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- Title
ERROR BOUNDS FOR SURFACE AREA ESTIMATORS BASED ON CROFTON'S FORMULA.
- Authors
Kiderlen, Markus; Meschenmoser, Daniel
- Abstract
According to Crofton's formula, the surface area S(A) of a sufficiently regular compact set A in ℝd is proportional to the mean of all total projections PA (u) on a linear hyperplane with normal u, uniformly averaged over all unit vectors u. In applications, pA(u) is only measured in k directions and the mean is approximated by a finite weighted sum Ŝ (A) of the total projections in these directions. The choice of the weights depends on the selected quadrature rule. We define an associated zonotope Z (depending only on the projection directions and the quadrature rule), and show that the relative error Ŝ (A) /S (A) is bounded from below by the inradius of Z and from in above by the circumradius of Z. Applying a strengthened isoperimetric inequality due to Bonnesen, we show that the rectangular quadrature rule does not give the best possible error bounds for d = 2. In addition, we derive asymptotic behavior of the error (with increasing k) in the planar case. The paper concludes with applications to surface area estimation in design-based digital stereology where we show that the weights due to Bonnesen's inequality are better than tile usual weights based on the rectangular role and almost optimal in the sense that the relative error of the surface area estimator is very, close to the minimal error.
- Subjects
SCIENTIFIC method; WEIGHTS &; measures; METRIC projections; APPROXIMATION theory; FUNCTIONAL analysis; COMPACTING; FUNCTIONAL equations; ISOPERIMETRIC inequalities; PLANE geometry
- Publication
Image Analysis & Stereology, 2009, Vol 28, Issue 3, p165
- ISSN
1580-3139
- Publication type
Article