Geometric models have received increasing attention in medical imaging for tasks such as segmentation, reconstruction, restoration, and registration. In order to determine the best configuration of the geometric model in the context of any of these tasks, one needs to perform a difficult global optimization of an ``energy" function that may have many local minima. Explicit models of geometry, also called deformable models, snakes, or active contours, have been used extensively to solve image segmentation problems in a non-Bayesian framework. Researchers have seen empirically that multi-scale analysis is useful for convergence to a configuration that is near the global minimum. In this type of analysis, the image data are convolved with blur functions of increasing resolution, and an ``optimal" configuration of the snake is found for each blurred image. The configuration obtained using the highest resolution blur is used as the solution to the global optimization problem. In this article, we use explicit models of geometry for a variety of Bayesian estimation problems, including image segmentation, reconstruction and restoration. We introduce a multi-scale approach that blurs the geometric model, rather than the image data, and show that this approach turns a global, highly nonquadratic optimization into a sequence of local, approximately quadratic problems that converge to the global minimum. The result is a deterministic, robust, and efficient optimization strategy applicable to a wide variety of Bayesian estimation problems in which geometric models of images are an important component.
Keywords: snakes, active contours, deformable models, Bayesian analysis, global optimization, multiscale, multiresolution
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