Most of the material in this book has been presented in lectures at Brown University, either in courses taught in the Division of Applied Mathematics or in the author's Re­ search Seminar in Pattern Theory. I would like to thank the several members of the Division of Applied Mathematics that have participated in the discussions and in particular w. Freiberger, S. Geman, C.-R. Hwang, D. McClure and P. Thrift. I would also like to thank F. John, J. P. LaSalle, and L. Sirovich for accepting the manuscript for the Series Applied Mathematical Sciences published by Springer-Verlag. The research reported here has been supported by the National Science Foundation, Office of Naval Research and the Air Force Office of Scientific Research. I am grateful for the active interest and help given in various ways by Dr. Eamon Barrett, Dr. Kent Curtis, Dr. Robert Grafton and Dr. I. Shimi of these agencies. I also thank C.-R. Hwang and P. Thrift for help with proofreading. I am indebted to Mrs. E. Fonseca for her careful pre­ paration of the manuscript, to Miss E. Addison for helping me with the many diagrams, and to Mrs. K. MacDougall for the final typing of the manuscript. Ulf Grenander Providence, Rhode Island October 1980 v TABLE OF CONTENTS Page INTRODUCTION . . . . .