Presented at *IEEE Int. Conf. on Image Processing* Kobe, Japan, Oct. 24-28, 1999.

## Uncertainties in Bayesian geometric models

K. M. Hanson, G. S. Cunningham, and R. J. McKee

*Los Alamos National Laboratory*

### Abstract

Deformable geometric models fit very naturally into
the context of Bayesian analysis. The prior proba-
bility of boundary shapes is taken to proportional to
the negative exponential of the deformation energy
used to control the boundary. This probabilistic in-
terpretation is demonstrated using a Markov-Chain
Monte-Carlo (MCMC) technique, which permits one
to generate configurations that populate the prior. One
of many uses for deformable models is to solve ill-
posed tomographic reconstruction problems, which we
demonstrate by reconstructing a two-dimensional ob-
ject from two orthogonal noisy projections. We show
how MCMC samples drawn from the posterior can be
used to estimate uncertainties in the location of the
edge of the reconstructed object.
Bayesian analysis is a model-based approach to an-
alyzing data with a strong emphasis placed on uncer-
tainty assessment. Every aspect of modeling is as-
signed a probability that indicates our degree of cer-
tainty in its value. At the lowest level of analysis,
the estimation of the values of parameters for a spec-
ified model, a probability density function (PDF) is
associated with each continuous parameter. Loosely
speaking, the range of a probability distribution in-
dicates the possible range of its associated parame-
ter. The benefit of Bayesian analysis over traditional
methods of uncertainty, or error, analysis is that it per-
mits the use of arbitrary probability distributions, not
just Gaussian distributions, and of arbitrary measures

**Keywords:** computed tomography, Bayesian analysis,
maximum a posteriori (MAP) reconstruction,
Markov chain Monte, Carlo, adjoint differentiation

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