Kalman filtering and Riccati equations for multiscale processes

Multiscale representations of signals and multiscale algorithms are addressed. A two-sweep smoothing algorithm is analyzed for fusing multiscale measurements of multiscale processes defined on trees. The algorithm is a generalization of the Rauch-Tung-Striebel algorithm for the smoothing of time series, and the filtering step differs from that of time series in that it consists of the successive fusing of data from level to level, thus introducing a new type of Riccati equation. The fusion step makes it necessary to view the optimal estimation as producing a maximum likelihood (ML) estimate which is then combined with prior statistics, and it is the dynamics of the ML estimate recursion which must be analyzed. Elements of a system theory required to derive bounds on the error covariance of the filter are developed. These results are then used along with a careful definition of stability on trees both to prove the stability of the filter and to give results on the steady-state filter