A general variance predictor for Cavalieri slices



    1. Department of Mathematics, Statistics and Computation, Faculty of Sciences, University of Cantabria, Avda. Los Castros s/n, E-39005 Santander, Spain
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L. M. Cruz-Orive. Tel.: +34 942 20 14 24; fax: +34 942 20 14 02; e-mail: lcruz@matesco.unican.es


A general variance predictor is presented for a Cavalieri design with slices of an arbitrary thickness t ≥ 0. So far, prediction formulae have been available either for measurement functions with smoothness constant q = 0, 1, … , and t ≥ 0, or for fractional q ∈ [0, 1] with t = 0. Because the possibility of using a fractional q adds flexibility to the variance prediction, we have extended the latter for any q ∈ [0, 1] and t ≥ 0. Empirical checks with previously published human brain data suggest an improved performance of the new prediction formula with respect to the hitherto available ones.

1. Introduction

Cavalieri sections are extensively used in stereology to estimate volumes, whereas Cavalieri slices may be used to estimate any other geometric measure, notably cell or particle number. The prediction of the variance of the corresponding estimators is therefore of interest. The problem is nontrivial because the observations are systematic, hence dependent in general. The state of the art has recently been reviewed in Cruz-Orive & García-Fiñana (2005) for Cavalieri sections, and in Cruz-Orive & Geiser (2004) for Cavalieri slices.

Suppose that the target is the volume Q of a bounded object. Fix a sampling axis and consider the area f(x) of the intersection between the object and a plane normal to the sampling axis at a point of abscissa x. Then, Q is the area under the graph of the function f : ℝ→ℝ+. In general, f is assumed to be square integrable, and it is called the measurement function. The ‘trend’ of the variance of the Cavalieri estimator of Q is well represented by a term of the variance called the extension term, VarE(). It turns out that this term is proportional to T2q+2, where T is the distance between the sampling planes and q is the ‘smoothness constant’ of f, namely a non negative real constant that represents the order of the first non-continuous (possibly fractional) derivative of f. So far, the following cases have been considered:

Kiêu (1997), Kiêu et al. (1999) and Gundersen et al. (1999) consider q = 0, 1, … , (denoted by ‘m’ in these papers) and, although they contemplate the case t > 0, their variance predictors do not involve the parameter t. As a consequence, their variance predictors do not tend to zero (as they should) when the slice thickness t approaches the sampling period T (namely the distance between slice midplanes). Earlier predictors (Matheron, 1965, 1971; Gundersen & Jensen, 1987; Cruz-Orive, 1989) involve neither t nor q (they implicitly assume t = 0 and q = 0).

The problem of predicting the variance of the estimator of Q from Cavalieri slices of thickness t > 0 for measurement functions with smoothness constant q ∈ [0, 1], has hitherto remained open. The purpose of this paper is to fill this gap.

The new prediction formulae are displayed in Section 3, the relevant coefficient α(q, τ) is computed in Section 4, and the performance of the formulae is checked in Section 5 for the brain data used in García-Fiñana & Cruz-Orive (2000) and in McNulty et al. (2000). For practical applications the reader may concentrate on the formulae (3.3)(3.5), resorting to Table 1 to compute the α(q, τ). The derivation details are given in the Appendixes.

Table 1.  Numerical values of the reciprocal 1/α(q, τ) of the coefficient α(q, τ), see Eq. (3.5), entering in the right hand side of Eq. (3.3), with up to four significant digits. q = smoothness constant of the measurement function; τ = t/T = slice thickness/sampling period.
τ = q0.

2. Sampling with slices: background and purpose

2.1. Target quantity

Consider a bounded and fixed subset X ⊂ ℝ3. The quantity of interest is the total measure Q = µ(X), where µ(X) may represent volume, surface area, curve length, or number of particles contained in X. All the measures considered are assumed to be finite.

To describe the sampling procedure, we need the following definitions and notation, borrowed from Cruz-Orive (2004) and Cruz-Orive & Geiser (2004). Choose a convenient sampling axis Ox, fixed with respect to X. We define:

Lx: plane normal to Ox at a point of abscissa x.

Lx(t): Slab of thickness t > 0 at a point of abscissa x, namely the portion of space between the two parallel planes Lx and Lx+t.

Lx(t) ∩ X:Slice determined in the object X by the slab Lx(t).

µt(x): =µ(Lx(t) ∩ X): Measure of the slice Lx(t) ∩ X (for examples see above). Thus, if the slice is empty then

µ(0) = 0.

If µ represents volume then the target quantity may be represented as follows,


where f is defined in the Introduction. However, if µ represents a measure other than volume, then it is necessary to assume that there exists a square integrable function f : ℝ→ℝ+ such that


in which case we can write


In the subsequent theory it is convenient to consider the regularization of the function f by the segment T0 := [0, t), namely




(Gual-Arnau & Cruz-Orive, 1998). After all, sampling the object X with a slice of thickness t is equivalent to sampling the function f with a straight line segment of length t.

2.2. Sampling design

We intersect the subset X with a Cavalieri series of slabs of thickness t > 0 and period T ≥ t, namely by the systematic series


where ℤ denotes the set of integers and U∼ UR(0, 1), namely U is a uniform random variable in the interval (0, 1), which ensures a random start of the series in an interval of length T. The gap between consecutive slabs is T − t ≥ 0. For an illustration see Cruz-Orive & Geiser (2004; fig. 2). The corresponding Cavalieri sample is the systematic set of slice contents




Let µt,1, µt,n denote the first and the last non-zero slice contents from the series (2.7), respectively. Then


is an unbiased estimator of Q.

When µ is a volume and non-invasive, computerized scanning is used, then the contents of each slice µt,i may be measured almost exactly by exhaustive pixel counting (as in McNulty et al., 2000). When µ represents cell or particle number, however, then each µt,i will usually have to be estimated by subsampling the corresponding slice by means of disectors, see for instance West et al. (1991, 1996). Slice subsampling may also be necessary when µ represents surface area, or curve length (Cruz-Orive, 1997). The corresponding estimator may be expressed as follows,


where et,i is the ‘local error’, namely the error of estimation of µt,i. Following Kiêu (1997), a plausible model has the following properties:

  • (i) ��(et,i) = 0, so that inline image is unbiased for µt,i.
  • (ii) inline image, constant for each pair (t, i).
  • (iii) Cov(et,i, et,j) = 0, i ≠ j, uncorrelated errors.

Thus, when local errors are present,


is the corresponding unbiased estimator of Q.

3. Variance prediction for measurement functions with fractional smoothness

3.1. Covariogram model and corresponding variance predictor

Our purpose is to estimate the ‘trend’ of Var(), namely its extension term VarE(), as explained for instance in Cruz-Orive & García-Fiñana (2005). Because the extension term depends only on the behaviour of the covariogram g of the measurement function f near the origin, a sensible model for g is,


(García-Fiñana & Cruz-Orive, 2000, 2004), where q is the smoothness constant of f and {b0b1b2} are numerical coefficients that can be estimated from the data displayed in Eq. (2.7). In Appendix I, we show that, under the above model,

image( (3.2))

where Γ(x) is the Gamma function. Next we need an estimator of the coefficient b1 from the data, with which we obtain an estimator of the preceding variance model.

3.2. Estimation of the variance predictor from a sample

In Appendix II we show that an estimator of the variance model (3.2) for the Cavalieri estimator (2.11), namely from a Cavalieri sample {inline image}, (n ≥ 3), with local errors, is,

image(  (3.3))



In turn, the total local variance νt,n, has to be estimated – different approaches to do this are reviewed in Cruz-Orive & Geiser (2004). The first term in the right hand side of Eq. (3.3) estimates the slices component, whereas the second term estimates the local error component. On the other hand,


Note that the right hand side of Eq. (3.3) cannot be evaluated directly for τ = 0, for instance. The computation of the coefficient α(q, τ) is discussed in Section 4 below. The cases of interest are considered next.

Case q = 0, 1, … . Although we have adopted q ∈ [0, 1] in Eq. (3.5), the following expression is valid for any q = 0, 1, … ,


where Br(x) is the Bernoulli polynomial of integer order r (see, e.g. Abramowitz & Stegun, 1965), whereas D(q, τ) is given by the second Eq. (3.5). In particular,


(Gual-Arnau & Cruz-Orive, 1998).

Casesτ = 0, 1. A passage to the limit when τ → 0 in the expression (3.5) yields,


(García-Fiñana & Cruz-Orive, 2000, 2004). On the other hand, naturally,

α(q, 1) = 0.((3.9))

Case q = 1/2. Again, a passage to the limit when q → 1/2 in the expression (3.5) yields

image( (3.10))

A further passage to the limit when τ → 0 in the preceding expression yields


where inline image is Riemann's Zeta function (e.g. Abramowitz & Stegun, 1965). The preceding result may be obtained directly from Eq. (3.8), as in García-Fiñana & Cruz-Orive (2000, 2004).

4. A table of the coefficient α(q, τ)

The coefficient α(q, τ) cannot be evaluated directly from Eq. (3.5) for all q ∈ [0, 1]. In fact, for q = 0, 1 we should use Eq. (3.7); for q = 1/2 and τ > 0 we should use Eq. (3.10); for q ≠ 1/2 and τ = 0 we should use Eq. (3.8), whereas α(1/2, 0) is given by Eq. (3.11). For the remaining values q ∈ (0, 1)∖{1/2} and τ ∈ (0, 1), we can use the expression (3.5).

The summation in the right hand sides of Eq. (3.5), (3.10) may be evaluated quickly with the aid of the software packages maple® or mathematica® using the polylogarithm function,


In particular Lia(1) =ζ(a) is Riemann's Zeta function. Thus,


where Re() denotes the real part of the expression within parentheses.

Table 1 and Fig. 1 have been obtained bearing the preceding recommendations in mind.

Figure 1.

Graph of the reciprocal 1/α(q, τ) of the coefficient α(q, τ) entering in the right hand side of Eq. (3.3). (τ = t/T = slice thickness/sampling period).

The graph of 1/α(q, τ) plotted in Fig. 1 is intended to give some idea of its pattern of variation; for instance, for τ beyond 0.7 (namely when the slice thickness approaches the sampling period), the value of the coefficient α(q, τ) rapidly approaches zero for any q ∈ [0, 1]. Numerical values of 1/α(q, τ) are better obtained from Table 1. The graph extends the particular case τ = 0 considered in García-Fiñana & Cruz-Orive (2000, 2004).

5. Empirical performance of the new variance predictors on real data

As a test example we consider the grey matter data of a human brain (including the cerebellum), as used by McNulty et al. (2000). Details on the data source are given in the latter paper. The basic data set was obtained from N0 = 183 consecutive coronal slices of t = 1 mm thickness, obtained by noninvasive magnetic resonance imaging techniques with the aid of the Analyze software (Mayo Foundation, MA, U.S.A.). The corresponding data vector can be found in appendix 4 from García-Fiñana & Cruz-Orive (2000). The volume of each basic slice, see Eq. (2.2), was measured with a negligible error by automatic pixel counting (pixel side = 1 mm). Following McNulty et al. (2000) we consider four slice thicknesses, namely t = 1, 3, 9, 27 mm. For t > 1 mm, slice volumes were computed by pooling the pertinent consecutive slices of t = 1 mm thickness, see McNulty et al. (2000), eq. (13). Thus, in this application local errors were discarded, and consequently we set νn = 0 in the right hand side of Eq. (3.3). The slice data were analysed without previous smoothing.

For a given slice thickness, t, the maximum number of non empty slices is


where ‘⌈⌉’ denotes ‘ceiling’ (e.g. ⌈3.1⌉ = 4, ⌈3.0⌉ = 3). For each t we considered an initial integer sequence of mean slice numbers n0 ranging from 1 to nmax. For each member n0 of this sequence the adopted sampling period T, namely the distance between consecutive slice midplanes, was forced to be an integer, namely


where ‘⌊⌋’ denotes ‘floor’, (e.g. ⌊3.9⌋ = 3, ⌊3.0⌋ = 3). Therefore the total number of available Cavalieri slice samples of period T was precisely T. Moreover, with the adjustment described in Remark 1 below, the actual number of slices of thickness t in each of the T samples of period T was also forced to be an integer and it was computed as


The number of nonempty slices in a sample could be either n, or n − 1. For each of the T samples the corresponding estimator of the grey matter volume was computed via Eq. (2.9), and then the variance between the T available volume estimates was also computed. This empirical variance was adopted to represent Var() for the pertinent pair (tT). The full thinner curves plotted in Fig. 2 represent the corresponding coefficients of error, namely the empirical version of

Figure 2.

Coefficient of error (%) of the Cavalieri slices estimator of human brain grey matter volume (data described in Section 5) with slice thicknesses of 1, 3, 9 and 27 mm, respectively. The thinner continuous curves represent the empirical CE() obtained by resampling from the basic data set. The upper and lower broken curves represent the corresponding theoretical predictions of CE() computed with q = 0, 1 via Eq. (3.3) with the previously known coefficients (3.7), respectively. The thicker continuous curves represent the new predictors computed with q = 0.42 via Eq. (3.3) with the general coefficient given by Eq. (3.5).


as a function of n, see Eq. (5.3). Further, Var() was computed via Eq. (3.3) with νn = 0 for each of the T samples and for q = 0, 0.42, 1, respectively. For each q the corresponding T estimates were averaged. The corresponding predictors of CE() are also plotted in Fig. 2. The estimate = 0.42 was the average of the two estimates (namely 0.44 and 0.40) computed, respectively, from the 92 odd– and the 91 even–numbered slices (that is with T = 2 mm), from the basic set of 183 slices of 1 mm thickness. The estimator used was Kiêu–Souchet's,


where the Ck are computed via the second Eq. (3.4) – see also García-Fiñana & Cruz-Orive (2000, 2004).

The results are commented in the Discussion.

Remark 1. As explained above, for each pair (tT) the total number of Cavalieri samples of size n [see Eq. (5.3)], is precisely T. Because nT ≥ N0, we adjusted t zeros at the beginning and nT − N0 − t zeros at the end of the initial vector of N0 primary slice volumes. In this way we proceeded with a suitable ‘extended’ vector of size nT (McNulty et al., 2000). Here it helps to realize that, in order to subsample a fixed segment [0, H] with a uniform random test segment [xx + t), the range of x is [– tH].

6. Discussion

The new variance predictor (3.3), with the general coefficient (3.5), covers the long standing need for a free choice of the smoothness constant q in the interval [0, 1], combined with an arbitrary slab thickness t > 0.

The performance of the new variance predictor has been tested for the data set described in Section 5. The results displayed in Fig. 2 look fairly satisfactory in the sense that, for each of the four slab thicknesses considered, the variance prediction (thicker continuous curve) seems to conform to the concept of ‘extension term’ or ‘trend’ of the variance. This impression should, however, be confirmed with further experiments on different objects.

The main snag in the prediction procedure lies, however, in the estimation of q. Thus:

  • (a) As explained in the last paragraph of Cruz-Orive (1999), the estimation of q is precluded in practice for t > 0. Thus, to estimate q by Eq. (5.5) it is necessary to use a Cavalieri sample of n ≥ 5 sections, namely slices of ‘zero thickness’ or ‘as thin as possible’.
  • (b) As shown in García-Fiñana & Cruz-Orive (2004), fig. 4(c), the estimator (5.5) of q is rather unstable for low or moderate sample sizes, and it may vary as a function of the sample size.

The best strategy to circumvent the preceding shortcomings is still open.

For the ‘splitting’ Cavalieri design described in Cruz-Orive (2004), the first factor in the right hand side of Eq. (2.27) – which holds for q = 0 – could be updated in the light of the new variance model (3.2). The task, however, would be unlikely to be worthwhile in view of the comments made in the last paragraph of the Discussion of that paper.


I wish to thank the two referees for useful comments, as well as my department colleague Professor L. González Vega for drawing my attention to the computing representation (4.2). This research was supported by the Spanish Ministry of Education and Science I+D Project no. MTM2005-08689-C02-01.


Appendix I. Variance predictor: proofs

In order to derive Eq. (3.2) we start from the well known representation


(Matheron, 1965; Gual-Arnau & Cruz-Orive, 1998), where inline image is the Fourier transform of the covariogram inline image of the regularization inline image of the measurement function f, see Eq. (2.4), by the test segment T0 := [0, t), (0 < t < T < ∞). Thus,




is the covariogram of f, whereas


is the geometric covariogram of T0.

The approach suggested by Matheron (1971) consists in submitting the covariogram model (3.1) to the expression (I.1), implicitly assuming that h∈ℝ. As a consequence we obtain a model of Var() that excludes the oscillating term (called the ‘Zitterbewegung’) and higher order terms; in fact, we obtain a model of the extension term VarE() (see also the discussion in García-Fiñana & Cruz-Orive, 2004; section 6.3). The relevant term in the right hand side of Eq. (3.1) is the second one; the Fourier transform of each of the other two terms is proportional to Dirac's delta function, and therefore their contribution to the right hand side of Eq. (I.1) will be zero. Distribution theory (e.g. Guelfand & Chilov, 1962) yields the following result,


On the other hand,


Therefore, (ignoring the aforementioned terms involving Dirac's delta function), the model version of Eq. (I.2) becomes,

image( (I.7))

which inserted into the right hand side of Eq. (I.1) yields the right hand side of Eq. (3.2). As a cursory check, when q is a non negative integer then the cosine series definition of the Bernoulli polynomial (Abramowitz & Stegun, 1965) yields the following identity,

image( (I.8))

which, substituted into the right hand side of Eq. (3.2), and bearing in mind that cos(πq) = (−1)q, (q = 0, 1, … ), yields the expression (II.3) from Cruz-Orive (1999), as rewritten from Gual-Arnau & Cruz-Orive (1998).

Appendix II. Estimation of the variance predictor and special cases: proofs

Our purpose is to derive the estimator (3.3) of the variance model (3.2) from a sample. Initially, we assume that there are no local errors involved, that is, νt,n = 0. The basic task is to express the coefficient b1 in terms of the observable data, see Eq. (2.7). The adopted covariogram model is,


with q ∈ [0, 1]. The covariogram we can observe is that of the regularized measurement function, namely the convolution inline image. Setting | h | = r and following Gual-Arnau & Cruz-Orive (1998), we obtain,


The preceding integrals were evaluated explicitly with maple®. The next step was to solve the following system of three equations with three unknown parameters {b0, bl, b2}, namely,




Again, the preceding system of equations was solved with the aid of maple®, yielding in particular the following expression for the relevant coefficient,

image( (II.5))

where D(q, τ) is given by the second Eq. (3.5). Substituting the preceding expression of b1 into the right hand side of Eq. (3.2), the estimator (3.3) (without local errors) is obtained. The actual predictor (3.3), involving local errors, is obtained in a manner entirely analogous to that used in Appendix II from Cruz-Orive (1999).

Next we consider the particular cases listed in Section 3.2.

For q = 0, 1, … , the first Eq. (3.5) becomes Eq. (3.6) by a direct application of the result (I.8).

For small τ, the summation in the right hand side of the first Eq. (3.5) becomes



image( (II.7))

from which Eq. (3.8) is obtained.



which is Eq. (3.10).