Image Anal Stereol 2008;27:17-22 Original Research Paper COMPARATIVE PRECISION OF THE PIVOTAL ESTIMATORS OF PARTICLE SIZE Luis M Cruz-Orive Department of Mathematics, Statistics and Computation, Faculty of Sciences, University of Cantabria, Avda. Los Castros s/n, E-39005 Santander, Spain (Accepted January 14, 2008) ABSTRACT The pivotal estimators of the surface area and the volume of a generic “particle” are based on a point sampled test line on an isotropic pivotal plane through a fixed point called the pivotal point. The purpose of this paper is to compare the precision of the pivotal estimator of surface area against the surfactor on the one hand, and of the pivotal estimator of volume against the nucleator on the other. For the sake of a tractable but informative model we compute exact variances for a spherical particle with an eccentric nucleolus. Keywords: convex body, invariant test line, particle, pivotal point, pivotal section, point sampled test line, stereology, support function, support set, surface area, volume. INTRODUCTION A new representation has been described for the invariant density of a straight line L31 in R3 (Varga, 1935; Cruz-Orive, 2005). With reference to Fig. 1, consider a fixed point O, called the pivotal point, an isotropic plane L32[0] through O, called the pivotal plane, and a point z endowed with the uniform density on L32[0]. The classical construction establishes that a straight line L31 through z and normal to the pivotal plane is invariant with respect to motions. The new representation implies that the density of a straight line L21[·] normal to the axis Oz at the point z, and contained in the pivotal plane, is also invariant with respect to motions in R3. In other words, a point sampled test line L21[·] in a pivotal plane has exactly the same properties as an invariant test line L3 in R3. Fig. 1. A point sampled test line L21[·] in an isotropic pivotal plane L32[0] through a fixed pivotal point O is equivalent to the classical invariant test line L31 in three dimensions. As described in Cruz-Orive (2005), point sampled test lines (not to be confused with point sampled intercepts, see for instance Howard and Reed, 2005) have multiple applications, notably to estimate individual particle surface area and volume simultaneously. It seems therefore opportune to explore the relative precision of the pivotal estimators against their current competitors, namely the surfactor (Jensen and Gundersen, 1989) and the nucleator (Gundersen, 1988): this is the purpose – and the new contribution – of the present paper. For the sake of a tractable but still orientating treatment, we adopt a simple geometric particle model. The subsequent results are not intended to be directly applicable to real objects; they rather constitute a tentative analysis of the relative efficiency of the pivotal estimators of surface area and volume. BACKGROUND AND MODEL PIVOTAL ESTIMATORS Here we summarize some results from Cruz-Orive (2005). Consider a particle Y, namely a compact and connected subset of R3 with piecewise smooth boundary, with (finite) surface area S(¶Y) and positive volume V(Y). Embed Y in a ball B3 of equatorial area a centred at a pivotal point O. Take a pivotal plane L32[0] through O with isotropic normal direction t, and generate a uniform random (UR) point z within the equatorial intersection circle B2,t = B3 n L32[0]. 17 Cruz-Orive LM: Precision of the pivotal estimators Fig. 2. Theoretical construction of a point sampled test line hitting a particle. Let (r, a) denote the polar coordinates of z with respect to O in the pivotal plane. Finally, draw a point sampled test line L2,-, through z and normal to the axis Oz (see Fig. 2). The joint probability element of the triplet (r,a,t) is P(dr,da,dt) rdrda dt a 2p because z is UR on B2,t and independent from the direction t, which is UR on the unit hemisphere. The test line L2,, is effectively isotropic uniform random (IUR) hitting B3, and therefore, if L2,, hits Y C B3, it is also IUR hitting Y (Miles and Davy, 1976). Define the random observationsI := number of intersections between L2,-, and dY, and I := intercept length determined in Y by L2,y Resorting to the pertinent Crofton formulae it follows that S(dY) = 2aI, V{Y) = aL (1) of the support function h(w) of the section. Then it holds that S(¶Y)=4E area H < area I HYnL3 ) > (2) the expectation being over the isotropic orientation distribution of the pivotal plane. J0 are unbiased estimators of the particle surface area and volume, respectively. The preceding construction will be useful in the sequel to compute the variances of the estimators (Eq. 1) for a simple model. In practice, however, we may lay a system of test points with a fundamental tile of area a uniformly at random with an arbitrary orientation on the pivotal plane, and then draw the relevant point sampled test lines (Fig. 3). The corresponding unbiased estimators have the same form (Eq. 1). A special case of interest is that of a convex particle Y with an interior pivotal point or “nucleolus” O. Upon the isotropic pivotal section YnL 320, which is almost surely convex, construct the corresponding support set HYnL3 , namely the set of points enclosed by the graph respectively Fig. 3. Point sampled test lines upon a test system on an isotropic pivotal plane, with pivotal point at O. The point z is UR within the fundamental tile J0 of area a. By Eq. 1 the surface area of the particle would be estimated by 12a, and its volume by the total intercept length times a. A SIMPLE MODEL FOR VARIANCE COMPARISONS Our particle model is the unit ball Y with an eccentric nucleolus O at a fixed distance k € [0,1] from the centre of the ball (see Fig. 4). Take an isotropic pivotal plane L320i through O. The corresponding pivotal section is a circle which depends only on k and on the colatitude 0 of the normal to the pivotal plane. Isotropy means that the probability element of 0 is P(d0)= sinodo, 0o and therefore, 4 1 CV2 Ssm(w,6;k) >--k2^------k4 3 15 >CV2(Sw{w,e-k)), see Fig. 7, in which the constant c(k) in the right hand side of Eq. 9 has been evaluated numerically. _ _ _ 19 Cruz-Orive LM: Precision of the pivotal estimators Remark. It is possible to consider more precise versions of the estimators (Eq. 4) and (Eq. 7) by averaging out with respect to the marginal probability element P(dw) = dw/(2p) for each q. For the pivotal estimator we obtain 4 times the area of the support set H mL3 namely, S ˆpiv(0;k) = 4A(0;k) 4n1-k2 + 3 k2sin2e (10) as expected by virtue of Eqs. 2 and 3. Less obvious, however, is the fact that the corresponding expectation of the surfactor estimator (Eq. 7) with respect to P(dw) yields the same result (Eq. 10). VOLUME ESTIMATORS THE PIVOTAL ESTIMATOR OF VOLUME We adopt the second estimator in Eq. 1 with a = S(¶Y)/4. This choice implies that only positive intercept lengths are recorded, namely L > 0. Equivalently, the sampling point z is chosen uniformly of at random in the interior of the support set H the pivotal section, see Fig. 6. Thus, Yr\L 2[0] V ˆpiv(p,w, Q;k) = -S(dY) -L(p,w,ô;k) , (11) is an unbiased estimator of V(Y), where L(p,w,d;k) = [r2(e;k)-(p + l(e;k)cosw)2]1/2 . (12) On the other hand, the joint probability element of (p,w,0)is, P(dp,dw,d0|L>0) pdpdw S(dY)/4 • sinodo, pe[0,h(w,6;k)) ,d£[0,n/2],w£[0,2n). (13) Note that the unbiasedness of the estimator (Eq. 11) is a consequence of the well known stereological relation EL = 4V/S, (Hostinsky, 1925). The estimator (Eq. 11) is of course not practical because, for the ball particle model considered here, if the factor S(dY) in the right hand side of Eq. 11 is known, then V(Y) is also known. The estimator is nonetheless useful for the intended purposes. Nucleator Fig. 6. Pivotal section circle of the particle model shown in Fig. 4, with the relevant parameters involved in the pivotal and in the nucleator estimators of particle volume, respectively. With the aid of Mathematica®, a direct evaluation from Eqs. 11-13 yields, CV 2 V ˆpiv(p,w,e;k)) (14) independently of the distance k of the pivotal point from the particle centre. The preceding result could have been anticipated bearing in mind that the estimator (Eq. 11) is proportional to the length of the intercept determined in Y by an IUR test line hitting it. This is because a point sampled test line on the pivotal plane is effectively an invariant test line in R3 for any location of the pivotal point, as stressed in the Introduction. The probability element of an IUR chord length L of the unit ball is P(dL) = LdL/2, 0 < L < 2, whereby CV2(L) = 1/8. The argument is general: CV2 V ˆpiv(p,w, 6;k)) is independent of the position of the pivotal point for any arbitrary particle Y; it only depends on the first two moments of the IUR intercept length of Y. If Y is not convex, then the relevant intercept may consist of several separate segments, in which case L(p,w,ô;k) is the sum of the corresponding lengths. Remark. It is instructive to realise that P(dp,dw|0,L>0) P(d0|L>0) pdpdw A(6;k)sin6d6 S(dY)/4 and the product of the preceding two conditional probability elements yields the joint one in Eq. 13. Here A(q;k) is defined as in Eq. 3. _ _ _ _ 20 Image Anal Stereol 2008;27:17-22 0.0 0.2 0.4 0.6 0.8 1.0 k = Distance from centre of unit ball to pivotal point 3 2 1 0.0 0.2 0.4 0.6 0.8 1.0 k = Distance from centre of unit ball to pivotal point Fig. 7. Upper panel: Graphs of the right hand sides of Eqs. 6 and 9, respectively, showing that the pivotal estimator of surface area is more efficient than the surfactor for the particle model described above (Fig. 4). Lower panel: Graphs of the right hand sides of Eqs. 14 and 16, respectively, showing that the pivotal estimator of volume is more efficient than the nucleator when k > 0.203···. THE NUCLEATOR With reference to Fig. 6 we consider the “one-ray” nucleator estimator V ˆnuc(w,0;k) 47T 3 ·r3(w,q;k) (15) which is unbiased for V(Y), (Gundersen, 1988). The expression forr(w,q;k) is given in Eq. 8. The relevant probability element is given by Eq. 5, whereby we obtain 3 1 CV2 V ˆnuc(w,6;k) = 3k2 + -k4------k6 , 5 35 k G [0,1] , (16) see Fig. 7. In practice it is much more efficient to consider two rays p(w,6;k) and p(w -\-n,0;k), but in Eq. 15 we have chosen one because then the efficiency comparison with the pivotal estimator (Eq. 11) looks more fair. For the two ray nucleator we obtain CV V ˆ 9 4 18 6 -k------k , k€ 0,1 , (17) 5 35 c(w,q;k) for the ball particle model considered here. DISCUSSION At least for the unit ball particle model considered here with a nucleolus at a distance k from the centre: (i) The one-ray pivotal estimator (Eq. 4) of surface area is simpler and more efficient than the surfactor. (ii) The one-intercept pivotal estimator (Eq. 11) of volume is more efficient than the one-ray nucleator for k > 0.203···. Even for the two-ray nucleator (Eq. 17) the pivotal estimator is more efficient for k > 0.542···. If Y is a convex body, then for the first estimator (Eq. 1) it is easy to show that CV 2 (S(dY)') S(dY)/4 -1 (18) On the other hand, if Y is the unit ball then for the second estimator (Eq. 1) we have CV 2 V ˆ (dY)') S(dY)/4 -1 (19) The preceding formulae take into account that the estimators (Eq. 1) may be equal to zero if the test line fails to hit Y. If we set a = S(¶Y)/4, then Sˆ(¶Y) = S(Y) because I = 2 with probability 1, and trivially CV2(Sˆ(¶Y)) = 0. On the other hand Vˆ (Y) is in this case the right hand side of Eq. 11, and Eq. 19 becomes Eq. 14. Finally we note that, unlike Eq. 18, Eq. 6 depends on the position of the pivotal point because the estimator (Eq. 4) is not a direct function of the intersection between the test line and Y. ACKNOWLEDGMENTS This research was supported by the Spanish Ministry of Education and Science I+D Project no. MTM2005-08689-C02-01. REFERENCES Cruz-Orive LM (2005). A new stereological principle for test lines in 3D. J Microsc 219:18–28. 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