Dynamic imaging of magnetic sources by the optimal constrained linear inverse method

The optimal constrained linear inverse method (OCLIM) for magnetic source imaging uses prior statistical information about the expected source amplitudes to yield a minimum mean square error estimate of the source amplitudes, plus standard errors for the reconstruction. It also includes a statistical test for the validity of the priors. A sequence of field measurements can be inverted instant by instant to yield a sequence of current distributions but it is desirable to exploit temporal coherence of the sources to reduce noise and improve the reconstruction accuracy. OCLIM applies in principle to the spatio-temporal reconstruction problem but a direct implementation is computationally expensive. The computational cost can be reduced from tens of hours to a few minutes by making some assumptions about the structure of the priors. The key assumption is that the prior source and noise covariance functions are separable; that is, that each covariance can be written as the product of a function of space only and a function of time only. Then there is a fast algorithm that approximates the optimal reconstruction. Assuming in addition that the noise statistics are the same for all detectors and uncorrelated between detectors yields a fast algorithm for the optimal reconstruction. Assuming that the temporal covariance is stationary allows the use of the fast Fourier transform and provides some additional speedup. These fast algorithms have been tested by simulation; they are almost as fast as instant by instant reconstruction and provide improved spatial and temporal resolution