# Precision of the Cavalieri method on systematic sections

The following is précised from García-Fiñana et al. (2003).

The volume of an object can be expressed as

$$V = \int_{a}^{b} A(x) \mathrm{d}x,$$

where $$A(x)$$, called the area function, denotes the intersection area between the object and a plane normal to a fixed, conveniently oriented sampling axis, at a point of abscissa $$x.$$ $$A(x)$$ is bounded and integrable in a bounded domain $$[a, b]$$, which represents the orthogonal linear projection of the object on the sampling axis.

To estimate $$V$$ intersect the object by a series of parallel planes a distance $$T$$ apart, to produce $$n$$ section areas $${A_1, A_2, ..., A_n}$$. More precisely, the $$i$$th section area is $$A_i = A(z + (i - 1)T)$$, $$i = 1, 2, ..., n$$, where $$z$$ is a uniform random abscissa in the interval $$[a, a + T)$$. The latter condition ensures that

$$\hat{V} = T \cdot (A_1 + A_2 + ... + A_n)$$

is an unbiased estimator of $$V$$ irrespective of the shape of the object.

The section areas can be estimated via point counting (Mathieu et al. 1981). A convenient test system is a square grid of side $$u$$. The grid is superimposed uniformly at random on each section image. Let $$P_i$$ denote the number of test points hitting a section of unknown area $$A_i$$. An unbiased estimator of $$A_i$$ is $$\hat{A_i} = u^2 P_i$$, whereby

$$\tilde{V} = T \cdot u^2 \cdot (P_1 + P_2 + ... + P_n)$$

is an unbiased estimator of $$V$$ based on two sampling stages, namely Cavalieri sectioning and point counting.

From Gundersen et al. (1999), Kiêu et al. (1999), García-Fiñana (2000), García-Fiñana and Cruz-Orive (2000a,b), it can be shown that

$$\mathrm{CE}^2(\tilde{V}) = \mathrm{CE}^2(\hat{V}) + \mathrm{CE}^2_{PC}(\tilde{V}),$$

where $$\mathrm{CE}^2(\hat{V})$$ is the true contribution of the variability among sections and $$\mathrm{CE}^2_{PC}(\tilde{V})$$ is the true mean variability due to point counting within sections.

The estimator of each term in the previous equation from the observed data is denoted in lower case, namely

$$\mathrm{ce}^2(\tilde{V}) = \mathrm{ce}^2(\hat{V}) + \mathrm{ce}^2_{PC}(\tilde{V}).$$

From Cruz-Orive (1999, equation 2.7), the sections and point counting contributions may be predicted as

\begin{align*} \mathrm{ce}^2(\hat{V}) &= \alpha(q) \cdot (3(C_0 - \hat{v}) - 4C_1 + C_2) \cdot (\sum P_i)^{-2}, \\ \mathrm{ce}^2_{PC}(\tilde{V}) &= \hat{v} \cdot (\sum P_i)^{-2}, \end{align*}
$$\mathrm{ce}^2(\hat{V}) = \alpha(q) \cdot (3(C_0 - \hat{v}) - 4C_1 + C_2) \cdot (\sum P_i)^{-2}$$ and $$\mathrm{ce}^2_{PC}(\tilde{V}) = \hat{v} \cdot (\sum P_i)^{-2}$$,

where

$$\sum P_i = P_1 + P_2 + ... + P_n$$

and

$$C_k = \sum^{n - k}_{i = 1} P_i P_{i+k}, k = 0, 1, ..., n - 1.$$

The numerical coefficient $$\alpha(q)$$, updated in García-Fiñana and Cruz-Orive (2000b, equation 26b), depends on the fractional smoothness constant $$q$$ of the area function and has the expression

$$\alpha(q) = \frac{\Gamma(2q + 2) \cdot \zeta(2q + 2) \cdot \cos(\pi q)}{(2 \pi)^{2q + 2} \cdot (1 - 2^{2q - 1})},$$
$$\alpha(q) = \Gamma(2q + 2) \cdot \zeta(2q + 2) \cdot \cos(\pi q) / (2 \pi)^{2q + 2} \cdot (1 - 2^{2q - 1})$$,

where $$q \in [0, 1]$$ and $$\Gamma()$$ and $$\zeta()$$ denote gamma and Riemann zeta functions respectively.

From a sample of at least five sections, the only estimator of $$q$$ available at the time of writing is Kiêu-Souchet’s described in Kiêu et al. (1999). Namely,

$$\hat{q} = \mathrm{max} \begin{Bmatrix} 0, \begin{matrix} 1 \\ \overline{2 \log(2)} \end{matrix} \cdot \log \begin{bmatrix} 3(C_0 - \hat{v}) - 4 C_2 + C_4 \\ \overline{3(C_0 - \hat{v}) - 4 C_1 + C_2} \end{bmatrix} - \begin{matrix} 1 \\ \overline{2} \end{matrix} \end{Bmatrix}.$$
$$\hat{q} = \mathrm{max}\{0, \frac{1}{2 \log(2)} \cdot \log[\frac{3(C_0 - \hat{v}) - 4 C_2 + C_4}{3(C_0 - \hat{v}) - 4 C_1 + C_2}] - \frac{1}{2} \}.$$

For $$n < 5$$, $$q$$ can be set equal to 1 for fairly regular, quasi-ellipsoidal objects and 0 for irregular objects.

Finally, as described in Matheron (1971) and Gundersen and Jensen (1987), $$\hat{v}$$ is an estimator of the point counting or nugget variance within sections. Namely,

$$\hat{v} = 0.0724 \cdot (\overline{B} / \sqrt{\overline{A}}) \cdot (n \cdot \sum P_i)^{1/2},$$

where $$\overline{B}$$, $$\overline{A}$$ are estimates of the mean boundary length and the mean area of the sections, respectively. Thus, $$\overline{B} / \sqrt{\overline{A}}$$ is a dimensionless shape coefficient of the sections and, for practical purposes, may be estimated from a few sections.

## References

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GARCÍA-FIÑANA, M. Muestreo Sistemático en R. Aplicaciones a la Estereología, Tesis doctoral. Universidad de Cantabria (2000).

GARCÍA-FIÑANA, M., CRUZ-ORIVE, L.M., New approximations for the efficiency of Cavalieri sampling. J. Microsc. 199, 224–238 (2000a).

GARCÍA-FIÑANA, M., CRUZ-ORIVE, L.M., Fractional trend of the variance in Cavalieri sampling. Image Anal. Stereol. 19, 71–79 (2000b).

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