An analytical comparison of three spatio-temporal regularization methods for dynamic linear inverse problems in a common statistical framework

Abstract

Dynamic inverse problems, which occur in medical imaging and other fields, are inverse problems in which the quantities to be reconstructed vary in time, although they are related to the measurements through spatial operators only. Traditional methods solve these problems by frame-by-frame reconstruction, then extract temporal behaviour of the objects or regions of interest through curve fitting and other image-based processing. These approaches solve the inverse problem while exploiting only the spatial relationship between the object and the measurement data at each time instant, without using any temporal dynamics of the underlying process, and thus are not optimal unless the solution is temporally uncorrelated. If the spatial operators are linear, and if one, by contrast, solves the whole spatio-temporal process jointly, it falls into the category of general linear least-squares problems. Such approaches are generally difficult, both due to the challenge of modelling the temporal dynamics appropriately as well as to the high dimensionality of the associated large linear system. Several recent reports have approached this problem in different ways, making different prior assumptions on the spatial and temporal behaviour. In this paper we discuss three such approaches, which have been introduced from different points of view, in a common statistical regularization framework, and illuminate their relationships. The three methods are a state-space model, the separability condition and a multiple constraints model. The key result is that there is a clear relationship among the three methods; specifically, the inverse of the spatio-temporal autocovariance matrix has a block tri-diagonal form, a Kronecker product form or a Kronecker sum form, respectively. Some simple simulation examples are presented to illustrate the theoretical analysis.