anisotropy, connectivity, efficiency, global stereology, local stereology, particle counting and sizing, second order stereology, size and orientation distributions


Stereology is the science of geometric sampling, with applications to the statistical analysis of microstructures in biology and materials science. Subsidiary disciplines are image analysis, quantitative microscopy, and radiology. This survey is organized chronologically within a series of topics which cover most aspects of stereology. Each topic is described informally to make it accessible to scientists of different disciplines.


Andersen IT, Hahn U, Jensen EBV (2015). Optimal PPS sampling with

vanishing auxiliary variables --- with applications in microscopy. Scand

J Statist 42:1136--1148.

Artacho-Perula E, Roldan-Villalobos R, Cruz-Orive LM (1999). Application


the fractionator and vertical slices to estimate total capillary length

in skeletal

muscle. J Anat 195:429--437.

Auneau J, Jensen EVB (2010). Expressing intrinsic volumes as rotational

integrals. Adv Appl Math 45:1--11.

Bach G (1963). Grundung einer internationalen Gesellschaft fur

Stereologie. Z wiss Mikrosk 65:190.

Baddeley AJ (1984). Vertical sections. In: Weil W, Ambartzumian RV eds,

Stochastic geometry and stereology, Oberwolfach 1983. Berlin:Teubner.

Baddeley AJ (1985). An anisotropic sampling design. In: Nagel N, ed.

Geobild '85, pp 92--97, Jena:Friedrich- Schiller-Universitat.

Baddeley AJ (1999). A crash course in stochastic geometry. In:

Barndorff-Nielsen OE, Kendall WS, van Lieshout MNM, eds. Stochastic

geometry: Likelihood

and computation, pp. 1--35. London: Chapman and Hall.

Baddeley AJ, Cruz-Orive LM (1995). The Rao-Blackwell theorem in


and some counterexamples. Adv Appl Prob 27:2--19.

Baddeley AJ, Gundersen HJG, Cruz-Orive LM (1986). Estimation of surface

area from vertical sections. J Microsc 142:259--276.

Baddeley AJ, Howard CV, Boyde A, Reid S (1987). Three-dimensional

analysis of the spatial distribution of particles using the tandem

scanning reflected light microscope. In: Miles RE, ed., pp. 87--100.

Baddeley A, Jensen EBV (2005). Stereology for statisticians. Boca Raton:

Chapman and Hall/ CRC.

Baddeley AJ, Moyeed RA, Howard CV, Boyde A (1993). Analysis of

three-dimensional point pattern with replication. Appl Statist


Baddeley AJ, Rubak E, Turner R (2015). Spatial point patterns:

Methodology and applications with R. Boca Raton: Chapman and Hall/CRC.

Baldwin JP, Tinker PB, Marriott FHC (1971). The measurement of length

and distribution of onion roots in the field and the laboratory. J Appl

Ecol 8:543--54.

Barbier E (1860). Note sur le probleme de l'aiguille et le jeu du joint

couvert. J Math Pures Appl 5:273--286.

Batra S, Konig MF, Cruz-Orive LM (1995). Unbiased estimation of

capillary length from vertical slices. J Microsc 178:152--159.

Bendtsen TF, Nyengaard JR (1989). Unbiased estimation of particle number

using sections --- an historical perspective with special reference to

the stereology of glomeruli. J Microsc 153:93--102.

Benes V, Gokhale AM, Slamova M (1996). Unfolding the bivariate

size-orientation distribution. Acta Stereol 15:9--14.

Benes V, Rataj J (2004). Stochastic geometry: Selected topics. Boston:


Bertrand J (1889). Calcul des probabilites. Paris: Gauthier-Villars.

Bhanu Prasad P, Jernot JP (1991). Topological description of the

densification of a granular medium. J Microsc 163:211--220.

Bhanu Prasad P, Lantuejoul C, Jernot JP, Chermant JL (1989).

Unbiased estimation of the Euler-Poincare characteristic. Proc. ECS V

Freiburg 1989. Acta Stereol 8/2:101--6.

Blaschke W (1936--37). Vorlesungen uber Integralgeometrie. Hamburg:

Hamburger Math Einzelschr 20 und 22.

Braendgaard H, Gundersen HJG (1986). The impact of recent

stereological advances on quantitative studies of the nervous system. J

Neurosci Meth 18:39--78.

Brungger A, Cruz-Orive LM (1987). Nuclear morphometry of

nuclear malignant melanoma and benign nevocytic nevy. Arch Dermatol Res


Buffon, G-L L, Comte de (1777). Essai d'arithmetique morale. In:

Supplement a l'Histoire Naturelle 4:46--123. Paris: Imprimerie Royale.

Cartan E (1896). Le principe de dualite et certaines integrales

multiples de l'espace tangentiel et de l'espace regle. Bull Soc Math Fr


Cavalieri B (1635). Geometria Indivisibilibus Continuorum. Bononi:

Typis Clementis Ferronij. Reprinted 1966 as: Geometria degli

Indivisibili. Torino: Unione Tipografico-Editrice Torinese.

Cochran WG (1977). Sampling techniques, 3rd edn., New York: J Wiley.

Crofton MW (1868). On the theory of local probability. Phil Trans Roy

Soc London 158:181--199.

Cruz M, Gomez D, Cruz-Orive LM (2015). Efficient and unbiased estimation

of population size. PLoS ONE 10(11):e0141868.


Cruz-Orive LM (1976). Particle size-shape distributions: the general

spheroid problem. I. Mathematical model. J Microsc 107:235--253.

Cruz-Orive LM (1979). Estimation of sheet thickness distributions from

linear and plane sections. Biom J 21:717--30.

Cruz-Orive (1980a). Best linear unbiased estimators for stereology.

Biometrics 36:595--605.

Cruz-Orive LM (1980b). On the estimation of particle number. J Microsc


Cruz-Orive (1982). The use of quadrats and test systems in stereology,

including magnification corrections. J Microsc 125: 89--102.

Cruz-Orive (1983). Distribution-free estimation of sphere size

distributions from slabs showing overprojection and truncation, with a

review of previous methods. J Microsc 131:265-290.

Cruz-Orive LM (1987a). Stereology: historical notes and recent

evolution. In: Miles ER, ed. ISS Commemorative-Memorial Volume, Acta

Ster 6/II:43--56.

Cruz-Orive LM (1987b). Particle number can be estimated using a disector

of unknown thickness: the selector. J Microsc 145:121--142.

Cruz-Orive LM (1989a). Second-order stereology: estimation of second

moment volume measures. Proc.ECS V Freiburg 1989. Acta Ster


Cruz-Orive LM (1989b). On the precision of systematic sampling: a review

of Matheron's transitive methods. J Microsc 153:315--333.

Cruz-Orive LM (1993). Systematic sampling in stereology. Proc 49th

Session, Florence. Bull Intern Statis Inst 55(2):451--468.

Cruz-Orive LM (1997). Stereology of single objects. J Microsc


Cruz-Orive LM (2002). Stereology: meeting point of integral geometry,

probability, and statistics. In memory of Professor Luis A.Santalo

(1911--2001). Special issue (Homenaje a Santalo), Mathematicae Notae


Cruz-Orive, L.M. (2004). Precision of the fractionator from Cavalieri

designs. J Microsc 213:205--211.

Cruz--Orive LM (2005). A new stereological principle for test lines in

D. J Microsc 219:18--28.

Cruz-Orive LM (2006). A general variance predictor for Cavalieri slices.

J Microsc 222:158--165.

Cruz-Orive LM (2009). Stereology: old and new. In: Capasso V, Aletti G,

Micheletti A, eds. Proc 10th Eur Congr Stereol Image Anal. Bologna:


Cruz-Orive LM (2012). Uniqueness properties of the invariator, leading

to simple computations. Image Anal Stereol 31:87--96.

Cruz-Orive LM (2013). Variance predictors for isotropic geometric

sampling, with applications in forestry. Stat Methods Appl 22:3--31.

Cruz-Orive LM, Geiser M (2004). Estimation of particle number by

stereology: an update. J Aerosol Med 17:197--212.

Cruz-Orive LM, Gelsvartas J, Roberts N (2014). Sampling theory and

automated simulations for vertical sections, applied to human brain. J

Microsc 253:119--150.

Cruz-Orive LM, Gual-Arnau X (2002). Precision of circular systematic

sampling. J Microsc 207:225--242.

Cruz-Orive LM, Gual-Arnau X (2015). The invariator design: an update.

Image Anal Stereol 34:147--159.

Cruz-Orive LM, Hoppeler H, Mathieu O, Weibel ER (1985). Stereological

analysis of anisotropic structures using directional statistics. J Roy

Statist Soc C (Appl Statist) 34:14--32.

Cruz-Orive LM, Howard CV (1991). Estimating the length of a bounded

curve in three dimensions using total vertical projections. J Microsc

: 101--113.

Cruz-Orive LM, Howard CV (1995). Estimation of individual feature

surface area with the vertical spatial grid. J Microsc 178:146--151.

Cruz-Orive LM, Hunziker EB (1986). Stereology for anisotropic cells:

application to growth cartilage. J Microsc 143:47--80.

Cruz-Orive LM, Insausti A, Insausti R, Crespo D (2004). A case study

from neuroscience involving stereology and multivariate analysis. In:

Evans SM, Janson AM, Nyengaard JM, eds., Quantitative methods in

neuroscience, pp.16--64. Oxford: Oxford University Press.

Cruz-Orive LM, Karlsson LM, Larsen SE, Wainschtein F (1992).

Characterizing anisotropy: a new concept. Micron Microsc Acta 23:75--6.

Cruz-Orive LM, Ramos-Herrera ML, Artacho-Perula E (2010). Stereology of

isolated objects with the invariator. J Microsc 240:94--110.

Cruz-Orive LM, Roberts N (1993). Unbiased volume estimation with coaxial

sections: an application to the human bladder. J Microsc 170:25--33.

Cruz-Orive LM, Weibel ER (1981). Sampling designs for stereology. J

Microsc 122:235-272.

Czuber E (1884). Geometrische Wahrscheinlichkeiten und Mittelwerte.

Leipzig: Teubner.

Davy PJ, Miles RE (1977). Sampling theory for opaque spatial specimens.

J Roy Statist Soc B, 39:56--65.

De Groot DMG (1988). Improvements of the serial section method in

relation to estimation of the numerical density of complex-shaped

synapses. In: Reith A, Mayhew TM, eds. Stereology and morphometry in

electron microscopy, pp. 135--158. New York: Hemisphere Publ. Co.

DeHoff RT (1968). Curvature and the topological properties of

interconnected phases. In: DeHoff RT, Rhines FN, eds., pp. 291--325.

DeHoff RT, Rhines FN (eds.) (1968). Quantitative microscopy. New York:


Delesse A (1847). Procede mecanique pour determiner la composition des

roches. C R Acad Sci Paris 25:544--545.

Deltheil R (1926). Probabilites geometriques. Paris: Gauthier-Villars.

Dorph-Petersen KA, Nyengaard JR, Gundersen HJG (2001). Tissue shrinkage

and unbiased stereological estimation of particle number and size. J

MIcrosc 204:232--246.

Elias H (1962). Stereology: scope and limits. Stereologia 1:1--2.

Elias H (1963). The birth of the stereological society. Proc First

Intern Congress for Stereology, pp. 2/1--2/3. Vienna: Vienna medical

academy for postgraduated medical training.

Elias H (ed.) (1967). Stereology. Proc Second Intern Cong for

Stereology, Chicago, April 8--13, 1967. New York: Springer.

Elias H, Hennig A, Schwartz E (1971). Stereology: applications to

biomedical research. Physiol Rev 51:158--200.

Evans SM, Gundersen HJG (1989). Estimation of spatial distributions

using the nucleator. Proc.ECS V Freiburg 1989. Acta Stereol


Garcia-Finana M (2006). Confidence intervals in Cavalieri sampling. J

Microsc 222:146--157.

Garcia-Finana M, Cruz-Orive LM (2004). Improved variance prediction for

systematic sampling on R. Statistics 38:243--272.

Glagolev AA (1933). On the geometrical methods of quantitative

mineralogical analysis of rocks [in Russian]. Trans Inst Econ Min,

Moskva 59:1--47.

Gokhale AM (1990). Unbiased estimation of curve length in 3D using

vertical slices. J Microsc 159:133--141.

Gokhale AM (1996). Estimation of bivariate size and orientation

distribution of microcracks. Acta mater 44:475--85.

Gokhale AM, Evans RA, Mackes JL, Mouton PR (2004). Design-based

estimation of surface area in thick tissue sections of arbitrary

orientation using virtual cycloids. J Microsc 216:25--31.

Gomez AI, Cruz M, Cruz-Orive LM (2016). On the precision of curve length

estimation in the plane. Image Anal Stereol 35:1--14.

Gonzalez-Villa J, Cruz M, Cruz-Orive LM (2017). On the precision of the

nucleator. Image Anal Stereol, in press.

Gridgeman NT (1960). Geometric probability and the number . Scripta Math


Gual Arnau, Cruz-Orive LM (1998). Variance prediction under systematic

sampling with geometric probes. Adv Appl Prob 30:889--903.

Gual-Arnau X, Cruz-Orive LM (2000). Systematic sampling on the circle

and on the sphere. Adv Appl Prob 32:628--647.

Gual-Arnau X, Cruz-Orive LM (2002). Variance prediction for

pseudosystematic sampling on the sphere. Adv Appl Prob 34:469--483.

Gual-Arnau X, Cruz-Orive LM (2009). A new expression for the density of

totally geodesic submanifolds in space forms, with stereological

applications. Diff Geom Appl 27:124--128.

Gual-Arnau X, Cruz-Orive LM (2016). New rotational integrals in space

forms, with an application to surface area estimation. Appl Math


Gual-Arnau X, Cruz-Orive LM, Nuno-Ballesteros JJ (2010). A new

rotational integral formula for intrinsic volumes in space forms. Adv

Appl Math 44:298--308.

Gundersen HJG (1977). Notes on the estimation of the numerical density

of arbitrary profiles: the edge effect. J Microsc 111:219--223.

Gundersen (1979). Estimation of tubule or cylinder LV, SV and VV on

thick sections. J Microsc 117:333--45.

Gundersen HJG (1986). Stereology of arbitrary particles. A review of

unbiased number and size estimators and the presentation of some new

ones, in memory of William R. Thompson. J Microsc 143:3--45.

Gundersen HJG (1988). The nucleator. J Microsc 151:3--21.

Gundersen HJG (1992). Stereology: the fast lane between neuroanatomy and

brain function --- or still only a tightrope? Acta Neurol Scand


Gundersen HJG (2002) The smooth fractionator. J Microsc 207:191--210.

Gundersen HJG, Bagger P, Bendtsen TF, Evans SM, Korbo L, Marcussen N,

Mller A., Nielsen K, Nyengaard JR, Pakkenberg B, Srensen FB, Vesterby A,

West MJ (1988). The new stereological tools: Disector, fractionator,

nucleator and point sampled intercepts and their use in pathological

research and diagnosis. APMIS 96:857--881.

Gundersen HJG, Boyce RW, Nyengaard JR, Odgaard A (1993). The conneulor:

unbiased estimation of connectivity using physical disectors under

projection. Bone 14:217--222.

Gundersen HJG, Boysen M, Reith A (1981). Comparison of semiautomatic

digitizer-tablet and simple point counting performance in morphometry.

Virchows Arch (Cell Pathol) 37:317--325.

Gundersen HJG, Jensen EB (1983). Particle sizes and their distributions

estimated from line- and point-sampled intercepts. Including graphical

unfolding. J Microsc 131:291--310.

Gundersen HJG, Jensen EB (1985). Stereological estimation of the

volume-weighted mean volume of arbitrary particles observed on random

sections. J Microsc 138:127-142.

Gundersen HJG, Jensen EB (1987). The efficiency of systematic sampling

in stereology and its prediction. J Microsc 147:229--263.

Gundersen HJG, Jensen EB, sterby R (1978). Distribution of membrane

thickness determined by lineal analysis. J Microsc 113:27--43.

Gundersen HJG, sterby R (1981). Optimizing sampling efficiency of

stereological studies in biology: or 'Do more less well!'. J Microsc


Hadwiger H (1957). Vorlesungen uber Inhalt,

Oberflache und Isoperimetrie. Berlin:


Harding EF, Kendall DG (eds.) (1974). Stochastic

geometry. A tribute to the memory of Rollo Davidson.

London: J Wiley.

Haug H (1985). Gibt es Nervenzellverluste wahrend der Alterung in der


Hirnrinde? Ein morphometrischer Beitrag zu dieser Frage. Nervenheilkunde


Haug H (1986). History of neuromorphometry. J Neurosci Meth 18:1--17.

Hilliard JE (1962). Specification and measurement of structural

anisotropy. Trans Metall Soc AIME 224:1201--11.

Hilliard JE (1967) Determination of structural anisotropy. In: Elias H,

ed., pp. 219--227.

Hobolth A, Jensen EBV (2002). A note on design-based versus model-based

variance estimators in stereology. Adv Appl Prob 34:484--90.

Horvith DG, Thompson DJ (1952). A generalization of sampling without

replacement from a finite universe. J Amer Statist Ass 47:663--685.

Howard V (1986). Rapid nuclear volume estimation in malignant melanoma

using point sampled intercepts in vertical sections: a review of the

unbiased quantitative analysis of particles of arbitrary shape. In: Mary

JY, Rigaut JP, eds. Quantitative image analysis in cancer cytology and

histology, pp. 245--256. Amsterdam: Elsevier.

Howard CV, Cruz-Orive LM, Yaegashi H (1992). Estimating neuron dendritic

length in 3D from total vertical projections and from vertical slices.

Acta Neurol Scand Suppl 137:14--19.

Howard CV, Reed MG (2005). Unbiased stereology. Three-dimensional

measurement in microscopy, 2nd ed, Oxford: Bios/ Taylor & Francis.

Howard CV, Reid S, Baddeley AJ, Boyde A (1985). Unbiased estimation of

particle density in the tandem scanning reflected light microscope. L

Microsc 138:203--212.

Hyksova M, Kalouskova A, Saxl I

(2012). Early history of geometric probability and stereology. Image

Anal Stereol 31, 1--16.

Jensen EB (1987). Design- and model-based stereological analysis of

arbitrarily shaped particles. Scand J Statist 14:161--180.

Jensen EB (1991). Recent developments in the stereological analysis of

particles. Ann Inst Statist Math 43:455--68.

Jensen EBV(1998). Local stereology. Singapore: World Scientific.

Jensen EB, Gundersen HJG (1982). Stereological ratio estimation based on

counts from integral test systems. J Microsc 125:51--66.

Jensen EB, Gundersen HJG (1985). The stereological estimation of moments

of particle volume. J Appl Prob 22:82--98.

Jensen EB, Gundersen HJG (1987). Stereological estimation of surface

area of arbitrary particles. Acta Stereol 6, Suppl. III:25--30.

Jensen EB, Gundersen HJG (1989). Fundamental stereological formulae

based on isotropically orientated probes through fixed points with

applications to particle analysis. J Microsc 153:249--67.

Jensen EBV, Gundersen HJG (1993). The rotator. J Microsc 170:35--44.

Jensen EB, Gundersen HJG, sterby R (1979). Determination of membrane

thickness distribution from orthogonal intercepts. J Microsc 115:19--33.

Jensen EBV, Kiderlen M (eds.) (2017) Tensor valuations and their

applications in stochastic geometry and imaging. Berlin: Springer.

Jensen EB, Sundberg R (1986). Statistical models for stereological

inference about spatial structures: on the applicability of best linear

unbiased estimators in stereology. Biometrics, 42:735--751.

Kanatani KI (1984). Stereological determination of structural

anisotropy. Int J Engng Sci 22:531--46.

Karlsson LM, Cruz-Orive LM (1997). Estimation of mean particle size from

single sections. J Microsc 186:121--32.

Kendall MG, Moran PAP (1963). Geometrical

Probability. London: Ch.Griffin.

Kiang T (1972). An old Chinese way of finding the volume of a sphere.

Math Gazette 56:88--91.

Kieu K (1997). Three Lectures on systematic geometric sampling. Memoirs

No. 13/1997. Aarhus: Department of Theoretical Statistics, University of


Kieu K, Jensen EBV (1993). Stereological estimation based on isotropic

slices through fixed points. J Microsc 170:45--51.

Kieu K, Mora M (2006). Precision of stereological planar area

predictors. J Microsc 222: 201--211.

Kieu K, Mora M (2009). Advances on the precision of several

stereological volume estimators. In: Capasso V, Aletti G, Micheletti A,

eds. Proc 10th Eur Congr Stereol Image Anal. Bologna: Esculapio.

Kieu K, Souchet S, Istas, J (1999). Precision of systematic sampling and

transitive methods. J Statist Plan Inf 77:263--279.

Kubinova L, Janacek J (2001). Confocal microscopy and stereology:

estimating volume, number, surface area and length by virtual test

probes applied to three-dimensional images. Microsc Res Techn 53,


Kubinova L, Mao XW, Janacek J, Archembeau JO (2003). Stereology

techniques in radiation biology. Radiation Research 160:110--19.

Lantuejoul C (1991). Ergodicity and integral range. J Microsc


Lantuejoul C (2002). Geostatistical simulation. Models and algorithms.

Berlin: Springer.

Maletti GM, Wulfshon D 2006). Evaluation of variance models for

fractionator sampling of trees. J Microsc 222:228--41.

Mardia KV (1972). Statistics of directional data. London: Academic


Matern B (1960). Spatial variation. Meddelanden fran Statens

Skogsforskningsinstitut, 49(5):1--114. 2nd ed. Berlin: Springer-Verlag,

Matheron G (1965). Les variables regionalisees et leur estimation.

Paris: Masson et Cie.

Matheron G (1967). Elements pour une theorie des milieux poreux. Paris:

Masson et Cie.

Matheron G (1971). The theory of regionalized variables and Its

applications. Les Cahiers du Centre de Morphologie Mathematique de

Fontainebleau, No.5. Fontainebleau: Ecole Nationale Superieure des Mines

de Paris.

Matheron G (1975). Random sets and integral geometry. New York: J Wiley.

Mathieu O, Cruz-Orive LM, Hoppeler H, Weibel ER (1981). Measuring error

and sampling variation in stereology: comparison of the efficiency of

various methods for planar image analysis. J Microsc 121:75-88.

Mathieu O, Cruz-Orive LM, Hoppeler H, Weibel ER (1983). Estimating

length density and quantifying anisotropy in skeletal muscle

capillaries. J Microsc 131:131-46.

Mattfeldt T, Clarke A, Archenhold G (1994). Estimation of the

directional distribution of spatial fibre processes using stereology and

confocal scanning laser microscopy. J Microsc 173:87--101.

Mattfeldt T, Frei H, Rose C (1993). Second-order stereology of benign

and malignant alterations of the human mammary gland. J Microsc


Mattfeldt T, Mall G (1984). Estimation of length and surface of

anisotropic capillaries. J Microsc 135:181--90.

Mattfeldt T, Mall G, Gharehbaghi H, Moller P (1990). Estimation of

surface area and length with the orientator. J Microsc 159:301--17.

Mayhew TM, Cruz-Orive LM (1974) Caveat on the use of the Delesse

principle of

areal analysis for estimating component volume densities. J Microsc


McMillan AM, Srensen FB (1992). The efcient and unbiased estimation of

nuclear size variability using the `selector'. J Microsc 165:433--37.

Mecke J (1981). Stereological formulas for manifold processes. Prob Math

Statist 2:31--35.

Michel RP, Cruz-Orive LM (1988). Application of the Cavalieri

principle and vertical sections method to lung: estimation of volume and

pleural surface area. J Microsc 150:117--136.

Miles RE (1972). Multidimensional perspectives in stereology. J Microsc


Miles RE (1974) On the elimination of edge effects in planar sampling.

In: Harding EF, Kendall DG, eds., pp.228--47.

Miles RE (1978a). The sampling, by quadrats, of planar aggregates. J

Microsc 113:257--267.

Miles RE (1978b). The importance of proper model specification in

stereology. In: Miles RE, Serra J, eds., pp. 115--136.

Miles RE (1979) Some new integral geometric formulae, with stochastic

applications. J Appl Prob 16:592--606.

Miles RE (ed.) (1987). The Commemorative-Memorial Volume. Twenty-five

years of stereology. Acta Stereol 6/II.

Miles RE, Davy PJ (1976). Precise and general conditions

for the validity of a comprehensive set of stereological

fundamental formulae. J Microsc 107:211--26.

Miles RE, Davy PJ (1977). On the choice of quadrats in

stereology. J Microsc 110:27--44.

Miles RE, Serra J (eds.) (1978). Geometrical Probability and Biological

Structures: Buffon's 200th Anniversary. Proceedings, Paris 1977. Lect

Not Biomath No.23. Berlin: Springer.

Moran PAP (1950). Numerical integration by systematic

sampling. Proc Camb Phil Soc 46:111--115.

Mouton PR, Gokhale AM, Ward NL, West MJ (2002) Stereological length

estimation using spherical probes. J Microsc 206:54--64,

Murthy NN (1967). Sampling theory and

methods. Calcutta: Statistical Publishing Society.

Nyengaard JR, Gundersen HJG (1992). The isector: a simple and direct

method for generating isotropic, uniform random sections from small

specimens. J Microsc 165:427--431.

Nyengaard JR, Marcussen N (1993). The number of glomerular

capillaries estimated by an unbiased and efficient stereological method.

J Microsc 171:27--37.

Ogbuihi S, Cruz-Orive LM (1990). Estimating the total number of

lymphatic valves in infant lungs with the fractionator. J Microsc


Ohser J (1981). A remark on the estimation of the rose of directions of

fibre processes. Math Operationsforsch Statist, Ser Statistics


Ohser J, Muckilch F (2000). Statistical

analysis of microstructures in materials science.

Chichester: J Wiley.

Ohser J, Nagel W (1996). The estimation of the Euler-Poincare

characteristic from observations on parallel sections. J Microsc


Ohser J, Schladitz K (2009) 3D images of materials structures.

Processing and Analysis. Weinheim: Wiley-VCH.

Pache JC, Roberts N, Zimmerman A, Vock P,

Cruz-Orive LM (1993). Vertical LM sectioning and parallel CT scanning


for stereology: application to human lung. J Microsc 170: 9--24.

Petkantschin P (1936). Zusammenhange zwischen den Dichten der linearen

Unterraume im n-dimensionalen Raum. Abh Math Sem Hamburg 11:249--310.

Rafati AH, Ziegel JF, Nyengaard JR, Jensen EBV (2016). Stereological

estimation of particle shape and orientation from volume tensors. J

Microsc 263:229--37.

Redenbach C, Wirjadi O, Rief S, Wiegmann A (2011). Modeling of ceramic

foams for filtration simulation. Adv Engng Mat 13:171--7.

Ripley BD (1977). Modelling spatial patterns. J Roy Statist Soc B


Ripley BD (1981). Spatial statistics. New York: J. Wiley.

Roberts N, Howard CV, Cruz-Orive LM, Edwards RHT (1991). The application

of total vertical projections for the unbiased estimation of the length

of blood vessels and other structures by magnetic resonance imaging.

Magn Res Imaging 9:917--25.

Rosiwal AK (1898). Uber geometrische Gesteinsanalysen. Ein einfacher Weg

zur ziffermassigen Feststellung des Quantitasverhaltnisses der

Mineralbestandtheile gemengter Gesteine. Verhandlungen KK Geol

Reichsanstalt 5-6:143--75.

Saltykov SA (1946). The method of intersections in metallography.

Zavodskaya Laboratorija,

(9--10):816--825. (In Russian).

Saltykov SA (1958). Stereometric metallography (in Russian), 2nd ed.

Moscow: State Publishing House for Metals and Sciences. (German

translation of 3rd ed, 1974, Leipzig: VEB Deutscher Verlag fur


Saltykov (1967). A stereological method for measuring the specific

surface area of metallic powders. In: Elias H, ed., pp. 63--4.

Sandau K (1987). How to estimate the area of a surface using the spatial

grid. Acta Stereol 6/III:31--36.

Santalo LA (1939). Geometria integral de figuras ilimitadas. Publ Inst

Mat Univ Litoral I/2:1--58. Rosario, Rep.Argentina.

Santalo LA (1956). Sur la mesure des espaces lineaires qui coupent un

corps convexe et problemes qui s'y rattachent. Colloque sur les

questions de realite en geometrie, Liege 23--26 mai 1955, pp.177--190.

Paris: Masson.

Santalo LA (1976). Integral geometry and geometric

probability. Addison-Wesley, Reading, Massachusetts.

Santalo LA (2009). Selected Works. Naveira AM, Reventos A, eds. Berlin:


Schneider R, Weil W (2008). Stochastic and integral

geometry. Berlin: Springer.

Scriven RA, Williams HD (1965). The derivation of angular distributions

of planes by sectioning methods. Trans Metall Soc AIME 233:1593--1602.

Serra J (1982) Image analysis and mathematical morphology. London:

Academic Press.

Shay J (1975). Economy of effort in electron microscope morphometry.

Amer J Pathol 81:503--512.

Smith CS, Guttman L (1953) Measurement of internal boundaries in

threedimensional structures by random sectioning. Trans AIME (J Metals)


Snedecor GW, Cochran WG (1980). Statistical methods. 8th ed. Ames, Iowa:

The Iowa State University Press.

Sommerville DMY (1958).An introduction to the geometry of N dimensions.

New York: Dover.

Souchet S (1995). Precision de l'estimateur de Cavalieri. Rapport de

stage, D.E.A. de Statistiques et Modeles Aleatoires Appliques a la

Finance, Universite Paris-VII. Laboratoire de Biometrie,


Spektor AG (1960). Estimation of the surface of structural components by

the method of intersections with a cycloid (in Russian). Zavodskaja

Laboratorija 26:303--305.

Steinhaus H (1930) Zur Praxis der Rektifikation und zum Langenbegriff.

Ber Akad Wiss Leipzig 82:120--130.

Sterio DC (1984). The unbiased estimation of number and sizes of

arbitrary particles using the disector. J Microsc 134:127--136.

Stoyan D (1985). Stereological determination of orientations,

second-order quantities and correlations for random spatial fibre

systems. Biom J 27:411--25.

Stoyan D, Kendall WS, Mecke J (1995). Stochastic geometry and its

applications, 2nd.ed. Chichester: J. Wiley.

Srensen FB (1991). Stereological estimation of mean and variance of

nuclear volume from vertical sections. J Microsc 162:203--229.

Thioulouse J, Mathy B, Ploye H (1985). Estimation d'un volume a partir

de coupes seriees: sous-echantillonnage, covariogramme transitif et

calcul de precision. Mikroskopie 42:215--224.

Thompson WR (1932). The geometric properties of microscopic

congurations. I. General aspects of projectometry. Biometrika 24:21--26.

Thompson WR, Hussey R, Matteis JT, Meredith WC, Wilson GC, Tracy FE

(1932). The geometric properties of microscopic congurations. II.

Incidence and volume of islands of Langerhans in the pancreas of a

monkey. Biometrika, 24:27 38.

Thomson E (1930). Quantitative microscopic analysis. J Geol 38:193--222.

Thorisdottir O, Kiderlen M (2014). The invariator principle in convex

geometry. Adv Appl Math 58:63--87.

Thorisdottir O, Rafati AH, Kiderlen M (2014). Estimating the surface

area of nonconvex particles from central planar sections. J Microsc


Tinajero-Bravo M, Eslava-Gomez G, Cruz-Orive LM (2014). Conditions for

exact Cavalieri estimation. Image Anal Stereol 33:219--224.

Tobias RT (1981). Volume: some historical perspectives. Math Gazette


Tomkeieff ST (1945). Linear intercepts, areas and volumes. Nature,


Correction in p. 107.

Underwood EE (1970). Quantitative

stereology. Reading, Mass.: Addison-Wesley.

Varga O (1935). Integralgeometrie 3. Croftons Formeln fur den Raum. Math

Z 40:387--405.

Vesterby A, Kragstrup J, Gundersen HJG, Melsen F (1987). Unbiased


estimation of surface density in bone using vertical sections. Bone


Voss F, Cruz-Orive LM (2009). Second moment formulae for geometric

sampling with test probes. Statistics 43:329--365.

Weibel ER (1963). Morphometry of the human lung. New York: Academic


Weibel ER (1967). Structure in space and its appearance on sections. In:

Elias H, ed. pp. 15--26.

Weibel ER (1969). Stereological principles for morphometry in electron

microscopic cytology. Int Rev Cytol 26:235--302.

Weibel ER (1979). Stereological Methods. Vol. 1: Practical methods for

biological morphometry. London: Academic Press.

Weibel ER (1980). Stereological Methods. Vol. 2: Theoretical

Foundations. London: Academic Press.

Weibel ER, Gehr P, Cruz-Orive LM, Muller AE, Mwangi DK, Haussener V

(1981) Design of the mammalian respiratory system. IV. Morphometric

estimation of pulmonar diffusing capacity; critical evaluation of a new

sampling method. Resp Physiol 44:39--59.

Weibel ER, Knight BW (1964). A morphometric study on the thickness of

the pulmonary air-blood barrier. J Cell Biol 21:367--84.

West MJ (2012). Basic stereology for biologists

and neuroscientists. New York: CSH Laboratory Press.

West MJ, Slomianka L, Gundersen HJG (1991). Unbiased

stereological estimation of the total number of neurons in the

subdivisions of the rat hippocampus using the optical fractionator. Anat

Record 231:482--497.

Wicksell SD (1925). The corpuscle problem. A mathematical study of a

biometric problem. Biometrika 17:84--89.

Wicksell SD (1926). The corpuscle problem. Second Memoir. Case of

ellipsoidal corpuscles. Biometrika 18:152--172.

Wirjadi O, Schladitz K, Easwaran P, Ohser J (2016). Estimating fibre

direction distributions of reinforced composites from tomographic

images. Image Anal Stereol 35:167-79.

Wulfsohn D, Aravena Zamora F, Potin Tellez C, Zamora Lagos I,

Garcia-Finana M (2012). Multilevel systematic sampling to estimate total

fruit number for yield forecastst. Precision Agric 12:256--275.

Wulfsohn D, Knust J, Ochs M, Nyengaard JR, Gundersen HJG (2010a).

Stereological estimation of the total number of ventilatory units in

mice lungs. J Microsc 238:75--89.

Wulfsohn D, Nyengaard JR, Gundersen HJG, Cutler AJ, Squires TM (1999).

Non-destructive, stereological estimation of plant root lengths,

branching pattern and diameter distribution. Plant and Soil 214:15--26.

Wulfsohn D, Sciortino M, Aaslyng JM, Garcia-Finana M (2010b).

Non-destructive, stereological estimation of canopy surface area.

Biometrics 66:159--168.

Youngs TA, Boyce TW, Mosekilde L, Sgaard CH,

Paddock CL, Gundersen HJG (1994). Direct stereological estimation of

--D connectivity density in human illiac cancellous bone: the

effect of age and sex. Proc 6ECS

Prague 1993. Acta Stereol 13/1:55--60.




How to Cite

Cruz-Orive, L. M. (2017). STEREOLOGY: A HISTORICAL SURVEY. Image Analysis and Stereology, 36(3), 153–177.



Review Article