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 \(\mathrm{ce}^2_{PC}(\tilde{V}) = \hat{v} \cdot (\sum P_i)^{-2}\),


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


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

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.


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