We present an algorithm to generate samples from probability distributions on the space of curves. Traditional curve evolution
methods use gradient descent to find a local minimum of a specified energy functional. Here, we view the energy
functional as a negative log probability distribution and sample from it using a Markov chain Monte Carlo (MCMC) algorithm.
We define a proposal distribution by generating smooth perturbations to the normal of the curve, update the curve
using level-set methods, and show how to compute the transition probabilities to ensure that we compute samples from the
posterior. We demonstrate the benefits of sampling methods (such as robustness to local minima, better characterization
of multi-modal distributions, and access to some measures of estimation error) on medical and geophysical applications.
We then use our sampling framework to construct a novel semi-automatic segmentation approach which takes in partial
user segmentations and conditionally simulates the unknown portion of the curve. This allows us to dramatically lower the
estimation variance in low-SNR and ill-posed problems.
We present a method for the simultaneous deconvolution and interpolation of remote sensing data in a single joint inverse problem. Joint inversion allows sparsely sampled data to improve deconvolution results and, conversely, allows large-scale blurred data to improve the interpolation of sampled data. Geostatistical
interpolation and geostatistically damped deconvolution are special cases such a joint inverse problem. Our method is posed in the Bayesian framework and requires the definition of likelihood functions for each data set involved, as well as a prior model of the parameter field of interest. The solution of such a problem is the posterior probability distribution. We present an algorithm for finding the maximum of this distribution. The particular application we apply our algorithm to is the fusion of digital elevation model and global positioning system data sets. The former data is a larger scale blurred image of topography, while the latter represent point samples of the same field. A synthetic data set is constructed to first show the performance of the method. Real data is then inverted.