Variance of the Isotropic Uniform Systematic Sampling


  • Jiri Janacek Institute of Physiology, ASCR, v.v.i.
  • Daniel Jirak Institute for Clinical and Experimental Medicine First Faculty of Medicine



isotropic design, spatial statistics, stereology, systematic sampling, variance


The integral of a smooth function with bounded support over a set with finite perimeter in Euclidean space ℝd is estimated using a periodic grid in an isotropic uniform random position. Extension term in the estimator variance is proportional to the integral of the squared modulus of the function over the object boundary and to the grid scaling factor raised to the power of d+1. Our result generalizes the Kendall-Hlawka-Matheron formula for the variance of the isotropic uniform systematic estimator of volume.


Ambrosio L, Fusco N, Pallara D (2000). Functions of bounded variation and free discontinuity problems, Oxford University Press.

Bochner S, Chandrasekharan K (1949). Fourier transform, Princeton University Press.

Brandolini L, Hofmann S, Iosevich A (2003). Sharp rate of average decay of the fourier transform of a bounded set, Geom Funct Anal 13:671--80.

Galerne B (2011). Computation of the perimeter of measurable sets via their covariogram. Applications to random sets. Image Anal Stereol 30:39--51.

Gundersen HJG, Jensen EB (1987). The efficiency of systematic sampling in stereology and its prediction. J Microsc 147 (3):229--63.

Hlawka E (1950). "{U}ber integrale auf konvexen k"{o}rpern I. Monatsh f"{u}r Math 54 1-36.

Jan'av{c}ek J (2006). Variance of periodic measure of bounded set with random position. Comment Math Univ Carolinae 47:473--82.

Jan'av{c}ek J (2008). An asymptotics of variance of the lattice points count. Czechoslovak Mathematical Journal, 58 (133),751--75.

Kendall DG (1948). On the number of lattice points inside the random oval. Q J Math Oxford Ser 19 (3):1--26.

Kieu K, Mora M (2004). Asymptotics for geometric spectral densities and stochastic approach of the lattice--point problem. Math Notae XLII:77--93.

Kieu K, Mora M (2006). Precision of stereological planar estimators. J Microsc 222 (3):201--11.

Matheron G (1965). Les variables r'egionalis'ees et leur estimations. Masson et Cie, 'Editeurs.

Rudin W (1987). Real and complex analysis. 3rd ed., McGraw-Hill.

Wiener N (1933). The fourier integral and certain of its applications.

Dover Publications Inc.




How to Cite

Janacek, J., & Jirak, D. (2019). Variance of the Isotropic Uniform Systematic Sampling. Image Analysis and Stereology, 38(3), 261–267.



Original Research Paper

Most read articles by the same author(s)