Area and Volume

Consider a bounded object in ². An example is shown in Figure 1. The area, A, of the object can be efficiently estimated by point counting. Onto the object, overlay a test system of points uniform randomly and count the number of test points hitting the object. The sample space or set of all possible outcomes, W, is {0 points hit object, 1 point hits object, 2 points hit object, …}. Let P be the random variable P(w) = w. If the area per test point is a/p, then

   (1)

An estimate of A can be obtained from a single throw of the test system of points:

   (2)

Figure 1: Area estimation using a test system of points. The area per point is a/p. A test point is a zero dimensional probe; to remove the influence of line thickness, Weibel (1979, p.113) defines a test point as the true point of intersection between the upper edge of the horizontal line of a cross, and the right-hand side edge of the vertical line of the cross. In this example, P = 9.

Now consider a bounded object in ³. An example is shown in Figure 2. The volume, V, of the object can be estimated if the object is systematically sampled by plane sections. Fix a convenient axis relative to the object. Next, intersect the object by a UR systematic set of parallel planes, a given distance T apart, and normal to the chosen axis. Let the random variables A₁, A₂, …, A_m denote the successive cross-sectional areas formed by the intersection of the bounded object with the parallel planes. If the random variable A_T represents the sum of the cross section areas,

   (3)

An estimate of V can be obtained from a single throw of the parallel plane test system:

   (4)

A further approximation of V is possible if each cross sectional area is estimated with a uniform random grid of points:

   (5)

So that

   (6)

Figure 2: Volume estimation using the Cavalieri method.

This method of volume estimation is called the Cavalieri method. The method is named after the Italian mathematician Bonaventura Cavalieri (1598-1647), a pupil of Galileo. To apply the Cavalieri method without incurring bias, sections have to be truly planar or infinitely thin. If the distance, T, between parallel sections varies, then T may be replaced with its mean value. There are many examples where the Cavalieri method has been used to estimate volume. Michel and Cruz-Orive (1988) estimated lung volume. Roberts et al. (1993) and (1994) used MRI to estimate, respectively, human body composition and fetal volume.

Buffon's Needle

Imagine a room whose floor is made up of planks of wood. The planks have uniform width, h, and are parallel to one another. A needle, length b (b < h), is thrown into the air. What is the probability, P, that the needle intersects one of the joints between the planks? This question was first asked by Georges Louis Leclerc, Comte de Buffon (1777).

Suppose the joints are represented by the lines y = nh (n = 0, ±1, ±2, …). Let (P, Q) be the coordinates of the centre of the needle and let F be the angle, modulo p, made by the needle and the x-axis. Denote the distance from the needle's centre and the nearest line beneath it as Z. The needle is assumed to be IUR. In other words, Z Î UR[0, h) and F Î UR[0, p). Z has density function f_Z(z) = 1/h. F has density function f_F(f) = 1/p. Z and F are independent variables and so f_Z,F(z, f) = f_Z(z) f_F(f). Thus, the pair Z, F has joint density function f(z, f)=1/ph.

Figure 3 suggests an intersection between the needle and a joint occurs if and only if z £ (b/2)sinf or z ³ h - (b/2)sinf. Let B₂ = {(z, f) : z £ (b/2)sinf or z ³ h - (b/2)sinf}. An intersection occurs for all (Z, F) Î B₂. From RandomVar 1-(11),

   (7)

Figure 3: Buffon's needle.

When b and h are known quantities, (7) can be used to estimate the numerical value of p. This was the original intention of Buffon. The following work assumes h and p are known with b to be estimated.

Suppose the needle is thrown and lands on the floor. The needle intersects a joint with probability 2b/ph and falls between joints with probability 1 - 2b/ph. Consider the discrete random variable I that maps the outcomes needle intersects a joint and needle falls between joints to the real numbers 1 and 0, respectively. According to RandomVar 1-(1), I has expected value EI = 2b/ph and therefore

   (8)

An estimate of b can be obtained from a single throw of the needle:

   (9)

Curve Length 2D

Consider the curve, C, shown in Figure 4. C can be approximated by a union of line segments:

If the length of y_i is b_i (1 £ i £ n), an approximation of the curve length of C is

Figure 4: A curve approximated by line segments.

The methodology of the previous section allows the length of each line segment and hence B to be estimated. Once more, imagine a room whose floor is made up of planks of wood. The planks have uniform width, h (with b_i < h), and are parallel to one another. Y is thrown into the air and lands on the floor. Y is assumed to be IUR in the plane. A sufficient condition for this to be true is that y₁ is IUR in the plane. By comparison with (7), the probability that the line segment y_i intersects a joint is 2b_i /ph. Furthermore, (8) implies the length of y_i is b_i = (p/2)×h×EI_i. I_i is a discrete random variable that maps the outcomes y_i intersects a joint and y_i falls between joints to the real numbers 1 and 0, respectively. Let the total number of intersections, I, between the joints and Y be denoted as the sum I₁ + I₂ + … + I_n. Taking into account all line segments:

   (10)

An estimate of B can be obtained from a single throw of Y:

   (11)

As the number of line segments is increased, the approximation of C is refined. In the limit (n -> ¥), Y becomes C and (10) gives the exact curve length of C.

Consider again the array of joints onto which Y is randomly thrown. The joints can be thought of as an unbounded, systematic set of parallel test lines a distance h apart. As such, the average length of test line likely to fall within a certain area can be calculated. Figure 5(a) shows a length of test line, L, and the area it occupies. The test system of lines has length per unit area L/A = L/Lh = 1/h. Next, substitute h = A/L into (10). The result is a well-known formula originally due to Steinhaus (1930):

   (12)

An estimate of B can be obtained from a single throw of Y:

   (13)

(a)                                                      (b)

Figure 5: Test systems of (a) parallel lines (L/A = 1/h) and (b) mutually orthogonal lines (L/A = 2/h). The figures shows each test system with the area, A, a length of test line, L, occupies.

Suppose a needle, length B (B > h), is thrown with IUR position onto a test system of parallel lines. is greatest when the needle is perpendicular to the test lines and smallest when the needle is parallel to the test lines. A better estimation of B is obtained if the test system of parallel lines is replaced by a test system of mutually orthogonal lines (see Figure 5(b)). Such a test system has twice as much length per unit area as the parallel line test system so that L/A = 2×(1/h).

One way of determining a test system's length per unit area, L/A, is by tiling the plane (²) with fundamental tile, J₀, of known area A where the mean length of test curve falling within each tile is L. The test line elements of the test systems described in Figure 5 may be rearranged into any arbitrary curve shape whose mean length per unit area, L/A, is known. This is because an IUR test line element is invariant under arbitrary translations and rotations. Therefore, the test system of rolling circles, shown in Figure 6, may be used in place of either of the test systems shown in Figure 5 to estimate curve length in ².

Figure 6: A test system of rolling circles, radius r. The figure shows the area, A, a length of test line, L, occupies. The test system has length per unit area, L/A = p/4r.

References

BUFFON, G. L. L. Comte de. Essai d'Arithmétique Morale. In: Supplément à l'Histoire Naturelle, v. 4. Paris: Imprimerie Royale (1777).

MICHEL, R. P. and CRUZ-ORIVE, L. M. Application of the Cavalieri principle and vertical sections method to lung: estimation of volume and pleural surface area. J. Microsc., 150, 117-136 (1988).

ROBERTS, N., CRUZ-ORIVE, L. M., REID, M., BRODIE, D., BOURNE, M. and EDWARDS, R. H. T. Unbiased estimation of human body composition by the Cavalieri method using magnetic resonance imaging. J. Microsc., 171, 239-253 (1993).

ROBERTS, N., GARDEN, A. S., CRUZ-ORIVE, L. M., WHITEHOUSE, G. H. and EDWARDS, R. H. T. Estimation of fetal volume by MRI and stereology. The British Journal of Radiology, 67, 1067-1077 (1994).

STEINHAUS, H. Zur Praxis der Rektifikation und zum Längenbegriff. Berichte Sächsischen Akad. Wiss. Leipzig, 82, 120-130 (1930).

WEIBEL, E. R. Stereological Methods. Vol. 1: Practical Methods for Biological Morphometry. London - New York - Toronto: Academic Press (1979).