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

## Uncertainties in
tomographic reconstructions based on deformable geometry

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 probability of boundary shapes is taken to proportional to the negative exponential of the deformation energy used to control the boundary. This probabilistic interpretation 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 illposed tomographic reconstruction problems, which we demonstrate by reconstructing a two-dimensional object 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 analyzing data with a strong emphasis placed on uncertainty assessment. Every aspect of modeling is assigned a probability that indicates our degree of certainty in its value. At the lowest level of analysis, the estimation of the values of parameters for a specified model, a probability density function (PDF) is associated with each continuous parameter. Loosely speaking, the range of a probability distribution indicates the possible range of its associated parameter. The benefit of Bayesian analysis over traditional methods of uncertainty, or error, analysis is that it permits the use of arbitrary probability distributions, not just Gaussian distributions, and of arbitrary measures. This work relies on our radiographic modeling tool, the Bayes Inference Engine.

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

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Send e-mail to coauthor at kmh@hansonhub.com