Consistent set estimation in k-dimensions : An efficient approach

Abstract

Determining the shape of a point pattern is a problem of considerable practical interest and has applications in many branches of science related to Pattern Recognition. Set estimators of a nonparametric nature which may be used as shape descriptors should have several desirable properties. The estimators should be (a) consistent, i.e. Lebesgue measure of the symmetric difference of the actual region and the estimated should go to zero in probability; (b) computationally efficient; and (c) automatic, in the sense that the method should be able to detect the number of independent disjoint components in the region even when this number is unknown. None of the currently known estimators combine all these properties. A new shape descriptor called s-shape in the context of perceived border extraction of dot patterns in 2-D has been recently proposed. Here, a class of set estimators based on the s-shape is developed in k-dimensions, which combine all the above properties. These estimators are consistent not just under the uniform distribution, but also when samples are drawn under any continuous distribution. The order of error in estimation is independent of the dimensionality k. To illustrate the effectiveness of the proposed approach, a linear order algorithm readily derived from the definition is applied in digital domain. The role of δ that controls the structure of the estimator, is analyzed.

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