Taken from Mike Puddephat's PhD, this article describes the vertical spatial grid and a method for estimating curve length in 3D from total vertical projections.
Using exhaustive vertical sections, the surface area, S, of a bounded object, M, can be estimated from two series of vertical sections (n = 2), separated by a distance T, whose orientations are φ0 ∈ UR[0, π/2) and φ1 = φ0 + π/2. Let Ii be the number of intersections between the boundary of M on the sections of series i and the rolling cycloids. The surface area of M is given by
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 I0 and I1 to be observed from a single series of vertical sections through M. 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 16, consists of rolling cycloids parallel to the observation series that ride upon rolling cycloids normal to the observation series.
Figure 16: The vertical spatial grid and its fundamental tile, J0. The vertical spatial grid is a test probe comprising two systems of rolling cycloids that are contained in mutually orthogonal sets of planes (xy and yz). These systems of rolling cycloids sweep out the grey surface shown here.
Suppose a single (n = 1) observation series of m parallel vertical sections (T0, ..., Tm-1) is constructed that cuts through M and whose orientation is φ0 ∈ UR[0, π/2). As shown in Figure 17, let the distance between successive vertical sections be T. On the first vertical section, T0, overlay a UR test system of points and rolling cycloids (see Figure 14(b)). Let Ixy,0 be the number of intersections between the boundary of M on T0 and the rolling cycloids. Let P0 be the number of test points hitting the two dimensional cross-section of M on T0. Test points that contribute to P0 are considered "switched on" or highlighted.
Figure 17: An observation series of m parallel vertical sections. The object of interest M, through which the series cuts, is omitted for clarity.
On the second section, T1, of the observation series, overlay the same test system that appeared on T0, but shifted vertically by the signed distance Δ0. The calculation of the vertical shift, Δ0, is described below. In the same way as described in the previous paragraph, count Ixy,1 and P1. On T1 (and subsequent sections) make an additional count P0,1 of highlighted points on T0 that remain highlighted on T1. Each of the three counts described is illustrated in Figure 18.
Figure 18: Point and intersection counting on the observation series. In this example, Ixy,i = 6, Pi = 8, Ixy,i+1 = 6, Pi+1 = 9 and Pi,i+1 = 5.
This process is repeated for all sections T0, ..., Tm-1 belonging to the observation series. The total number of intersections, I0, between the boundary of M on the observation series sections and the test systems of rolling cycloids is
Consider a second constructed series of parallel vertical sections (shown in Figure 19), perpendicular to the observation series and horizontal plane, whose orientation is φ1 = φ0 + π/2. Let the distance between successive vertical sections be πr. Position the constructed series so that test points on the observation series also lie on the new series. The vertical shifts Δi are calculated so that test points on the constructed series trace out constructed rolling cyloids that are normal to the observation series.
Figure 19: 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.
The vertical shifts Δi are calculated as follows. Let Z0 ∈ UR[0, T) and Zi = Z0 + (i⋅T) (i = 0, ..., m - 1) be systematic abscissae along a rolling cycloid, radius r, whose corresponding ordinates are Yi. As shown in Figure 20, the vertical shifts Δi (i = 0, ..., m - 2) are given by
However, given a cycloid abscissa Zi, the corresponding ordinate Yi is not available explicitly. (23) must be solved numerically for θ and this value of θ substituted into (24). For each i = 0, ..., m - 1 let
The solutions θi are taken to be θi,N as soon as | θi,N - θi,N - 1 | < ε or N > N0, where ε is a small positive number and N0 is the maximum number of iterations. Cruz-Orive and Howard (1995) suggest ε = 0.001 and N0 = 100. Greater accuracy is readily achieved (by decreasing ε and increasing N0) when modern computers are employed. Finally the solutions θi are substituted into (24) to obtain (i = 0, ..., m - 2)
Figure 20: A constructed vertical section through M. Test points on the observation series T0, ...,Tm-1 trace constructed rolling cycloids. Test points within M are represented by rectangles. Test points highlighted for more than one section (moving in the direction T0 to Tm-1) are circled. The number of intersections between the cross-section of M and the constructed rolling cycloids is I1 = 2⋅(4 - 2) = 4.
If a test point is highlighted on T0 and switched off on T1, then a constructed rolling cycloid has traversed the boundary of M (moving from the inside to the outside of M). For every inside-outside intersection, there must be a corresponding outside-inside intersection. Therefore, the total number of intersections, I1, between the boundary of M on the new series of sections and the constructed rolling cycloids is
Equation (26) cannot be used directly to calculate the surface area of M 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 (21). If the boundary of M is intersected with an unbounded VSG then
L/V is the mean length of rolling cycloids per unit volume of space. As shown in Figure 17, the VSG has as its fundamental tile, J0, a rectangular box with volume V = 64πr3. The mean length of rolling cycloid, EL, that falls within the volume is made up of two components.
The first component, EL0, is the length of rolling cycloid contributed by the observation series. Each observation section that cuts through J0 contains one full period of rolling cycloid within J0. The expected number of observation sections that intersect J0 is 4πr/T - i.e., the length of J0 in the constructed series direction divided by the distance between observation series. A proof of this fact will be written up in the future. Since one period of rolling cycloid has length 16r, EL0 = 16r⋅4πr/T = 64πr2/T.
The second component, EL1, is the length of constructed rolling cycloid contributed by the constructed series. Four constructed sections, a distance πr apart, cut through J0 and each section contains one full period of rolling cycloid within J0. Therefore EL1 = 4⋅16r = 64r, so that
The rolling cycloids that comprise the VSG have test area per test point, a/p = πr⋅4r = 4πr2 and test length per test point, l/p = 4r. When these values are substituted into (27), Cruz-Orive and Howard's result for estimating surface area using the VSG is obtained:
Implementing the VSG is not straightforward. Some guidelines are stated here. Care should be taken to avoid periodicity between the step length T and the horizontal width of a cycloid, πr, as this causes unwanted repetitions of the vertical shifts Δi and decreases precision. Large values of T are also discouraged as this causes constructed cycloids (shown in Figure 20) to be poorly approximated and thus intersections contributing to I1 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/V) than the other. Figure 19 illustrates a VSG that has L/V identical for both components. The observation and constructed series intersect each other in a square lattice. Unfortunately, this configuration implies T = πr with unwanted repetitions of the vertical shifts Δi resulting. These conflicting guidelines suggest the following VSG implementation:
Set T considerably smaller than πr, e.g. T < πr/4, (but not πr divided by some positive integer). Count intersections, I1, contributed by the constructed cycloids as described in the text. Intersections, I0, on the observation series should only be counted on every [πr/T]'th section (where [x] means the integer part of x). The distance between successive sections comprising the observation series is now T2 = [πr/T]⋅T, and this value should be used instead of T in (28).
When the VSG is implemented using tools that allow vertical scanning, such as a confocal scanning laser microscope, the intersection count I1 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 18).
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 I0 and I1 recorded in the normal way.
Total vertical projections (TVPs) can be obtained by fixing a vertical axis with respect to the object under study. Then, with a random starting angle and continuing at uniform intervals, the whole object is projected onto a plane in a systematic set of directions between 0 and π radians. An estimator of curve length from TVPs can be obtained with the help of (19). The following derivation is illustrated in Figure 22.
Figure 22: Estimating curve length from total vertical projections. Redrawn from Cruz-Orive (1997).
Consider a curve in ℜ3, of length L, in a bounded reference space of volume V. Choose an arbitrary horizontal plane with vertical axis perpendicular. Next, sit a vertical slab of known thickness t and width w on the horizontal plane. The vertical slab must be positioned so that its rectangular base is IUR in the horizontal plane. Construct a test plane perpendicular to the slab. The test plane must be IUR in ℜ3. Therefore, the angle θ between the normal to the plane and the vertical axis must have probability density function sinθ. Furthermore, given θ, the position of the test plane must be UR on any bounded interval along the normal to the plane. The intersection of the test plane with the vertical slab forms an IUR rectangle, R, whose area is t⋅l, where l = w/cosθ. If there are Q transects between the curve and R, then from (19):
Project the curve and R onto an observation plane parallel to the vertical slab. R becomes a line segment of length l. The line segment is cosine-weighted because its normal is sine-weighted. If there are no overlapping problems the number of transects, Q, between the curve and R is equal to the number of intersections, I, between the projections of R and the curve. Equation (29) becomes
Instead of repeated sampling of cosine-weighted straight lines, it is convenient to use a test system of cycloids with the minor principal axis perpendicular to the vertical axis. Let t → ∞. The projection is now a total vertical projection (TVP). Onto the TVP superimpose a uniform random test system of (cosine-weighted) cycloids, with known length per unit area l/a. From (30) Cruz-Orive and Howard (1991) obtained:
I is the total number of intersections between the TVP of the curve and the test system of cycloids on the projection plane. When m TVPs are obtained, curve length L is estimated by:
TVPs have been used by Roberts et al. (1991) with MRI to estimate the length of blood vessels, while Howard et al. (1992, 1993) and Roberts and Cruz-Orive (1993) have estimated neuron dendritic length.
HOWARD, C. V., JOLLEYS, G., STACEY, D., FOWLER, A., WALLÉN, P. and BROWNE, M. A. Measurement of total neuronal volume, surface area, and dendritic length following intracellular physiological recording. Neuroprotocols: A Companion to Methods in Neurosciences, 2, 113-120 (1993).
MCMILLAN, P. J., ARCHAMBEAU, J. O., GOKHALE, A. M., ARCHAMBEAU, M. H. and OEY, M. Morphometric and stereologic analysis of cerebral cortical microvessels using optical sections and thin slices. Proc. 6ECS Prague 1993. Acta Stereol., 13/1, 33-38 (1994).
ROBERTS, N., HOWARD, C. V., CRUZ-ORIVE, L. M. and EDWARDS, R. H. T. The application of total vertical projections for the unbiased estimation of the length of blood vessels and other structures by magnetic resonance imaging. Magnetic Resonance Imaging, 9, 917-925 (1991).