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 as C-G = (epsilon[G'G] circle times epsilon[G'G])/(epsilon[parallel toGparallel to(2)]) = ((epsilon[H'H] - epsilon[N'N]) circle times C-g)/((F'F) circle dot C-g) where C-g 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, circle times is the Kronecker product, parallel to(.)parallel to is Frobenius norm, and the "matrix scalar product" circle dot indicates the inner product of the two vectors formed by the entries of the two adjacent matrices (i.e., A circle dot B drop 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.