Multiscale signal processing: isotropic random fields on homogeneous trees

In this paper we consider isotropic processes indexed by the nodes of a homogeneous tree of order q. An oriented (“hanging”) version of the qth order tree appears naturally when we consider successive filtering-and-decimation (by a factor of q) operations, as in multirate filtering and wavelet transforms. We derive Levinson and Schur recursions which provide us with a parametrization of an isotropic process via its reflection (or PARCOR) coefficient sequence. This can be done in an elegant way in the non-oriented setting, by making use of some general prediction errors. We state the counterpart of these results for the case of the oriented tree, where we have forward and backward prediction errors, defined according to a notion of causality “from coarse to fine scales”. These “oriented” results represent generalizations of the Levinson and Schur recursions derived in for the case of the dyadic tree (i.e., q=2), which were used in to develop modeling and whitening filters for isotropic processes