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<p><b>Abstract</b>—We propose a new image segmentation technique called strings. A string is a variational deformable model that is learned from a collection of example objects rather than built from a priori analytical or geometrical knowledge. As opposed to existing approaches, an object boundary is represented by a one-dimensional multivariate curve in functional space, a feature function, rather than by a point in vector space. In the learning phase, feature functions are defined by extraction of multiple shape and image features along continuous object boundaries in a given learning set. The feature functions are aligned, then subjected to functional principal components analysis and functional principal regression to summarize the feature space and to model its content, respectively. Also, a Mahalanobis distance model is constructed for evaluation of boundaries in terms of their feature functions, taking into account the natural variations seen in the learning set. In the segmentation phase, an object boundary in a new image is searched for with help of a curve. The curve gives rise to a feature function, a string, that is weighted by the regression model and evaluated by the Mahalanobis model. The curve is deformed in an iterative procedure to produce feature functions with minimal Mahalanobis distance. Strings have been compared with active shape models on 145 vertebra images, showing that strings produce better results when initialized close to the target boundary, and comparable results otherwise.</p>
Machine learning, deformable models, energy minimization, multivariate statistics, shape analysis, functional data analysis, chemometrics, active shape models.

S. Ghebreab and A. W. Smeulders, "Strings: Variational Deformable Models of Multivariate Continuous Boundary Features," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 25, no. , pp. 1399-1410, 2003.
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