Volume 10, Number 2, Summer 2002
STATISTICAL ESTIMATION OF THE PARAMETERS OF A PDE
COLIN FOX AND GEOFF NICHOLLS
Abstract. We consider the class of image recovery problems
where the desired image appears as the spatiallyvarying
coeffcients of a partial differential equation (PDE), the data
consists of measured values on the boundary, and the forward
map can only be adequately simulated by solving the PDE subject
to boundary conditions. This is the natural mathematical
setting for noninvasive imaging using strongly scattered waves.
We give a gentle introduction to the statistical (Bayesian) approach
to solving this class of inverse problems, and present
some computational examples using sampling algorithms. The
statistical approach quantifies the inherent uncertainty in images
recovered from incomplete noisy data using knowledge of
the forward map the measurement process and noise statistics.
The single most likely image, usually found by applying the inverse
of the forward map to the measured data, does not give a
good reconstruction in this class of problems because that image
is unrepresentative of the bulk of feasible images. Instead
it is necessary to summarize the feasible images by calculating
expectations over the posterior distribution. One advantage of
this route to solving inverse problems is the ability to quantify
accuracy within the recovered image, while a major disadvantage is
computational expense. We use the illustrative example
of imaging electrical conductivity and give examples of reconstruction
of an unknown conductivity from simple synthetic
data. In principle, highlevel models may be incorporated relatively
easily in the sampling algorithms, and we give a glimpse
of some image models that are currently used.
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