MULTISCALE SIGNAL-PROCESSING - ISOTROPIC RANDOM-FIELDS ON HOMOGENEOUSTREES

In this paper we consider isotropic processes indexed by the nodes of a homogeneous tree of order q. An oriented (''hanging'') version of th e qth order tree appears naturally when we consider successive filteri ng-and-decimation (by a factor of q) operations, as in multirate filte ring [8] and wavelet transforms [9]. We derive Levinson and Schur recu rsions 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 f or the case of the oriented tree, where we have forward and backward p rediction errors, defined according to a notion of causality ''from co arse to fine scales.'' These ''oriented'' results represent generaliza tions of the Levinson and Schur recursions derived in [1] for the case of the dyadic tree (i.e., q = 2), which were used in [2] to develop m odeling and whitening filters for isotropic processes.