Curvature measures and random sets II

This paper deals with isotropic randoni mosaics of Sd-' whose ce!!s are piecewise smooth. As geometric measures concentrated on the i-skeleton (i = 0. . . . , d -I ) the i-dimensional surface area (volume) measure and (i -I ) different curvature measures are chosei;. Relations be!ween the corresponding densities are found. This leads to a parameier systein from which 211 mean values can be calculated. The curvature intensities are alsu compiited for the intersec!im with a subsphere, for superposition and for great hypersphere rnmaiis

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“…we also consider which is said to be k-th total curvature measure of the i-cell process pi. (In [9] it is s h~w a that C,(pt, .) and C,@f, .)…”

Section: Random Mosaics Of S D -'mentioning

confidence: 98%

Geometric measures for random mosaics in spherical spaces

Arbeiter

Zähle

1994

Stochastics and Stochastic Reports

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“…Let ~ be a random closed set of R d (in the sense of Matheron [3]) whose realizations a.s. have positive reach. Then similarly as in [6], it can be shown that C~,b=(4, ") is a random measure. If we assume additionally that 4 is motion invariant then the absolute curvature intensity measure ab= ECk (4, ") has the same property and by the uniqueness of Haar measure we obtain l:c~,b=(4, ')=2~L d for Lebesgue measure L e and certain constant 2~E [O, oo] which is said to be the kth absolute curvature intensity of 4.…”

Section: Appendixmentioning

confidence: 99%

“…(Corresponding results for Lipschitz-KillingFederer curvature measures may be found in [6].) Let ~ be a random closed set of R d (in the sense of Matheron [3]) whose realizations a.s. have positive reach.…”

Section: Appendixmentioning

confidence: 99%

Absolute curvature measures, II

Rother

Zhle

1992

Geom Dedicata

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The absolute curvature measures for sets of positive reach in R d introduced in [7] satisfy the following kinematic relations: Their integrated values on the intersections with (or on the tangential projections onto) uniformly moved p-planes are constant multiples of the corresponding absolute curvature measures of the primary set. In the special case of convex bodies the first result is the so-called Crofton formula. An analogue for signed curvature measures is well known in the differential geometry of smooth manifolds, but the motion of absolute curvatures used there does not lead to this property. For the special case of smooth compact hypermanifolds our absolute curvature measures agree with those introduced by Santal6 [4] with other methods.In the appendix, the section formula is applied to motion invariant random sets. EQUIVALENT DEFINITIONSThe present paper is a continuation of [7], which also contains motivations and references to the related literature. We first recall some notions introduced there. At the same time we correct an error in formulas (3)-(6) of [7], where certain projection Jacobians have to be inserted. In the sequel, X denotes an arbitrary subset of R d of positive reach (in the sense of Federer [1]) and B a bounded Borel set in R d x S d-1 (or in Rd). Note that convex bodies and compact C2-submanifolds are special cases for X. In [7] the kth absolute curvature measure c~bs(x, B), k = 0 ..... d-1, is introduced by means of the natural invariant measure of the set of all (d-1 -k)-planes locally colliding with X inside B. Thereby affine and linear subspaces of R d are considered as dements of sufficiently large Euclidean spaces via representation by multivectors. Let nor X be the unit normal bundle of X and G(X, d, k) its kth Grassmann bundle (denoted in [7] by G(X, k)), i.e. G(X, d, k)= {(x, n, V): (x, n)~nor X, V~G,(d-1, k)} where G,(d-1, k) is the Grassmann submanifold of G(d, k) of those kdimensional linear subspaces of R d which are orthogonal to the unit vector n ES d-1. Let v" be the normalized rotation invariant measure on d-l,k G,(d-l,k). 17 v is the orthogonal projection onto the subspace V The function f(x, n, V) = (IIv~ x, V) maps G(X, d, k) onto the set sO(X, d, k) of parametrized k-dimensional affine subspaces (k-planes) locally colliding with X. Let P be the coordinate Geometriae Dedicata 41: 229-240, 1992.

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“…(The case in which one of the sets is compact was shown in Federer [3], and the case of no compactness was shown in [14].) a second type of invariance of the curvature measures, where the lefthand side may be interpreted as a Radon-like transform.…”

Section: Curvature Measures For Sets Of Positive Reachmentioning

confidence: 99%

Curvatures and currents for unions of sets with positive reach

Zähle

1987

Geom Dedicata

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For locally finite unions of sets with positive reach in R a, generalized unit normal bundles are introduced in support of a certain set additive index function. Given an appropriate orientation to the normal bundle, signed curvature measures may be defined by means of associated locally rectifiable currents (with index function as multiplicity) and specially chosen differential forms. In the case of 'regular' sets this is shown to be equivalent to well-known classical concepts via former results. The present approach leads to unified methods in proving integral-geometric relations. Some of them are stated in this paper, Geometriae Dedieata 23 (1987) 155-171

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Curvature measures and random sets II

Cited by 37 publications

References 13 publications

Geometric measures for random mosaics in spherical spaces

Geometric measures for random mosaics in spherical spaces

Absolute curvature measures, II

Curvatures and currents for unions of sets with positive reach

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