Dynamic medical imaging as a partial inverse problem

Dynamic medical imaging problems are typically structured as "partial" inverse problems: the desired time series of images (a spatiotemporal matrix) is subject to a purely spatial transformation (its product with a transfer matrix) - so that the forward operator does not address all of the variables of the function to be transformed (analogous to partial differentiation, which is also a partial inverse problem). Given that the transfer matrix is ill-conditioned, the problem of providing the image sequence requires regularization. Under rather general conditions applicable to the setting of partial inverse problems, the regularization parameter of Tikhonov regularization generalizes to a regularization parameter operator, which in proper combination with a standard (spatial) Tikhonov regularizing operator describes the minimum-mean-square-error estimate analogously to that of the Bayesian interpretation of Tikhonov regularization as applied to a "complete" inverse problem. This is in distinction to usual methodology employed for dynamic imaging problems, including usual applications of Tikhonov regularization in this realm, which cannot similarly supply the minimum-mean-square-estimate (under the stated general conditions). The potential power of the implied methodology is illustrated with a numerical example - indicating that substantial improvements in solution estimation are possible.