• Luis Manuel Cruz-Orive University of Cantabria (E-Santander)
  • Ximo Gual-Arnau Department of Mathematics-INIT, University Jaume I, E-12071 Castell\'o



invariator, peak-and-valley formula, stereology, surface area, test line weighting, volume


The invariator is a method to generate a test line within an isotropically oriented plane through a fixed point, in such a way that the test line is effectively motion invariant in three dimensional space. Generalizations exist for non Euclidean spaces. The invariator design is convenient to estimate surface area and volume simultaneously. In recent years a number of new results have appeared which call for an updated survey. We include two new estimators, namely the a posteriori weighting estimator for surface area and volume, and the peak-and-valley formula for surface area.


Auneau J, Jensen EVB (2010) Expressing intrinsic volumes as rotational integrals. Adv Appl Math 45:1--11.

Cruz-Orive LM (1987) Particle number can be estimated using a disector of unknown thickness: the selector. J Microsc 145:121--42.

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 41:49--98.

Cruz--Orive LM (2005) A new stereological principle for test lines in 3D. J Microsc 219:18--28.

Cruz-Orive LM (2008) Comparative precision of the pivotal estimators of particle size. Image Anal Stereol 27:17--22.

Cruz-Orive LM (2009a) The pivotal tessellation. Image Anal Stereol 28:101--05.

Cruz-Orive LM (2009b) Stereology: old and new. In: Proceedings of the 10th European Congress of ISS, ( V.Capasso et al.). The MIRIAM Project Series, ESCULAPIO Pub Co, I-Bologna.

Cruz-Orive LM (2011) Flowers and wedges for the stereology of particles. J Microsc 243:86--102.

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, Ramos-Herrera ML, Artacho-Perula E (2010) Stereology of isolated objects with the invariator. J Microsc 240:94--110.

De-lin R (1994) Topics in integral geometry. Singapore: World Scientific.

Dvorak J, Jensen EBV (2013) On semiautomatic estimation of surface area. J Microsc 250:142--57.

Gual-Arnau X, Cruz-Orive LM (2002) Variance prediction for pseudosystematic sampling on the sphere. Adv Appl Prob 34:469--83.

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--28.

Gual-Arnau X, Cruz-Orive LM (2015). New rotational integrals in space forms, with an application to surface area estimation. Internal Report 1/2015, April 2015. Departamento de Matem'aticas, Estad'istica y Computaci'on, Universidad de Cantabria, E-Santander. Submitted to Applications of Mathematics.

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

Hansen LV, Nyengaard JR, Andersen JB, Jensen EBV (2011) The semi-automatic nucleator. J Microsc 242:206--15.

Jensen EBV (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 (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, Rataj J (2008) A rotational integral formula for intrinsic volumes. Adv Appl Math 41:530--60.

Lockwood EH (1961) A book of curves. CUP, Cambridge.

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.

Petkantschin B (1936) Zusammenhange zwischen den dichten der linearen unterraume im n-dimensionalen raum. Abh Math Sem Hamburg 11:249--310.

Santalo LA (1976) Integral geometry and geometric probability. Addison-Wesley, Reading, Massachusetts.

Schneider R, Weil W (2008) Stochastic and integral geometry. Springer-Verlag, Berlin.

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 255:49--64.

Varga O (1935) Integralgeometrie 3. Croftons formeln fur den raum. Math Zeitschrift 40:387--405.




How to Cite

Cruz-Orive, L. M., & Gual-Arnau, X. (2015). THE INVARIATOR DESIGN: AN UPDATE. Image Analysis and Stereology, 34(3), 147–159.



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