The main result of this Section is the following Theorem concerning the unfolding ..... [1] A. Baddeley and L. M. Cruz-Orive: The Rao-Blackwell theorem in ...
K Y B E R N E T I K A — V O L U M E 33 ( 1 9 9 7 ) , N U M B E R 3, P A G E S 2 4 5 - 2 5 8
DECOMPOSITION IN STEREOLOGICAL UNFOLDING PROBLEMS V I K T O R B E N E Š AND
PAVEL
KREJČIŘ
Some new results in an old problem of stereological unfolding in particle systems are presented. Under the conditional independence property of particle section parameters the multivariate unfolding problem can be decomposed into a series of simpler problems. The general idea is applied to the unfolding of trivariate size-shape-orientation distribution of spheroids using the vertical uniform random sampling design.
1. I N T R O D U C T I O N T h e conditional independence relation plays a key role in a number of topics in probability theory and its applications, cf. van P u t t e n and van Schuppen [14]. In the area of sufficient statistics this relation enters in a n a t u r a l way. Therefore it is not surprizing t h a t in this context the assumptions of conditional independence appeared in a stereological study Baddeley and Cruz-Orive [1]. We present here another application of conditional independence in stereology, namely in the classical unfolding problem. Consider a system of three-dimensional particles of similar shape spread in an opaque base and observe its planar section. T h e problem may be formulated either in the design-based approach where particles are fixed and the section plane r a n d o m or in the model-based approaches where particles form a stationary r a n d o m process. Defining some geometrical parameters of particles one would like to evaluate their joint distribution from the observed par ameters of planar particle sections. T h e problem typically leads to integral equations between corresponding joint probability densities, which are solved either analyti cally (Cruz-Orive [4], Gokhale [6]) or numerically (Ohser and Miicklich [12]). In the present paper it is shown how multivariate unfolding problems may be in vestigated using the probabilistic interpretation of the kernel function in the integral equation. When a suitably defined conditional independence property is satisfied the unfolding can be decomposed into a series of simpler problems. T h e general theory is applied finally to the trivariate size-shape-orientation distribution of ellipsoidal par ticles. Using the sampling design of vertical uniform r a n d o m sections the relation between planar and spatial p a r a m e t e r s is obtained. In a statistical study numerical
246
V. BENEŠ AND P. KREJČÍŘ
EM-algorithm (Silvermann et al [13]) is used for the real d a t a evaluation and the stability of solution is discussed. 2. UNFOLDING AND C O N D I T I O N A L I N D E P E N D E N C E d
A bounded closed convex set in R is called a particle. Let a fixed particle X be described by n real geometrical parameters x\,X2, • . • ,xn. A sampling design is represented by a r a n d o m hyperplane p with probability distribution Q on the para metric space of hyperplanes. Assume t h a t the intersection Y = X C\ p ^ 0, then y is a r a n d o m closed convex set in Rd~l called a particle section. Let y\, . . . , ym be geometrical parameters describing Y, such t h a t y\, . .. ,y^, k < min(n,m) cor respond to properties of x\, . .., x^, e.g. x\, y\ size, „2, y2 shape factor etc. Let p(y\, . . . ,ym\x\, . . ., Xn,^) be the conditional probability density of y\, . . ., ym given x\,...,xn and given t h a t the particle is hit by p. The upper arrow f emphasizes t h a t the distribution p depends not only on particle characteristics but also on the sampling design Q. Further assume t h a t particles are randomly dispersed in Rd with constant inten sity Nd. If particles are still fixed (just translates of X) denote Nd-\(x\,. . ., xn) the mean (with respect to Q) intensity of particle sections in p. In the following step given Nd let particles are r a n d o m with probability density f(x\, . . . ,xn) of parameters x\, . .., xn invariant with respect to translations in R . We are interested in particle sections observed in p. D e f i n i t i o n 1. The average particle section intensity N__i and probability density g(y\,... ,ym) of parameters y\,. .. ,ym are defined by
0)
Nd-\g(y\,...,ym) Nd_i(„i,..
,,xn)p(y\,..
. , y m | x i , . . .,xn,])f(x\,..
.,a?n)dxi • • • dxn.
The stereological unfolding problem consists in the estimation of unknown particle characteristics /, Nd from the particle section distribution g and N__i, which can be observed and estimated from realizations of p. T h e first p a r t of the solution is to establish the theoretical relations. P r o p o s i t i o n 1. Nd-\g(y\,.
..,ym)
T h e unfolding problem is decribed by an equation = Nd
• • • I k(x\,..
.,xn,y\,..
.,ym)f(x\,.
.. ,xn)
dxx • • • dxn,
for some nonnegative kernel function k.
(2)
P r o o f . Use Definition 1 and put ,, k(x\,...,xn,y\,...,]jm)
v
=
N__i(xi,....xn) — P(!/l.-- 0 / c centred in the origin. T h e orientation of the axis of rotation is 9, 4>. Let a vertical section plane p have normal orientation 0* = 7r/2,0* in spherical coordinates and the distance d from the origin. Under the condition t h a t the particle is hit by p denote the semiaxes of intersection ellipse (10) by A,C, A > C and by a the angle between the semiaxis A and vertical axis (it is correctly defined whenever C 7^ A). D e f i n i t i o n 3 . T h e shape factors of the particle, its section, are defined as s = c/a, S = CI A, respectively. It follows t h a t always 0 < s < 1, 0 < 5 < 1. L e m m a 2 . The spatial a n d planar parameters of a vertical section of given ellipsoid are related as shu>*-(/i)
=
cot 6 cot a
- = ^r^ľ
(11)
(12)
Decomposition
in Stereological
251
Unfolding Problems
2
A where
_ wn
2
-(a
2
/ d а J l - — ,
=
V
2
(13)
w
v.
2
2
- c ) s i n t f c o s ( 0 * - ).
(14)
P r o o f . A straightforward algebraic calculation using L e m m a 1.
•
We proceed by randomizing the sampling design to get conditional densities for size-orientation and size-shape problem of type (2). Denote S((3,z)
= J Jo
\J\-
z2sin2pdtp
the elliptic integral of second kind, specially S(^,z)
=
S(z).
P r o p o s i t i o n 2. Under the vertical uniform r a n d o m sampling design the condi tional distributions of particle section parameters for the size-orientation, size-shape unfolding problem have densities cosO 1 - (1 - s 2 ) (sin 2 9 - cos 2 0cot2 a ) fST—d r ^ r -, L V a 2 - A 2 sin a y sin 9 - cos 2 a 7r/2 — # < a < 7r/2, 0 < A < a, pi = 0 otherwise, and
,t . n ^ pl(A,a\a,9,s,])=fc
4
A
4 P2
( A , S\a, 0, s,])
=
J
^
2
A _ ^
s -
f / 92\ r 92 11 2 2 J {l - - J [ S s m lj + ? cos 2 g - ij j
"
'
r
,'x 15
_ 1 / 2
, (16)
for s < S < s / v s 2 sin 6'+ cos 2 t9, 0 < jl < a, p 2 = 0 otherwise, respectively. Here L = jr6(a ( C,«) = 4 a ^ ( \ / l — s 2 sinfj) is the perimeter of the ellipse of particle projection (in vertical direction), b(a,0,s) its mean breadth. P r o o f . Consists of the evaluation of Jacobians analogously to the proof of The orem 2. For the size-orientation problem we start from formula (11) and d = \Ja
2
- A
2
2
2
2
2
y j l - ( l - s ) (sin 0 - cos 9 cot a ) ,
for the size-shape problem we start from formula (12) and sin(* -4)
U-1
5 = Wl ' . 2 , y (s- — l ) s i n 0
obtained from ( 1 1 ) - ( 1 4 ) .
•
T h e main result of this Section is the following Theorem concerning the unfolding problem (2) of size-shape-orientation distribution. Let f(a,0,s), g(A,a,S) be the probability densities of spatial and planar parameters, respectively. Further denote
£( a , 0) = y ^ « E _ _ J
an d
B(s, 9) = sin flyT^2.
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V. BENEŠ AND P. KREJČÍŘ
T h e o r e m 3. The size parameter A is strongly conditionally independent on the shape factor S and orientation a and the outer size problem of the decomposition is / °° A / -=-• h(a, a, S) da (17) 2 2 J A Va - A for some nonnegative function h and any fixed a, S. Let H(a, a,S) = L fQ h(a, ft, T) d(3dT. The inner shape-orientation problem for any fixed a is H(a, a, S) = - / / K(a, S, 0, s) f(a, 9, s) sin 9 dOds, (18) NAg(A,
W h e i e
a, S) = 2NV
K(a,S,9,s)
= mm(Kl(a,0,s),K2(S,0,s)).
(19)
Here for each fixed 0, s Ki(a, 0, s) = £(arcsin D(a, 0), B(s, 0)) for
TT/2
- 6 < a < ir/2, Ki(a, 9,s) = 0 for a
b = c under the same notation as in the previous subsection. The unfolding problem for joint distribution of spatial parameters (a,9,s) from planar parameters (A,a,S) cannot be derived exactly in the same way as in the oblate case. It will be shown later that it holds P r o p o s i t i o n 3. In the prolate case the parameter A is not strongly conditionally independent of S and a given a, 9, s. However, still an analogous way exists, namely to replace in the analysis a, A by the shorter semiaxes c,C. In fact the triplet c,9,s yields the same information as a, 9, s. Therefore solution of the unfolding problem between joint probability densities f(c, 9, s) and g(C, a,S) of spatial, planar parameters, respectively, is satisfactory for practical statistical purposes. First let the particle be a fixed prolate rotational ellipsoid E3 centred in the origin. L e m m a 3. The spatial and planar parameters of a vertical section of given ellipsoid are related as s'm(-, (27) 2 2 Ь л/с - С соз а У зт 9 - зт а
254
V. BENEŠ AND P. K R E J Č Í Ř
for 0 < a < 9, 0 < C < c, p\ = 0 otherwise, and C
p2(C, S\c, 9, s,])=j 2
2
-
[(S2 - s2) (sin 2 9 + s2 cos 2 9 - S2)} ~ 1 / 2 , (28)
2
for s < S < v s cos 9 -f sin 9, 0 < C < c, p2 = 0 otherwise, respectively. Here L = nb(a,9,s) = 4cZ(s,9) E(M(s,9)) is the perimeter of the ellipse of particle projection (in vertical direction), b(a,9,s) its mean breadth. P r o o f . Consists of the evaluation of Jacobians analogously to the proof of The orem 2. For the size-orientation problem we start from formula (23) and 2
2
2
d= \/c -C \Jl
2
2
2
+ (s~ - I) (sin 9 - cos 9 tan a),
for the size-shape problem we start from formula (24) and S2-s< cos(ф*-ф) = x l - 2 , , ч _ , , 2 / > , (l-s )sin'9
(29)
obtained from (23)-(26).
•
Remark. For the longer semiaxes it holds d = -\/s2a2 — S2A2, the Jacobian of this transformation (including (29)) cannot be factorized and the negative result of Proposition 3 follows. Concerning the unfolding problem of size-shape-orientation distribution we get the following result. T h e o r e m 4. The size parameter C is strongly conditionally independent on the shape factor S and orientation a and the outer size problem of the decomposition is NAg(Ctat'S)
= 2NvJ
/ °°
C -j====h(c,a,S)dc
(30)
for some nonnegative function h and any fixed a,S. Let H(c,a,S) = f0 f0 h(c, (3, T) d(3dT. The inner shape-orientation problem for any fixed c is H(c, a, S) = - í í K(a, S, 9, s) f(c, 9, s) sin 9d9ds,
(31)
where K(a, S, 9, s) = max (o, K^a,
9, s) + K2(S, 9, s) - y/'Z(s,9) S(M(s, 0))) .
(32)
Here for each fixed 9, s Ki{atets)
=
/ v
z'(s,lj)^(arcsin(cot0tana),
M(s,9))
(33)
Decomposition
in Stereological Unfolding Problems
255
for 0 < a < 9, Kx(a, 9, s) = y/Z(s, $) £(M(s, 9)) for a > 9 and K2(S,9,s)
y/Z(s,$)£ aгcsm
=
(щ--ч1-i),щ',в)
(34)
Í - ^\lz(s,e) - ^ for s < S < y/s2 cos2 9 +sin2 9, K2(S,9,s) y/Z(s,9)£(M(s,9)) otherwise.
= 0 for S < s and K2(S,9,s)
=
P r o o f . Proposition 4 yields the strong conditional independence of size and formula (30) follows as in Theorem 2 and Corollary 1. From formulas (22)-(24) in Lemma 3 it follows that for fixed 9, s, S = S(a) = yfs2 + (l-s2)
(sin2 9 - cos2 9 tan 2 a),
(35)
WThich means that orientation and shape factor are conditionally functionally dependent and the joint conditional density p(a, S\9, s, f) is degenerate. Observe that the transformation S(a) in (35) is monotone decreasing on (0, 5) for each fixed s, 9. Again by Mikusinski et al [10], the joint conditional distribution function
P(a,S|9,*,T)-
A
> ^ . ^ - ! > » / ) - K(si,9j,Sk,ai) -K(si ,9j,Sk-i,ai-.i)
ai-i,Sk)
(37)
+ K(si, 9j, Sk, a.-i)]
using (19), i, j , k,l = l , . . . , m . The discrete analogue of (18) is the system of equations m m Hki-Y^Y^PijkiFij, (38) i=lj =l
which is solved by EM-algorithm with Ath iteration step -,(A + 1) _
Í-
—
F
,(A) ij V^V^
HklPijkl
f.. 2Lu2—< -A
(39)
kl
Л o _i-r_ Ш l ł l Q І i ł û Г Q f í л n JГ^ 1 where t{j = J2k HiPijki, r*. = £ i E ; Fij^ !«.., Pijkh, As an initial iteration Ft) = H8j is sufficient. In a statistical study, a sample of n = 10017 particle sections of a composite material (Benes et al [2]) was classified according to (36) with b = 1.756, j = 1 , . . . ,m; v — 1.5, m = 8. The matrix P of coefficients Pijki for the inner problem (38) has size 64 x 64 and condition number cond(P) = ||E|| ||E _ 1 || = 75.75 using 7
Decomposition
in Stereological Unfolding
257
Problems
the norm ||P|| = \/YlijkiPijki- This relatively low value (cf. Gerlach and Ohser [5]) justifies the use of the method. Histogram FHJ of estimated frequencies of spatial parameters a/, t?,-, Sj obtained from (39)for each fixed o.\ is in Figure 1. It enables further investigation of various kinds of dependencies within the particle system.
a i size
•*•••! • ' ? • : • : •
•:-:f:;. shape
M • • ; *r>^ orientation •t
Fig. 1. Histogram of estimated spatial size-shape-orientation distribution. The volume of three dimensional balls in the Figure is proportional to the estimated values of FUJ, 1, i, j = 1,..., 8. The axes intersect in the point I = i — j — 1.
6. DISCUSSION M0ller [11] proved that it is possible to reconstruct an ellipsoid completely from three parallel sections. His method is hardly applicable in quantitative metallography from two reasons. First the preparation of appropriate parallel sections in hard materials with small particles (cf. Benes et al [2]) is almost impossible, while a vertical plane (e.g. parallel to the deformation axis) is easily obtained. From the same reason also the assumption-free methods of stereology (Karlsson and Cruz-Orive [8]) may sometimes be useless. Secondly M0ller's method works for perfect ellipsoids while in practice the shape assumption is often an approximation, only. Therefore we revisited the 70 years old problem in order to pose a new three-parametric ill-posed problem, the solution of which is, thanks to modern numerical approaches, acceptable for practice.
258
V. BENES AND P. KREJCTR
ACKNOWLEDGEMENTS T h e research was s u p p o r t e d by t h e G r a n t Agency of the Czech Republic, projects No. 2 0 1 / 9 6 / 0 2 2 8 , 106/95/1482 and 2 0 1 / 9 4 / 0 4 7 1 . We thank to Prof. A . M . Gokhale for sending us a preprint of his p a p e r [6] before its publication. (Received March 29, 1996.)
REFERENCES [1] A. Baddeley and L. M. C r u z - O r i v e : T h e Rao-Blackwell t h e o r e m in stereology and some counterexamples. Adv. in Appl. P r o b a b . 2 7 ( 1 9 9 5 ) , 2-19. [2] V. Benes, A . M . Gokhale and M. Slamova: Unfolding the bivariate size-orientation distribution. A c t a Stereol. 15, (1996), 1, 9-14. [3] R. Coleman: Inverse problems. J. Microsc. 1 5 5 ( 1 9 8 9 ) , 3, 233-248. [4] L. M. C r u z - O r i v e : Particle size-shape distributions: T h e general spheroid problem, I., II. J. Microsc. 1 0 7 ( 1 9 7 6 ) , 3, 235-253, 1 1 2 ( 1 9 7 8 ) , 153-167. [5] W . Gerlach and J. Ohser: On the accurancy of numerical solutions for some stereological problems as t h e Wicksell corpuscule problem. J. B i o m a t h . 28 (1986), 7, 881-887. [6] A. M. Gokhale: E s t i m a t i o n of bivariate size and orientation distribution of microcracks. A c t a Metall. and Mater. 44 (1996), 2, 475-485. [7] I. S. G r a d s h t e j n and I. M. Ryzhik: Tables of Integrals, Sums, Series and P r o d u c t s (in Russian). G I F M L Moscow 1963. [8] L. M. Karlsson and L . M . C r u z - O r i v e : T h e new stereological tools in metallography: estimation of pore size and n u m b e r in aluminium. J. Microscopy 165 (1992), 3, 3 9 1 415. [9] A. Kleinwachter and M. Zahle: Size distribution stereology for quasiellipsoids in Rn. M a t h . Oper. S t a t . 1 7 ( 1 9 8 6 ) , 332-335. [10] P. Mikusinski, H. Sherwood and M. D. Taylor: Probabilistic i n t e r p r e t a t i o n s of copulas and their convex s u m s . In: Advances in Probability Distributions with Given Marginals (Dall'Aglio et al, eds.), Kluwer Acad. PubL, Dordrecht 1991, p p . 95-112. [11] J. M0ller: Stereological analysis of particles of varying ellipsoidal s h a p e . J. Appl. P r o b a b . 2 5 ( 1 9 8 8 ) , 322-335. [12] J. Ohser and F . Miicklich: Stereology for some classes of polyhedrons. Adv. in Appl. P r o b a b . 2 7 ( 1 9 9 5 ) , 2, 384-96. [13] B . W . Silvermann, M. C. Jones, D . W . Nychka and J. D. Wilson: A s m o o t h e d E M approach to indirect estimation problems, with particular reference to stereology and emission tomography. J. Roy. Statist. Soc. Ser. B 52 (1990), 271-324. [14] C. van P u t t e n and J. H. van Schuppen: Invariance properties of t h e conditional independence relation. A n n . P r o b a b . 13 (1985), 3, 934-945. [15] S. D. Wicksell: T h e corpuscule problem. A m a t h e m a t i c a l s t u d y of a biometrical problem. Biometrika 1 7 ( 1 9 2 5 ) , 84-88.
Prof. RNDr. Viktor Benes, DrSc, Katedra technicke matematiky, (Department of Mathematics, Faculty of Mechanical Engineering versity), Karlovo nam. 13, 121 35 Praha 2. Czech Republic. Mgr. Pavel Krejcir, KPMS, Matematicko-fyzikdlni partment of Statistics, Faculty of Mathematics Sokolovskd 83, 18600 Praha 8. Czech Republic.
Fakulta strojni CVUT - Czech Technical Uni-
fakulta Univerzity Karlovy (Deand Physics - Charles University),
