Image Anal Stereol 2000;19:15-18 Original Research Paper
ON THE VARIANCE OF LOCAL STEREOLOGICAL VOLUME ESTIMATORS
Eva B Vedel Jensen
Laboratory for Computational Stchastic Department of Mathematical Sciences University of Aarhus (Accepted November 4, 1999)
ABSTRACT
In the present paper, the variance of local stereological volume estimators is studied. For isotropic designs, the variance depends on the shape of the body under study and the choice of reference point. It can be expressed in terms of an equivalent star body. For a collection of triaxial ellipsoids the variance is determined by simulation. The problem of estimating particle size distributions from central sections through the particles is also discussed.
Keywords: local stereology, particle size distribution, star body, triaxial ellipsoid, variance.
INTRODUCTION
Local stereology is a collection of sampling designs based on sections through a reference point of the spatial structure under study. Such sections are usually called central sections. The local methods are used in themicroscopical study of biological tissue in cases where the tissue is transparent and physical sections can be replaced by optical sections.
An overview of local stereology has recently been given in Jensen (1998). A comprehensive treatment of local stereological volume estimators can be found in Jensen (1998, p. 105-111). In Jensen and Petersen (1999), it is discussed when the volume of a body in R" can be determined without error by a local stereological estimator and a general variance formula is derived.
In the present paper, we study the variance of local stereological volume estimators for bodies in R3. In particular, it is investigated when such estimators can be used in the stereological inference of particle volume distributions.
THE LOCAL VOLUME ESTIMATORS
Let us start by a short presentation of the local stereological volume estimators. For a body X c R", the local stereological estimator of its volume V(X), based on information in an isotropic /?-subspace Lp, containing a fixed r-subspace Lr, takes the form
n-p  T(P~r)
,p,  v         p       '             T^ n-r\JxnLp\   L r
V   2   /
where   || ||   is  the  Euclidean  norm,   n       is the
orthogonal projection onto £r and d^ is the element
of ^–dimensional volume measure in Lp. When convenient we will use the short notation Vnpr (X) for
this estimator.In this paper we will restrict attention to R3. In R3, there are three local stereological volume estimators which will be considered in more detail below. Without loss of generality, we can assume that the reference point through which the sections pass is the origin O.
The first local stereological estimator of V(X) is based on information in an isotropic line Z, through O and is given by
V3,1,0 ( Xf]L1;O ) = 27t\     |x|2    1.
If X n Z; consists of a countable number of line-segments, then
ˆ 3,1,0 (xn4;O ) = 23   X ( -1 )a(x)H3 ,
where a(x)xMnLl is a sequence of 0's and 1's; for details, see Jensen (1998, p. 107-108). In particular, if X is star-shaped at O and O e X, then
V3,1,0 ( Xnspan {o)} ;0) = -7t[px (co3 + px ( -(o3  (1)
where co e S2, the unit sphere in R3, and prfco) is the distance from O to the boundary of X in the direction co
es2.
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Vedel Jensen EB: On the variance of local stereological volume estimators
The second local stereological estimator of V(X), based on an isotropic plane L2 through O, is
V ˆ3,2,0(Xf)L2;O) = 2JX L 2\x\\dx2.
This estimator can be obtained as a rotational average of V3,1,0 (X)
V ˆ3,2,0(XfL2;O) = E(V ˆ3,1,0(XfL 1;O) | L2),
where the line L1 on the right-hand side of (2) is an isotropic line through O in L2.
The third local volume estimator in R3 is based on information in an isotropic plane L2, containing a fixed axis L1 through O. Such a plane is usually called a vertical plane, cf. Baddeley (1984). The estimator takes the form
V3,2,1{X^L2;L1 ) = KXnL 2nL x
dx2
THE ESTIMATOR VARIANCES
Any of the estimators V3,1,0(X), V3,2,0(X) and
V ˆ3,2,1 (X) are always non-negative and bounded above by
 p3   where /ma x is the maximal distance from O to the
3   i   max
boundary of X Accordingly, all three estimators have finite variance. (This appears to be a particularly nice feature of local stereological volume estimators compared to local estimators of lower-dimensional properties such as surface area.) Furthermore, the variance is zero if X is a ball centred at O.
Let us study the kind of shapes and choices of origin that imply large respectively small variances of ˆ 3,1,0 ( X ) To any body Xc R3 we can associate a star-shaped (at O) body star(X) which is symmetric around O (and therefore contains O) and has the property that the     distribution     of     its     volume      estimator
V ˆ3,1,0 (star (X)) is the same as for X This body star(X)
will be called the equivalent star body. The body star(X) is defined by
starfX)
(CO)
4n
V ˆ310(Xf]span{co};O)
coeS2 ,(3)
where ?star(X)(?) is the distance from O to the boundary of star(X) in the direction ? ! S2. If we let Br
be a ball with centre O and radius r, it is easy to see, using (1) with X replaced by star(X), that
V ˆ310(Xn span {co};O) =
V ˆ3,1,0 (star (X) PI span {co}; O) =
VB
coe S

pstar(X) (ffl)
because of the unbiasedness of volume estimators, (4) implies that
V(star(X)) = V(X).
If X is already star-shaped at O and O e X, the
mapping   X   ->   star(X)   is   a   particular  type   of symmetrization
star(X)
(co) =
-¦(pX (co3 + pX ( -co3
If X is both symmetric and star-shaped at O then star(X) = X. Note that star(X) does not need to be convex.
Pronounced elongation of X along some lines through O will also be seen in star(X) and implies large variances. If O is situated asymmetrically in X then star(X) may show elongation along some lines which again lead to large variances. If star(X) is a ball then the variance of V3,1,0 (X) is zero. In Figure 1, an example
of a planar section of a body X through O is shown together with the corresponding section of star(X).
Using (4), it is easy to see that the squared coefficient of error of V3,1,0 (X) can be expressed in
terms of the equivalent star body as
CE2(V310(Xf]L1;O)) = — f
4nS
( b        V
VB
V(Xf
dco2
^'
Fig. 1. A planar section of a body X through O (black boundary) together with the corresponding section of star(X) (red boundary). The lines emanating radially from O show that X is not star-shaped at O.
Let us now turn to V3,2,0 (X). Because of (2)
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Image Anal Stereol 2000;19:15-18
Varij0 {X n 4; O) < VarF3,1,0 (x n A; o).
Therefore, if VarF310(X) = 0 then VarK,20(X) = 0.
It can also be shown that the opposite statement is correct, cf. Jensen and Petersen (1999, p. 6-8). As for V310(X), it can be  shown that the variance of
^3,2,0 (X) depends only on the equivalent star body. We have, cf. Jensen and Petersen (1999, p. 15),
C£2JF320(lnI2;O)) = 8  f   f    i   1//          \
v b
x
star (x)(01 )           \      star (x)(02 )
V{X)            V{X)
dco2dco2
In Table 1, we have determined by simulation the coefficient of error of V310 (X) = 0 and V3,2,0 (X) for
the case where star(X) is a triaxial ellipsoid centred at O and with semiaxes of lengths ßi>ß2> ß3 and ß/ß* ß2/ß3
E {1,2,4}. We have also determined the coefficient of
error of an intermediate estimator, usually called the nucleator, cf. Gundersen (1988),
^3,1,0 {X) = 1[V3,1,0 {Xnspan{ü)1};0) + F3,1, 0 (xnspan{<u2};0)] ,
where co1 e S2 n L2 is an isotropic direction in an
isotropic plane L2 and co2 e S2 n Z2 is orthogonal to ©1.
Note that a prolate ellipsoid (f1 =a,ß2 =ß3 = ß) gives a larger C£ than the corresponding oblate ellipsoid
(ß1 = ß2 = a,ß3 =ß).
Table 1. Coefficient of error of local stereological volume estimators in various triaxial ellipsoids. For details, see text.
ß1/ß2      ß2/ß3	ce (ˆ3,1,0)	ce^0)	cE (ˆ3,2,0)
11 1             2 1            4	0 0.59 1.20	0 0.33 0.76	0 0.28 0.52
2            1 2            2 2            4	0.72 1.27 2.01	0.41 0.79 1.34	0.34 0.56 0.77
4            1 4            2 4            4	1.85 2.74 3.98	1.21 1.86 2.76	0.77 1.02 1.25
Similar results can be established for V3,2,1 (X). The equivalent star body is here replaced by an
equivalent cylinder with axis parallel to the vertical axis and a star-shaped, symmetric base.
In practice, the shape of X is unknown, however. If it is possible, using optical sectioning, for instance, to suggest an extreme shape such that the real distribution of V(X)/V(X) is more concentrated around 1 than
that   obtained   under   the    extreme    shape,   then conservative (1 - a) - confidence limits are
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(5)
where ua is the 100a-percentile, determined by simulation, in the distribution of V(X)/V(X) under
the extreme shape assumption for X
PARTICLE SIZE DISTRIBUTIONS FROM CENTRAL SECTIONS
Let us consider an aggregate of particles {X,}, the object being to estimate the particle size distribution from central sections through the particles. To be more
precise, we suppose that a point x, e X, associated to
each X. A central section through Xt is then a section containing x. As size parameter, we may take the volume, but other size parameters are possible, see below. For any of the volume estimators, described in the previous section, we have
v{xy
+ £,
where the associated point xt e Xt is acting as origin and e,- has mean zero and a distribution depending on the shape of Xt and the relative position of x, in Xt. In a vertical design, the distribution of e,- also depends on the choice of vertical axis relative to Xt. If the confidence limits (5) for each sampled particle are narrow compared to the variability in the volume distribution then the observed distribution of volume estimates can be regarded as a direct observation of the volume distribution. In any case the average of the volume estimates is an unbiased estimate of the mean particle volume.
If the particle aggregate consists of spheroids (either exclusively prolates or oblates) then more information can be obtained from sections through the centres of the spheroids. In contrast to Cruz-Orive (1976, 1978) we sample directly from the particle distribution. This fact appears to make the inference more simple and sound.
Consider a spheroid centred at O and with semiaxes of lengths a and ß, where a>ß. We will parametrize by (size, shape). For prolates, it is convenient to use
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Vedel Jensen EB: On the variance of local stereological volume estimators
(**)=
ß,

1-2
1/2
^>0, 0<z<1 ,
while for oblates
(**) =
a,
cT ^
1/2
y>0, z>0.
/J
1/2
A central section through the spheroid is an ellipse, with semiaxes of lengths /u andp, say, where /u>p. Again, we will parametrize by (size, shape), for prolates by
(u,v) =
for oblates by
(u,v) =
w>0, 0<v<1
ß,
P
1/2
w>0, v>0
Then, it can be shown, cf. Cruz-Orive (1976, formulae (2a) and (2b)) that
u = y,v = sin 0 ¦ z ,
(6)
where
K
0 - I is the angle that the normal of the
section makes with the rotary axis of the spheroid. From (6), it can be seen that the size of the spheroid is observed directly in a central section while the shape is not.
Let us now suppose that the central section is isotropic   with   a   random   angle   ©   with   density
n
sin0,0e
L    2
Using   (6),   we   can   express   the
conditional density of the observed shape in the section V given the size Y = y in terms of the conditional density of the 3D shape Z given the size Y = y
fvlr(v\y)= =rr=2 fzlr(z\y)dz. (7)
1                       zv z -v        '
The integral Eq. 7 is of Abel type and can be inverted. Apart from observing the size distribution, we can thus also in principal estimate the conditional distribution of shape given size.
Note also that (6) implies the following moment relationship between (size, shape) of a randomly chosen spheroid, (Y,Z), and that of an isotropic central section through the spheroid, (U,V),
E(uavb) = -B -+1,-Wyz*),
2    ^2       2) where B(·,·) is the Beta function.
A preliminary report of some of the data (Vedel Jensen, 1999) has been presented at the Xth International Congress for Stereology, Melbourne, Australia, 1-4 November 1999.
ACKNOWLEDGEMENTS
I want to thank Asger Hobolth and Michael Kjærgaard Sørensen for skilful technical assistance.
REFERENCES
Baddeley AJ. Vertical sections (1984). In: Ambartzumian RV, Weil W, eds. Stochastic Geometry, Geometric Statistics, Stereology. Leipzig: Teubner, 43-52.
Cruz-Orive LM (1976). Particle size-shape distributions: the general spheroid problem. I. Mathematical model. J Microsc 107:235-53.
Cruz-Orive LM (1978). Particle size-shape distributions: the general spheroid problem. II. Stochastic model and practical guide. J Microsc 112:153-67.
Gundersen HJG (1988). The nucleator. J Microsc 151:3-21.
Jensen EBV (1998). Local Stereology. London: World Scientific.
Jensen EBV, Petersen L (1999). When are local stereological volume estimators exact? Research report 4, Laboratory for Computational Stochastics, University of Aarhus.
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