A nonparametric approach to the estimation of lengths and surface areas. (English) Zbl 1124.62017
Summary: The Minkowski content \(L_0(G)\) of a body \(G\subset \mathbb R^d\) represents the boundary length (for \(d=2\)) or the surface area (for \(d=3\)) of \(G\). A method for estimating \(L_0(G)\) is proposed. It relies on a nonparametric estimator based on the information provided by a random sample (taken on a rectangle containing \(G\)) in which we are able to identify whether every point is inside or outside \(G\). Some theoretical properties concerning strong consistency, \(L_1\)-error and convergence rates are obtained. A practical application to a problem of image analysis in cardiology is discussed in some detail. A brief simulation study is provided.
MSC:
| 62G07 | Density estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62H35 | Image analysis in multivariate analysis |
Keywords:
Minkowski content; nonparametric set estimation; length estimation; statistical image analysisReferences:
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