the sources are localized and statistically uncorrelated, then a map of expected power dissipation is equivalent to the source covariance matrix. Prior geological or physiological knowledge can be used to determine such an expected power map and thus the source covariance matrix. The optimal constrained linear inverse method (OCLIM) derived in this paper uses this prior knowledge to obtain a minimum mean square error estimate of the current distribution. OCLIM can be efficiently computed using the Cholesky decomposition, taking about a second on a workstation-class computer for a problem with 64 sources and 144 detectors. Any source and detector configuration is allowed as long as their positions are fixed a priori. Correlations among source and noise amplitudes are permitted. OCLIM reduces to the optimally weighted pseudoinverse method of Shim and Cho if the source amplitudes are independent and identically distributed and to the minimum-norm least squares estimate in the limit of no measurement noise or no prior knowledge of the source amplitudes. In the general case, OCLIM has better mean square error than either previous method. OCLIM appears well suited to magnetic imaging, since it exploits prior information, provides the minimum reconstruction error, and is inexpensive to compute.
Date: September 1, 1993
Creator: Hughett, P.
Partner: UNT Libraries Government Documents Department

Detection geometry and reconstruction error in magnetic source imaging

Description: A recently developed reconstruction algorithm for magnetic source imaging exploits prior knowledge about source location, source power density, detector geometry, and detector noise power to obtain an explicit estimate of the reconstruction error. This paper demonstrates the application of the new algorithm to the optimal design of practical detector arrays to minimize the reconstruction error in specific applications. For a representative configuration for magnetocardiography, the optimal array width (for minimum reconstruction error) varies from 19 to 28 cm depending on the assumed source depth, number of detectors, source power and noise power. The reconstruction accuracy ranges from 5% of the a priori standard deviation for the sources nearest the detector plane to 95% of the a priori deviation for the deepest sources. The reconstruction error was found to depend on accidental alignments between dipole sources and point detectors, indicating that a more sophisticated model is required for accurate estimates of reconstruction error. The error calculation is fast, taking about a second for this problem on a workstation-class computer. The availability of a method for rapidly computing the reconstruction error for any given source characteristics and detector geometry will facilitate the optimal design of magnetometer array size, element spacing, and orientation for specific applications in biomagnetic and geomagnetic source imaging.
Date: November 1, 1993
Creator: Hughett, P. & Budinger, T. F.
Partner: UNT Libraries Government Documents Department

Algorithms for biomagnetic source imaging with prior anatomical and physiological information

Description: This dissertation derives a new method for estimating current source amplitudes in the brain and heart from external magnetic field measurements and prior knowledge about the probable source positions and amplitudes. The minimum mean square error estimator for the linear inverse problem with statistical prior information was derived and is called the optimal constrained linear inverse method (OCLIM). OCLIM includes as special cases the Shim-Cho weighted pseudoinverse and Wiener estimators but allows more general priors and thus reduces the reconstruction error. Efficient algorithms were developed to compute the OCLIM estimate for instantaneous or time series data. The method was tested in a simulated neuromagnetic imaging problem with five simultaneously active sources on a grid of 387 possible source locations; all five sources were resolved, even though the true sources were not exactly at the modeled source positions and the true source statistics differed from the assumed statistics.
Date: December 1995
Creator: Hughett, P. W.
Partner: UNT Libraries Government Documents Department

A comparison of vector and radial magnetometer arrays for whole-head magnetoencephalography

Description: The number of detectors in magnetometer arrays for magnetoencephalography (MEG) has been steadily increasing, with systems containing or more detectors now possible. It is of considerable interest to know how best to configure such a large array. In particular, is it useful to measure all three components of the magnetic field, rather than just the radial component? This paper compares the information content provided by three different magnetometer arrays for whole-head measurements, using a definition of information content developed by Kemppainen and Ilmoniemi.
Date: February 1, 1996
Creator: Hughett, P. & Miyauchi, S.
Partner: UNT Libraries Government Documents Department