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


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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