Hemisphere Sizing from Linear Intercept Measurement

The stereological problem of unfolding the hemisphere radius distribution from the chord length distribution is analyzed. Let a stationary isotropic process of hemispheres be given. The hemispheres have random diameters and are isotropically uniformly randomly orientated in space. A straight line probe yields a process of intercepts. The inverse problem of re-obtaining the size distribution of the hemispheres in terms of an experimental intercept length distribution is solved. The chord length distribution of a single hemisphere, known analytically, is approximated by piecewise polynomials in two intervals. The solution of the inverse problem is traced back to a simple recurrence equation. Numerical checks with exact and simulated data are performed to demonstrate the applicability. Data of "chord length sampling", resulting from image analysis procedures, from scattering methods or from other appropriate physical apparatuses, are applicable.


INTRODUCTION
One of the fundamental problems in stereology is the derivation of the size distribution of geometric objects (particles) from partial information which is contained in the length distribution of linear intersections of the particles (Fig. 1). Textbooks by Weibel [1] , Stoyan, Kendall and Mecke [2] , Serra [3] , Ohser and Mucklich [4] handle this problem for specific shapes. Information about the particle size via chord length measurement can be obtained by physical apparatuses.
The standard problem of tracing back the size distribution of hemispheres to linear intercept measurement occurs in different fields, involving specificities. The universal nomenclature used in this representation is explained in Appendix A.
Let L be the largest diameter of the largest hemisphere. For the analysis of the hemisphere sizes, isotropic uniform random chord lengths exclusively inside the hemispheres will be analyzed, L r ≤ ≤ 0 . No outside chord lengths or any other experimental information about the spatial arrangement of the hemispheres are included and no interaction between the hemispheres is considered.
For the following, the normalized, isotropized geometric covariogram of the hemispheres will serve as a universal working function (WF) ) (r γ , Here, the function |) (| ) ( r K r K = is the isotropized geometric covariogram of an ensemble of hemispheres. Some steps of explanation concerning the WF and the background of Eq. (1) are helpful.
The consideration starts with the geometric covariogram ) ( 1 r K for a single particle in a fixed direction [3] . The geometric covariogram is useful to characterize a single hemisphere R H of radius R , as well as an ensemble of hemispheres possessing a certain size distribution.
For a single hemisphere the geometric covariogram [3] is the overlapping integral Here, u d is the element of the surface measure, given by use of spherical coordinates ϕ and ϑ via . Considering a single hemisphere R H , it is useful to define a dimensionless working function is a strictly monotonously decreasing function on the interval , possessing the property are assumed to be exclusively influenced by the sizes of the hemispheres. Clearly, this is trivial in cases, if image material is available and exclusively chord lengths inside the hemispheres can be selected and measured. However, in the case of scattering experiments, the assumption of a so-called quasi-diluted particle arrangement must be additionally fulfilled, Appendix B (Fig. B1). For compact convex particles, the chord length distribution density look for example back at the parallelepiped case [10] . Thus, functions were determined for many basic geometric shapes [5,[11][12][13] .
Particularly, the analytical expression for the CLD of the hemisphere with a fixed radius R , , is known, Gille [10,14] (Fig. 2). The motivation of these calculations are experimental results, showing the existence of hemispherical micro-particles, Appendix C. Moreover, there is a theoretical interest to add a new result to the sequence of known cases with elementary particle shapes.
There are cases in which two-or three-dimensional images are available. The theory reported here, operating with the WF ) (r γ , can even be applied without an image. for the single hemisphere of radius R (Fig. 2). Section 2 considers the general formulas for the simulation of the WF and the CLD depending on

Simulation of
: For a single hemisphere of radius R (for more details Gille [14] and Appendix D), The abbreviation for the interval restrictions in Eq. (3) means Thus, H γ is split into three functions:

Averaging over different radii:
The volume ) , ( is proportional to the geometric covariogram, Serra [3] . Then, the averaged WF The integral in the denominator of Eq. (5) is the third moment M3 of is the weighing function for averaging. Eq. (5) satisfies . The lower integration limit depends on r . Here, , Eqs. (3,4). In more detail, for any assumed The mean hemisphere volume V is . The connection between Eq. (6) and A(l) is Eq. (2). Furthermore, the WF can be traced back to the isotropic scattering intensity, Appendix B. (two local maxima occur in Fig. 3).

An assumed density function
Consequently, 5 10 3 ≈ M and 5 10 1 . The insert of Fig. 3 is contained in the WF as well as in the CLD and as well as in the SAS curve. Now, the inverse problem consists in the determination of Depending on the respective practical problem to be solved, ) (l A can be measured directly or obtained from the WF. Furthermore, the WF can be traced back to the scattering intensity. by Eq. (6). Taking into account the current opinion of the specialists in the field of integral equations, for example Camko et al. [15] , Wiener [16] , including the actual developments in the field of computer algebra, Mathematica [17] , it seems difficult to find out an exact analytical solution. Fedorova & Schmidt [11] reached analytic solutions for particle sizing problems for a sequence of particle shapes. The hemisphere case is not included there. There are no simplifying integral transformations for the actual kernel type. Evidently, the only way is to apply numerical procedures, developed for solving a wide group of inverse problems.
Another way for solving the inverse problem is to apply a method recently developed by Ohser, also Ohser & Mucklich [4] . This procedure should work for all cases, when the kernel function is known explicitly with high precision (which is fulfilled by Eq. (3)). However, in the present paper the goal of developing a simple practicable solution was reached by puzzling over a suited approximation of the function ) , ( R r H γ . In the beginning, it seemed to be advisable to apply series expansions, Gille [10] , or their combination at well chosen r-positions. A two-dimensional series expansion of ) , ( R r H γ was also inserted into Eq. (6). A long series of trials has shown that the best suited approximation (with respect to the desired solution of Eq. (6)) is a piecewise linear approximation on two r-intervals. This is based on the fact that the CLD (Fig. 2) can be approximated by a straight line in the first r interval, . After two differentiation steps (operating with Eqs. (3-6)), the second derivative . Considering the intrinsic properties of the CLD (Fig. 2) in greater detail, two polynomials of third degree on the r-intervals,

A procedure for determining
) (R f : Because of experimental errors, a lower limit of information, 0 r r = , exists. This has to be taken into account for establishing a stable procedure, based on Eq. (12). For example in image analysis 0 r is given by the resolution limit of the image. On computer screens 0 r is connected with the number of pixels used. For micro-particles in the field of SAS, ≈ 0 r 2nm. On the other hand, there is an upper experimental limit max r . Consequently, limiting conditions In the simulated cases these parts have been obtained from random number generators.
The procedure involves these restrictions. The first step is the determination of M3 from the WF, followed by a recursive solution of the recurrence equation Eq. (12).
The algorithm is: π

Application and stability:
The given algorithm is applied to the simulated data set, section 2.3 and Fig. 3. The second derivative of the WF , whose numbers possess two digits. This represents a not normalized experimental CLD. Based on T, the function ) (R f should reappear. In fact, the dashed line (Fig. 4) is obtained, based on Eqs. (11,12) which are involved in the procedure. The full line in Fig. 4 is given by Eq. (7). Figure 4 demonstrates the stability of the method for the bimodal distribution type assumed. Furthermore, the procedure is checked for several conditions and for other size distribution models; Rayleigh-distribution, Normal-distribution, Log-Normal-distribution. The most sensitive point is the parameter 0 r . Exceptional situations can be constructed, in which the method fails (in the case of extremely narrow or not continuous distributions). Here, the approximation errors given by the difference between H γ and A γ (or H A and A A ) influence the stability. In most of the tested cases the assumed r is sufficiently small, compared with the abscissa value . Tests show that the relation 5 / 0 m R r < is sufficient. In most experimental cases, 0 r is known beforehand, however the parameter m R results (and can be checked) only after performing the data evaluation.
is a superposition of two Maxwell distribution densities (Fig. 3) ) values, each number with two digit precision). Here 2 0 = r and 200 max = r are inserted. Even in this case the procedure is applicable (logically, the smaller 0 r the better the dashed line agrees with Eq. (7)).

CONCLUSION
The problem of estimation of the radius distribution function and its density from the observation of chord lengths in IUR sections of a macroscopic isometric system of hemispheres with straight lines is solved. After writing the mean WF as an integral over the hemisphere size density, a procedure for determining is established. This is based on the analytic expression for the single hemisphere chord length distribution and further, on the simplifying piecewise approximation of the WF by polynomials of third degree. Thus, a simple recurrence equation follows. It traces the unknown parameters back to a data table T , involving the frequency of chord segments i l . In most practical cases, smooth distributions without singularities exist. Under this condition, a sequence of tests shows: A stable solution is obtained based on at least . The approximation error, inherent in the method, is unimportant in contrast to the errors resulting from the lower experimental limit

Appendix B: Scattering experiments
Particle sizing by use of SAS experiments is a standard method, Guinier & Fournet [18] . Here, the particle shape must be known a priori.  (13) However, in order to exclude the confusing influence of so-called interparticle interferences the conditions explained in Fig. B1 must be fulfilled (socalled quasi-diluted particle arrangement).
This figure shows an assumed configuration of hemispheres 1, 2, 3 with volumes 1 V , 2 V , 3 V and corresponding WFs 1 γ , 2 γ , 3 γ . The diameter l R = 2 is smaller than the minimum of all possible distances m between any points of the surface areas of the hemispheres, 1-2, 1-3, 2-3. In such a system, the WF in the interval is the volume averaged mean, whatever the particle shape, . The normalization

Appendix C: Hemispherical particles in aluminum alloys
There is an eminent physical background in materials research, Dutkievicz et al. [20] . This particle shape has already been observed in metal physics on a length scale of about 50 nm in several alloys. Lukac [21] investigated the influence of the strain rate on the plastic instabilities in aluminum based alloys (Fig. C1).
can be expressed by ratios of projection portions perpendicular to the α -direction. Finally, was performed, Gille [14] . The analytic representation of ) , ( R l A bc is more complicated than that of ) , ( R l A cc , The first moment of cc A is Fig. 2) takes into account the geometry of IUR-chords on a straight test line (Fig. 1 . However, simultaneously this complicates the integrands in Eq. (8). The effort is not worth it.
Any modification of H γ leads to a certain change of the moments of the CLD of the single hemisphere. Analyzing this effect carefully an optimization problem results. A normalization factor, which is indispensable, cannot correct all moments of the CLD simultaneously. Eqs. (9,15) yield an exact 0th moment of the CLD. However, as a consequence of the linearization applied, the higher moments of ) , ( R l A A differ somewhat from the exact ones. The differences, due to the application of Eq. is investigated (Fig. E1).
Eqs. (3,9) have been studied by a Mathematica program. The exact WF is defined in program parts 1. and 2. Eq. (9) is compiled in parts 3. and 4. Finally, part 5. investigates ) (r d . The insert of Fig. E1 results. . The maximum deviation is smaller than 2 10 − .