4 April 1997 Multivariate granulometric filters
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Abstract
As introduced by Matheron, granulometries depend on a single sizing parameter for each structuring element. The concept of granulometry has recently been extended in such a way that each structuring element has its own sizing parameter resulting in a filter (Psi) t depending on the vector parameter t equals (t1..., tn). The present paper generalizes the concept of a parameterized reconstructive (tau) -opening to the multivariate setting, where the reconstructive filter (Lambda) t fully passes any connected component not fully eliminated by (Psi) t. The problem of minimizing the MAE between the filtered and ideal image processes becomes one of vector optimization in an n- dimensional search space. Unlike the univariate case, the MAE achieved by the optimum filter (Lambda) t is global in the sense that it is independent of the relative sizes of structuring elements in the filter basis. As a consequence, multivariate granulometries provide a natural environment to study optimality of the choice of structuring elements. If the shapes of the structuring elements are themselves parameterized, the expected error is a deterministic function of the shape and size parameters and its minimization yields the optimal MAE filter.
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Sinan Batman, Edward R. Dougherty, "Multivariate granulometric filters", Proc. SPIE 3026, Nonlinear Image Processing VIII, (4 April 1997); doi: 10.1117/12.271129; https://doi.org/10.1117/12.271129
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