## Ulf Grenander

L. Herbert Ballou University Professor (Emeritus)

Division of Applied Mathematics

Phone: +1 401 863 2115

[email protected]

Ph.D., University of Stockholm, 1948

Professor Grenander's research involves a broad range of seminal contributions within the area of Applied Mathematics, and upon the ongoing development of a remarkable Pattern Theory and its applications to image analysis.

His current research interests focus on the the creation of a new theory of regular structure (pattern theory) and its application to the actual and man-made world. Also of interest is the development of the theory of statistical inference on stochastic processes and abstract spaces, by functional analytic methods. Work is directed toward the development of algorithms for medical image processing, and a unified theory of automatic target recognition.

### BIOGRAPHY

Most of my efforts since moving to Brown in 1966 have gone into creating a mathematical theory of patterns, starting from algebraic considerations, based on which are probabilistic results (metric pattern theory) and methods of inference (pattern inference). The latter are implemented as algorithms and computer programs intended to analyze and understand patterns of various types: pictures, syntactic patterns, logical structures, social systems, and patterns of doctrines.

Up to the early 1980's this work was of purely theoretical nature. Starting at this time I and my group in the Division of Applied Mathematics have tried to apply these ideas to practical tasks in pattern analysis, in particular to image restoration. The images can have been acquired using different technologies such as visible light digital cameras, infrared, X-rays, tomographic techniques, and more recently laser radar. Among current research topics that I am studying one can mention global models for highly variable shapes (such that occur in anatomy), 3-D recognition (restoration based on range data), and MRI and CT imaging based on mathematical models using background deformations. The main theme in this research has been to exploit the interaction between mathematics of a new type (pattern theory) and advances in high technology (computer architectures, languages, and new sensors).

### INTERESTS

Professor Grenander's research work involves inference for stochastic processes; insurance mathematics; probability theory on algebraic structures; nonparametric statistics; a general pattern theory and its applications to model-based image analysis. His work also involves a broad range of seminal contributions within the area of Applied Mathematics, and upon the ongoing development of a remarkable Pattern Theory and its applications to image analysis. He was among the first to show the utility of functional-analytic methods in the theory of inference for stochastic processes. Early work also includes fundamental contributions to the spectral analysis of time series and breakthrough developments in the application of mathematics to insurance. He was among the first to extend probability and its associated limit theory to general algebraic structures, a recently rediscovered and exciting branch of modern probability and statistics. In recent years he has pioneered the use of computers for experimental mathematics. Grenander's most recent, and ongoing, contributions are among his most innovative and important. Convinced that regularities of natural patterns reveal important common themes, he has devised an algebraic/probabilistic theory of regular structures. This theory, substantive in its practical, computational, and mathematical content, has inspired mathematical scientists throughout the world. The term "Pattern Theory" was introduced by Ulf Grenander in the 70's as a name for a field of applied mathematics which gave a theoretical setting for a large number of related ideas, techniques and results from fields such as computer vision, speech recognition, image and acoustic signal processing, pattern recognition and its statistical side, neural nets and parts of artificial intelligence. The problem that "Pattern Theory" aims to solve, may be described as follows - the analysis of the patterns generated by the world in any modularity, with all their naturally occurring complexity and ambiguity, with the goal of reconstructing the processes, objects and events that produced them and of predicting these patterns when they reoccur.

Professor Grenander's current research interests focus on the the creation of a new theory of regular structure (pattern theory) and its application to the actual and man-made world. Also of interest is the development of the theory of statistical inference on stochastic processes and abstract spaces, by functional analytic methods. Work is directed toward the development of algorithms for medical image processing, and a unified theory of automatic target recognition.

### AWARDS

Arhennius Fellow - 1948 Member, International Statistical Inst. Fellow, Inst. of Mathematical Statistics, 1953 Prize of the Nordic Actuarial Societies, 1961 Arnberger Prize, Royal Swedish Academy of Science, 1962 Member, Royal Swedish Academy of Science, 1965 Board Member, Sweden-America Foundation Member, Swedish Actuarial Society Guggenheim Fellowship, 1979 Honorary Fellow, Royal Statistical Society, London, 1989 D.Sc. (hon), University of Chicago, 1993 Fellow, American Academy of Arts & Sciences, 1995 Fellow, National Academy of Sciences, 1998

### WEB LINKS

A conversation with Ulf Grenander

Faculty Research Profile

SIAM Tribute to Ulf's 80th Birthday