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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).

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