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Ph.D. Theses
1985
Jose Luis Marroquin,
Probabilistic Solution of Inverse Problems.
Ph.D. Thesis, Department of Electrical Engineering and Computer
Science, Massachusetts Institute of Technology, Cambridge, MA, USA, September 1985.
In this thesis we study the general problem of
reconstructing a function, defined on a finite lattice, from a set of
incomplete, noisy, and/or ambiguous observations. The goal of this work
is to demonstrate the generality and practical value of a probabilistic
(in particular, Bayesian) approach to this problem, particularly in the
context of Computer Vision. In this approach, the prior knowledge about
the solution is expressed in the form of a Gibbsian probability
distribution on the space of all possible functions, so that the
reconstruction task is formulated as an estimation problem. Our main
contributions are the following:
beginenumerate
We introduction the use of specific error criteria for the design
of the optimal Bayesian estimators for several classes of problems, and
propose a general (Monte Carlo) procedure for approximating them. This
new approach leads to a substantial improvement over the existing
schemes, both regarding the quality of the results (particularly for
low signal to noise ratios) and the computational efficiency.
We apply the Bayesian approach to the solution of several
problems, some of which are formulated and solved in these terms for
the first time. Specifically, these applications are: the
reconstruction of piecewise continuous surfaces from sparse and noisy
observations; the reconstruction of depth from stereoscopic pairs of
images and formation of perceptual clusters.
For each one of these applications, we develop fast,
deterministic algorithms that approximate the optimal estimators, and
illustrate their performance on both synthetic and real data.
We propse a new method, based on the analysis of the residual
process, for estimating the parameters of the probabilistic models
directly from the noisy observations. This scheme leads to an
algorithm, which has no free parameters, for the restoration of
piecewise uniform images.
We analyze the implementation of the algorithms that we develop
in nonconventional hardware, such as massively parallel digital
machines, and analog and hybrid networks.
endenumerate