Estimation Theory and Statistical Physics.
Abstract
The construction of a non-linear filter involves an integration over function space which is exactly analogous to the construction of a measure on path space via the Feynman-Kac-Nelson Formula. In Kalman-Bucy filtering problem the filtering of Gauss-Markov processes in the presence of additive white Gaussian noise occupies the same role as the Ornstein-Uhlenbeck process (finite or infinite-dimensional) in Quantum Mechanics or Quantum Field Theory. That this analogy is borne out by the fact that a solvable Lie algebra, the oscillator algebra which contains the Heisenberg algebra as a derived algebra is intrinsically attached to the Kalman-Bucy filtering problem. The problem of non-linear filtering of diffusion processes was shown to admit a stochastic variational interpretation. The objective of this paper is to strengthen these analogies further with a view to showing the close relationship of estimation theory to statistical mechanics. The motivation for this comes from problems of estimation and inverse problems related to image processing. In order to carry out this program it is necessary to generalize these ideas to filtering problems for infinite-dimensional processes. There are two types of processes involved: continuous processes such as infinite-dimensional Ornstein-Uhlenbeck processes and their L2-functionals which represent intensities of images and processes of a discrete nature which will represent boundaries of images. The most interesting models are obtained when these processes are coupled according to a probabilistic mechanisms. The discrete processes should be thought of as gauge fields and will be a process on connection forms.
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1985
- Accession Number
- ADA164054
Entities
People
- Sanjoy K. Mitter
Organizations
- Massachusetts Institute of Technology