Abstract
Periodic spatial grids can be used for unbiased estimation of length and surface area of objects by counting or measuring intersections of the objects with the grids. The estimators are theoretically based on discrete approximation of well established integral geometric formulas. The variance of the estimates depends on properties of both the grid and the measured objects. Main results of the theory of variance of the isotropic uniform random (IUR) volume estimation by spatial grids, especially a formula relating the variance of the volume estimator with the object surface area and the grid constant, are recapitulated. To identify main features of length and surface area IUR estimates the variance due to rotation and simple asymptotic formulas for the residual variance of estimates of selected model objects is calculated. Surface area estimates by multiple grids of parallel lines in 3D and of the variance of length estimates by periodic grids of planes or spheres in ddimensional space are studied.
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