Multi-Scale Autoregressive Processes

In many applications (e.g. recognition of geophysical and biomedical signals and multiscale analysis of images), it is of interest to analyze and recognize phenomena occuring at different scales. The recently introduced wavelet transforms provide a time-and-scale decomposition of signals that offers the possibility of such analysis. At present, however, there is no corresponding statistical framework to support the development of optimal, multiscale statistical signal processing algorithms. In this paper we describe such a framework. The theory of multiscale signal representations leads naturally to models of signals on trees, and this provides the framework for our investigation. In particular, in this paper we describe the class of isotropic processes on homogenous trees and develop a theory of autoregressive models in this context. This leads to generalizations of Schur and Levinson recursions, associated properties of the resulting reflection coefficients, and the initial pieces in a system theory for multiscale modeling. 1M.B. is also with the Centre National de la Recherche Scientifique (CNRS) and A.B. is also with Institut National de Recherche en Informatique et en Automatique (INRIA). The research of these authors was also supported in part by Grant CNRS GO134 2The work of this author was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-88-0032, in part by the National Science Foundation under Grant ECS-8700903, and in part by the US Army Research Office under Contract DAAL03-86-K-0171. In addition some of this research was performed while A.S.W. was a visitor at and received partial support from INRIA. Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE JUN 1989 2. REPORT TYPE 3. DATES COVERED 00-06-1989 to 00-06-1989 4. TITLE AND SUBTITLE Multi-Scale Autoregressive Processes 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Massachusetts Institute of Technology,Laboratory for Information and Decision Systems,77 Massachusetts Avenue,Cambridge,MA,02139-4307 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT 18. NUMBER OF PAGES 81 19a. NAME OF RESPONSIBLE PERSON a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18