Vertical spatial grid

As shown in the previous section, the surface area, SS, of a bounded object, MM, can be estimated from two series of vertical sections (n=2n = 2), separated by a distance TT, whose orientations are ϕ0UR[0,π/2)\phi_0 \in \mathrm{UR}[0, \pi /2) and ϕ1=ϕ0+π/2\phi_1 = \phi_0 + \pi /2. Let IiI_i be the number of intersections between the boundary of MM on the sections of series ii and the rolling cycloids. The surface area of MM is given by

S=S0+S12 S = \frac{S_0 + S_1}{2}

and from (49)

S=(al)T(EI0+EI1).(50) S = \left(\frac{a}{l}\right) \cdot T \cdot \left(\mathbf{E}I_0 + \mathbf{E}I_1\right). \tag{50}

The vertical spatial grid (VSG), devised by Cruz-Orive and Howard (1995), is a test probe comprising two systems of rolling cycloids that allows the intersection counts I0I_0 and I1I_1 to be observed from a single series of vertical sections through MM. The rolling cycloids are contained in two mutually perpendicular sets of planes parallel to the sections in series 0 and series 1. Let series 0 be the observation series. The VSG, shown in Figure 23, consists of rolling cycloids parallel to the observation series that ride upon rolling cycloids normal to the observation series.

Vertical spatial grid and fundamental tile

Figure 23: The vertical spatial grid and its fundamental tile, J0J_0. The vertical spatial grid is a test probe comprising two systems of rolling cycloids that are contained in mutually orthogonal sets of planes (xyxy and yzyz). These systems of rolling cycloids sweep out the grey surface shown here.

Suppose a single (n=1n = 1) observation series of mm parallel vertical sections (T0,...,Tm1)(T_0, ..., T_{m-1}) is constructed that cuts through MM and whose orientation is ϕ0UR[0,π/2)\phi_0 \in \mathrm{UR}[0, \pi/2). As shown in Figure 24, let the distance between successive vertical sections be TT. On the first vertical section, T0T_0, overlay a UR test system of points and rolling cycloids (see Figure 21(b)). Let Ixy,0I_{xy,0} be the number of intersections between the boundary of MM on T0T_0 and the rolling cycloids. Let P0P_0 be the number of test points hitting the two dimensional cross-section of MM on T0T_0. Test points that contribute to P0P_0 are considered "switched on" or highlighted.

An observation series of parallel vertical sections

Figure 24: An observation series of mm parallel vertical sections. The object of interest MM, through which the series cuts, is omitted for clarity.

On the second section, T1T_1, of the observation series, overlay the same test system that appeared on T0T_0, but shifted vertically by the signed distance Δ0\Delta_0. The calculation of the vertical shift, Δ0\Delta_0, is described below. In the same way as described in the previous paragraph, count Ixy,1I_{xy,1} and P1P_1. On T1T_1 (and subsequent sections) make an additional count P0,1P_{0,1} of highlighted points on T0T_0 that remain highlighted on T1T_1. Each of the three counts described is illustrated in Figure 25.

Point and intersection counting on the observation series

Figure 25: Point and intersection counting on the observation series. In this example, Ixy,i=6I_{xy,i} = 6, Pi=8P_i = 8, Ixy,i+1=6I_{xy,i+1} = 6, Pi+1=9P_{i+1} = 9 and Pi,i+1=5P_{i,i+1} = 5.

This process is repeated for all sections T0,...,Tm1T_0, ..., T_{m-1} belonging to the observation series. The total number of intersections, I0I_0, between the boundary of MM on the observation series sections and the test systems of rolling cycloids is

I0=i=0m1Ixy,i. I_0 = \sum_{i=0}^{m-1} I_{xy, i}.

Consider a second constructed series of parallel vertical sections (shown in Figure 26), perpendicular to the observation series and horizontal plane, whose orientation is ϕ1=ϕ0+π/2\phi_1 = \phi_0 + \pi/2. Let the distance between successive vertical sections be πr\pi r. Position the constructed series so that test points on the observation series also lie on the new series. The vertical shifts Δi\Delta_i are calculated so that test points on the constructed series trace out constructed rolling cycloids that are normal to the observation series.

A constructed series of vertical sections perpendicular to the observation series

Figure 26: A constructed series of vertical sections perpendicular to the observation series. In this example, the distance between successive vertical sections on both series is πr\pi r.

The vertical shifts Δi\Delta_i are calculated as follows. Let Z0UR[0,T)Z_0 \in \mathrm{UR}[0, T) and Zi=Z0+(iT)Z_i = Z_0 + (i \cdot T) (i=0,...,m1)(i = 0, ..., m - 1) be systematic abscissae along a rolling cycloid, radius rr, whose corresponding ordinates are YiY_i. As shown in Figure 27, the vertical shifts Δi\Delta_i (i=0,...,m2)(i = 0, ..., m - 2) are given by

Δi=Yi+1Yi(i=0,...,m2). \Delta_i = Y_{i+1} - Y_{i} (i = 0, ..., m-2).

However, given a cycloid abscissa ZiZ_i, the corresponding ordinate YiY_i is not available explicitly. (47) must be solved numerically for θ\theta and this value of θ\theta substituted into (48). For each i=0,...,m1i = 0, ..., m - 1 let

θi,0=Zir \theta_{i, 0} = \frac{Z_i}{r}

so that

θi,k+1=sinθi,k+Zir,(k = 0, 1, 2, ...). \theta_{i,k+1} = \sin\theta_{i,k} + \frac{Z_i}{r}, \text{(k = 0, 1, 2, ...).}

The solutions θi\theta_i are taken to be θi,N\theta_{i,N} as soon as θi,Nθi,N1<ε| \theta_{i,N} - \theta_{i,N-1} | \lt \varepsilon or N>N0N \gt N_0, where ε\varepsilon is a small positive number and N0N_0 is the maximum number of iterations. Cruz-Orive and Howard (1995) suggest ε=0.001\varepsilon = 0.001 and N0=100N_0 = 100. Greater accuracy is readily achieved (by decreasing ε\varepsilon and increasing N0N_0) when modern computers are employed. Finally the solutions θi\theta_i are substituted into (48) to obtain (i=0,...,m2)(i = 0, ..., m - 2)

Δi=ξ(θi+1)ξ(θi) \Delta_i = \xi(\theta_{i+1}) - \xi(\theta_i)

where

ξ(θ)=((1cosθ)r)(1)[θ/(2π)]. \xi(\theta) = \left((1 - \cos\theta)r\right) \cdot (-1)^{[\theta/(2\pi)]}.

A constructed vertical section

Figure 27: A constructed vertical section through MM. Test points on the observation series T0,...,Tm1T_0, ...,T_{m-1} trace constructed rolling cycloids. Test points within MM are represented by rectangles. Test points highlighted for more than one section (moving in the direction T0T_0 to Tm1T_{m-1}) are circled. The number of intersections between the cross-section of MM and the constructed rolling cycloids is I1=2(42)=4I_1 = 2 \cdot (4 - 2) = 4.

If a test point is highlighted on T0T_0 and switched off on T1T_1, then a constructed rolling cycloid has traversed the boundary of MM (moving from the inside to the outside of MM). For every inside-outside intersection, there must be a corresponding outside-inside intersection. Therefore, the total number of intersections, I1I_1, between the boundary of MM on the new series of sections and the constructed rolling cycloids is

I1=2(i=0m1Pii=0m2Pi,i+1). I_1 = 2 \cdot \left( \sum_{i=0}^{m-1} P_i - \sum_{i=0}^{m-2} P_{i,i+1} \right).

Equation (50) cannot be used directly to calculate the surface area of MM when the VSG is employed. The extra constraints imposed on the positioning of the rolling cycloid test systems mean a new equation must be formulated. This is achieved by referring back to (45). If the boundary of MM is intersected with an unbounded VSG then

S=2(VEL)(EI0+EI1). S = 2 \cdot \left( \frac{V}{\mathbf{E}L} \right) \cdot (\mathbf{E}I_0 + \mathbf{E}I_1).

L/VL/V is the mean length of rolling cycloids per unit volume of space. As shown in Figure 24, the VSG has as its fundamental tile, J0J_0, a rectangular box with volume V=64πr3V = 64 \pi r^3. The mean length of rolling cycloid, EL\mathbf{E}L, that falls within the volume is made up of two components.

The first component, EL0\mathbf{E}L_0, is the length of rolling cycloid contributed by the observation series. Each observation section that cuts through J0J_0 contains one full period of rolling cycloid within J0J_0. The expected number of observation sections that intersect J0J_0 is 4πr/T4 \pi r/T - i.e., the length of J0J_0 in the constructed series direction divided by the distance between observation series. Since one period of rolling cycloid has length 16r16r, EL0=16r4πr/T=64πr2/T\mathbf{E}L_0 = 16 r \cdot 4 \pi r / T = 64 \pi r^{2}/T.

The second component, EL1\mathbf{E}L_1, is the length of constructed rolling cycloid contributed by the constructed series. Four constructed sections, a distance πr\pi r apart, cut through J0J_0 and each section contains one full period of rolling cycloid within J0J_0. Therefore EL1=416r=64r\mathbf{E}L_1 = 4 \cdot 16r = 64r, so that

VEL=64π2r364r+64πr2T=π2r2TT+πr.(51) \begin{align} \frac{V}{\mathbf{E}L} &= \frac{64 \pi^2 r^3}{64r + \frac{64 \pi r^2}{T}} \\ &= \frac{\pi^2 r^2 T}{T + \pi r}. \end{align} \tag{51}

The rolling cycloids that comprise the VSG have test area per test point, a/p=πr4r=4πr2a/p = \pi r \cdot 4r = 4 \pi r^2 and test length per test point, l/p=4rl/p = 4r. When these values are substituted into (51), Cruz-Orive and Howard's result for estimating surface area using the VSG is obtained:

S=2(a/p)T(l/p)+(4/π)T(EI0+EI1).(52) S = 2 \cdot \frac{(a/p) \cdot T}{(l/p) + (4 / \pi) \cdot T} \cdot (\mathbf{E}I_0 + \mathbf{E}I_1). \tag{52}
S=2(a/p)T(l/p)+(4/π)T(EI0+EI1).S = 2 \cdot \frac{(a/p) \cdot T}{(l/p) + (4 / \pi) \cdot T} \cdot (\mathbf{E}I_0 + \mathbf{E}I_1).
(52)(52)

Implementing the VSG is not straightforward. Some guidelines are stated here. Care should be taken to avoid periodicity between the step length TT and the horizontal width of a cycloid, πr\pi r, as this causes unwanted repetitions of the vertical shifts Δi\Delta_i and decreases precision. Large values of TT are also discouraged as this causes constructed cycloids (shown in Figure 27) to be poorly approximated and thus intersections contributing to I1I_1 may be omitted.

Another issue is oversampling on either the observation series or the constructed series. This will occur if either VSG component (observed or constructed) has significantly greater length per unit volume (L/VL/V) than the other. Figure 26 illustrates a VSG that has L/VL/V identical for both components. The observation and constructed series intersect each other in a square lattice. Unfortunately, this configuration implies T=πrT = \pi r with unwanted repetitions of the vertical shifts Δi\Delta_i resulting. These conflicting guidelines suggest the following VSG implementation:

Set TT considerably smaller than πr\pi r, e.g. T<πr/4T < \pi r/4, (but not πr\pi r divided by some positive integer). Count intersections, I1I_1, contributed by the constructed cycloids as described in the text. Intersections, I0I_0, on the observation series should only be counted on every [πr/T]th[\pi r/T]^{\text{th}} section (where [x][x] means the integer part of xx). The distance between successive sections comprising the observation series is now T2=[πr/T]TT_2 = [\pi r/T] \cdot T, and this value should be used instead of TT in (52).

When the VSG is implemented using tools that allow vertical scanning, such as a confocal scanning laser microscope or a computer programme with 3D digital image loaded, the intersection count I1I_1 need not be counted as described in the text. Instead virtual cycloid probes are imagined travelling normal to the observation series that pass through points on the observation cycloids (marked as vertical and horizontal lines in Figure 25).

Furthermore, 3D digital images can be reformatted using image analysis tools to produce 2D image sections that comprise observation and constructed series. For images sectioned in this way, true horizontal rolling cycloids can be overlain on both observation and constructed sections with intersection counts I0I_0 and I1I_1 recorded in the normal way.

References

CRUZ-ORIVE, L. M. and HOWARD, C. V. Estimation of individual feature surface area with the vertical spatial grid. J. Microsc., 178, 146-151 (1995).