A Fully Implicit Framework for Sobolev Active Contours and Surfaces
We present a convenient framework for Sobolev active contours and surfaces, which uses an implicit representation on purpose, in contrast to related approaches which use an implicit representation only for the computation of Sobolev gradients. Another difference to related approaches is that we use a Sobolev type inner product, which has a better geometric interpretation, such as the ones proposed for Sobolev active contours. Since the computation of Sobolev gradients for surface evolutions requires the solution of partial differential equations on surfaces, we derive a numerical scheme which allows the user to obtain approximative Sobolev gradients even in linear complexity, if desired. Finally, we perform several experiments to demonstrate that the resulting curve and surface evolutions enjoy the same regularity properties as the original Sobolev active contours and show the whole potential of our method by tracking the left ventricular cavity acquired with 4D MRI.
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