Signal Processing 30 (1993) 263 265 Elsevier
Thesis alerts Bernhard Claus*
Multiscale Signal Processing" Gaussian Random Fields on Homogeneous Trees** Universitk de Rennes L IRISA, Rennes, France
The wavelet transform represents a tool for the multi-scale analysis or synthesis of signals which appealed in the last few years the interest of people from a vast diversity of fields, mathematics as well as physics, signal processing or electrical engineering. However, this technique is completely deterministic and thereby the question arises naturally to establish a counterpart of that theory, dealing with stochastic processes. Obviously one could apply the wavelet transform to stochastic processes and examine the statistical properties of the coefficients in the transform domain. Another approach consists in considering stochastic multi-scale processes, i.e. processes which live naturally on different scales. In this thesis we adopt the latter approach and we hope to contribute to the theory, the modeling and processing of multiscale stochastic processes. In particular we consider processes which are indexed by the nodes of a homogeneous tree. This index set emerges as the index set associated with the coefficients of the wavelet transform, ani~ it seems therefore to be natural to consider stochastic processes on that index set. In this thesis we deal in particular with isotropic processes, where isotropy defines a notion of sta* Thesis advisor: Mich~le Basseville. ** Copies are available on demand from: Biblioth~que de I'IRISA, Campus de Beaulieu, 35042 Rennes Cedex, France. Elsevier Science Publishers B.V.
tionarity. An isotropic process on a homogeneous tree can be characterized by means of the associated covariance sequence r or, equivalently, by means of a sequence of reflection coefficients k. We show how these two characterizations are connected via Levinson and Schur recursions. We introduce a notion of causality on the homogeneous tree, which allows us to define a special class of isotropic processes, the autoregressive processes, and to derive properties of it. An AR(n) process is characterized by the fact that only the reflection coefficients up to the order n are not necessarily equal to zero, i.e. we have km= 0 i f m > n. We examine in detail the characterization of an AR(n) process via a regression equation, where we use relations of the Yule-Walker type. Furthermore we consider the problem of identifying the process on the tree from the observation of an ordinary signal, which we interpret as the restriction of our process on the tree to one level of resolution. The first question which arises in this context is to determine the position of the observed signal with respect to the indexing tree. Then we can estimate the evenindexed covariances r(2i) (as all the nodes on one level lie at an even distance from each other). We show how to interpolate the covariance sequence, i.e. how to calculate the associated odd-indexed covariances r(2i+ 1), and we reconstruct the process on the tree in the sense of the conditional expectation.