Stereology
Spheres
Other shapes
Wicksell’s Corpuscle Problem
Markus Kiderlen, CSGB
Kolloquium at Osnabrück University
June 18, 2014
Local Wicksell pb.
Stereology
Spheres
Other shapes
0. Stereology at the CSGB.
Local Wicksell pb.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Prologue: What is stereology?
Stereology is ‘spatial sampling theory’: estimation of geometric
charcteristics from section or projection samples of an object.
Typical problem: estimate the volume of the hippocampus.
Possible solutions:
Stereology
Spheres
Other shapes
Local Wicksell pb.
Prologue: What is stereology?
Stereology is ‘spatial sampling theory’: estimation of geometric
charcteristics from section or projection samples of an object.
Typical problem: estimate the volume of the hippocampus.
Possible solutions:
1. Archimedes:
Stereology
Spheres
Other shapes
Local Wicksell pb.
Prologue: What is stereology?
Stereology is ‘spatial sampling theory’: estimation of geometric
charcteristics from section or projection samples of an object.
Typical problem: estimate the volume of the hippocampus.
Possible solutions:
1. Archimedes:
2. Stereology.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Volume estimation with stereology
Let X ⊂ R3 be an (smooth) object in the unit cube [0, 1]3 .
Choose a randomized plane P parallel to the x-y -plane.
Mean value stereology: We have
E λ2 (X ∩ P) = λ3 (X ),
where λn is Lebesgue measure in Rn and E is usual expectation.
Intuitive reason:
The section is Resentative as the position of P is randomized.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Volume estimation with stereology
Let X ⊂ R3 be an (smooth) object in the unit cube [0, 1]3 .
Choose a randomized plane P parallel to the x-y -plane.
Mean value stereology: We have
E λ2 (X ∩ P) = λ3 (X ),
where λn is Lebesgue measure in Rn and E is usual expectation.
Intuitive reason:
The section is Resentative as the position of P is randomized.
Classical stereology:
position and orientation of P are randomized. ‘kinematic averages’.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Volume estimation with stereology
Let X ⊂ R3 be an (smooth) object in the unit cube [0, 1]3 .
Choose a randomized plane P parallel to the x-y -plane.
Mean value stereology: We have
E λ2 (X ∩ P) = λ3 (X ),
where λn is Lebesgue measure in Rn and E is usual expectation.
Intuitive reason:
The section is Resentative as the position of P is randomized.
Classical stereology:
position and orientation of P are randomized. ‘kinematic averages’.
Local stereology:
only orientation of P is randomized.
‘rotational averages’.
Stereology
Spheres
Other shapes
Local Wicksell pb.
What is needed for stereology?
If we want the section to be representative:
One of the two assumptions needed:
• X is deterministic and the plane is suitably randomized.
Design-based approach
• the section plane is deterministic and fixed,
but the the set is random and statistically homogeneous:
Model-based approach
Stereology
Spheres
Other shapes
I. The classical Wicksell problem
for spheres
Local Wicksell pb.
Stereology
Spheres
Other shapes
The size distribution of particles
Wicksell’s corpuscle problem:
Determine the size-distribution of spherical particles from planar sections.
Assume a model-based setting:
• P is deterministic and
• X a random collection of spheres;
statistically translation invariant.
‘stationary marked point process with radii as marks’.
• Wanted: ‘mark distribution’
FR = distribution function of spheres’ radii R in R3 .
• We can estimate:
Fr = distribution function of profiles’ radii r in E .
Densities: fR and fr .
Local Wicksell pb.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Why is this interesting?
The Swedish statistician S.D. Wicksell was the first to formulate
and solve this problem in the 1920ies.
His research was motivated by two applications:
Estimate size of ball-shaped
cells in planar microscopy.
Here: Hassal corpuscles in the
thymus (’im lymphatischen
System’).
Stereology
Spheres
Other shapes
Local Wicksell pb.
Why is this interesting?
The Swedish statistician S.D. Wicksell was the first to formulate
and solve this problem in the 1920ies.
His research was motivated by two applications:
Estimate size of ball-shaped
cells in planar microscopy.
Here: Hassal corpuscles in the
thymus (’im lymphatischen
System’).
Estimate the density of stars
in a globular cluster from telescope observations.
Stereology
Spheres
Other shapes
Relations between R and r
Two effects:
1. Given a sphere hits P, its radius Rw has
1
size weighted distribution: fRw (y ) = 2ER
2yfR (y ).
Local Wicksell pb.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Relations between R and r
Two effects:
1. Given a sphere hits P, its radius Rw has
1
size weighted distribution: fRw (y ) = 2ER
2yfR (y ).
2. Given sphere with radius Rw = y hits P,
x
fr (x|Rw = y ) = p
,
y y2 − x2
0 6 x 6 y.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Relations between R and r
Two effects:
1. Given a sphere hits P, its radius Rw has
1
size weighted distribution: fRw (y ) = 2ER
2yfR (y ).
2. Given sphere with radius Rw = y hits P,
x
fr (x|Rw = y ) = p
,
y y2 − x2
=⇒
fr (x) =
x
ER
R∞
x
√fR (y ) dy
2
2
y −x
0 6 x 6 y.
Stereology
Spheres
Other shapes
The influence of the two effects
• There are populations for which Er > ER (!)
Local Wicksell pb.
Stereology
Spheres
Other shapes
The influence of the two effects
• There are populations for which Er > ER (!)
• Z is Rayleigh(σ)-distributed
⇐⇒ Z =
√
X 2 + Y 2 , where (X , Y ) ∼ N(o, σ 2 )
Density: fZ (t) =
t2
t − 2σ 2
e
,
σ2
t > 0.
Reproducing property of the Rayleigh distribution
R ∼ Rayleigh(σ) ⇐⇒ r ∼ Rayleigh(σ)
• It is the only distribution with this property.
•
simple parametric models. [Wicksell 1925],
[Keiding et al. 1972], mixture of χ-distributions,
[Bach 1959] related distributions.
Local Wicksell pb.
Stereology
Spheres
Other shapes
Local Wicksell pb.
The integral equation and analytical unfolding
fr (x) =
x
ER
R∞
x
√fR (y ) dy
2
2
y −x
• First derived by [Wicksell 1925],
• Proof by [Kendall & Moran 1963] (independence assumpt.)
• General proof [Mecke & Stoyan 1980] .
Stereology
Spheres
Other shapes
Local Wicksell pb.
The integral equation and analytical unfolding
fr (x) =
x
ER
R∞
x
√fR (y ) dy
2
2
y −x
• First derived by [Wicksell 1925],
• Proof by [Kendall & Moran 1963] (independence assumpt.)
• General proof [Mecke & Stoyan 1980] .
This is an Abel integral equation with solution
R∞
1 − FR (y ) = π2 (2ER) y √fr 2(x) 2 dx.
x −y
• the Abel integral is smoothing,
(corresponds to “1/2 integration”)
• the unfolding problem is (moderately) ill posed
Stereology
Spheres
Other shapes
Numerical unfolding: a simple algorithm
Discretization of the direct integral equation
e.g. Scheil-Schwartz-Saltykov method
r
-
q q q q q
R
Local Wicksell pb.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Numerical unfolding: a simple algorithm
Discretization of the direct integral equation
e.g. Scheil-Schwartz-Saltykov method
q q q q q
-
r
R
Scheil−Schwartz−Saltikov, 8 bins
0.00
0.00
0.05
0.05
0.10
0.10
0.15
0.15
0.20
0.20
0.25
0.25
0.30
0.30
0.35
0.35
Scheil−Schwartz−Saltikov, 20 bins
0
2
4
R
6
8
0
2
4
6
8
R
R, r ∼ Rayleigh(2), n = 1000 with 20 (left) and 8 (right) bins.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Conclusion: Numerical unfolding
• subtle interplay of numerical and statistical problems,
• the lower tail of FR is not accessible,
• no method appears to be generally best,
• large samples required, n ∼ 1000,
in binning methods: low number of bins ∼ 7 − 10
or smoothing
• Suggestion:
pilot study as input for a non-parametric method
proper study with a fitted parametric model (max. likelihood)
Stereology
Spheres
Other shapes
Local Wicksell pb.
Variants
• Thin sections of opaque spheres: “tomato salad problem”
[Bach 1959, 1967,. . .]: Fr
FR ;
(ER + δ)Er k = ck ER k+1 + δER k ,
where 2δ = thickness of the slab.
• Thin sections of transparent spheres “swiss cheese”
[Coleman 1981,1982,1983]
• Truncated Fr and measurement errors
e.g. small sphere radii unobservable or inexact due to
preparation [Cruz-Orive 1983], [Coleman 1980], . . .
• Sections with lines:
[Spektor 1950, Lord & Willis 1951]
(more unstable than 2D sections)
Stereology
Spheres
Other shapes
Local Wicksell pb.
II. Wicksell’s problem for other shapes
Stereology
Spheres
Other shapes
Local Wicksell pb.
Ellipsoids
General Wicksell problem for ellipsoids:
Determine the joint distribution of the three principal axis in R3
from the joint distribution of two principal axis in the profiles.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Ellipsoids
General Wicksell problem for ellipsoids:
Determine the joint distribution of the three principal axis in R3
from the joint distribution of two principal axis in the profiles.
• restrict attention to spheroids (= ellipsoids of revolution)
Stereology
Spheres
Other shapes
Local Wicksell pb.
Ellipsoids
General Wicksell problem for ellipsoids:
Determine the joint distribution of the three principal axis in R3
from the joint distribution of two principal axis in the profiles.
• restrict attention to spheroids (= ellipsoids of revolution)
• In a population of prolate spheroids (“cigars”) we have
uniqueness.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Ellipsoids
General Wicksell problem for ellipsoids:
Determine the joint distribution of the three principal axis in R3
from the joint distribution of two principal axis in the profiles.
• restrict attention to spheroids (= ellipsoids of revolution)
• In a population of prolate spheroids (“cigars”) we have
uniqueness.
• In a population of oblate spheroids (“flying saucers”) we have
uniqueness.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Ellipsoids
General Wicksell problem for ellipsoids:
Determine the joint distribution of the three principal axis in R3
from the joint distribution of two principal axis in the profiles.
• restrict attention to spheroids (= ellipsoids of revolution)
• In a population of prolate spheroids (“cigars”) we have
uniqueness.
• In a population of oblate spheroids (“flying saucers”) we have
uniqueness.
• In a mixed population, we do not have uniqueness.
Stereology
Spheres
Other shapes
Local Wicksell pb.
III. Wicksell’s problem with central sections
for balls
(joint with Ólöf Thórisdóttir)
Stereology
Spheres
Other shapes
Local Wicksell pb.
Wicksell’s corpuscle problem in local stereology
Local stereology: Particle properties
from sections through a reference point P.
For spherical particles:
• In R3 :
R = radius of the sphere (“size”)
Q = relative distance of P from particle center (“shape”)
Stereology
Spheres
Other shapes
Local Wicksell pb.
Wicksell’s corpuscle problem in local stereology
Local stereology: Particle properties
from sections through a reference point P.
For spherical particles:
• In R3 :
R = radius of the sphere (“size”)
Q = relative distance of P from particle center (“shape”)
• In the isotropic section plane:
r = radius of the section profile
q = relative distance of P from
profile circle center
Problem:
Find distributions of R and Q
from those of r and q.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Integral relations
Assume that Q and R are independent, P(Q = 0) = 0
1 − Fq (x) =
Z
fr (y ) = y
y
√ 1
1−x 2
∞
√
1
t 2 −y 2
Z
1 √
x
R1
1 √
t
s 2 −x 2
dFQ (s),
s
1−(y /t)2
dFQ (s)
s
dFR (t).
Hence:
• q is independent of R
• Fq determines FQ uniquely (analytic solution FQ = . . .)
• The second integral is a generalized Abel integral.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Uniqueness in the local Wicksell problem
Assume that Q and R are independent, P(Q = 0) = 0
The distributions of r and q determine those of R and Q.
• There is no reproducing distribution like the
Rayleigh-distribution before.
• There are moment and extreme value relations for the radii.
• It is open if the distribution of (q, r ) determines the
distribution of (Q, R).
• There is a two-step algorithm that resonstructs fQ and fR .
Stereology
Spheres
Other shapes
Local Wicksell pb.
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
FR(u)
FQ(u)
Unfolding example for the local Wicksell problem
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
0.1
0
0.1
0.2
0.3
0.4
0.5
u
0.6
0.7
0.8
0.9
1
0
0
1
2
3
4
5
u
6
7
True cumulative distribution functions for R (dashed)
and step the functions from the unfolding.
Left: Q ∼ unif[0, 1] (not shown) and R ∼ exp(1).
Right: Q ∼ Beta(5, 2) (not shown) and R ∼ unif[0, 10].
8
9
10
Stereology
Spheres
Other shapes
Local Wicksell pb.
References.
R.S. Anderssen and A.J. Jakeman.
Abel type integral equations in stereology. ii. computational
methods of solution and the random spheres approximation.
Journal of Microscopy, 105(2):135–153, 1975.
G. Bach.
Über die Grössenverteilung von Kugelschnitten in
durchsichtigen Schnitten endlicher Dicke.
Zeitschrift wiss. Mikroskopie, 57:265–270, 1959.
G. Bach.
Kugelgrößenverteilung und Verteilung der Schnittkreise; ihre
wechselseitigen Beziehungen und Verfahren zur Bestimmung
der einen aus der anderen, pages 23–45.
Quantitative Methods in Morphology (ed.s E.R. Weibel and H.
Elias), Springer, New York, 1967.
V. Beneš, K. Bodlák, and D. Hlubinka.
Stereology of extremes; bivariate models and computation.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Methodol. Comput. Appl. Probab., 5:289–308, 2003.
LM. Cruz-Orive.
Particle size-shape distributions: the general spheroid problem.
i. mathematical model.
J Microsc., 107:235–53, 1976.
LM. Cruz-Orive.
Distribution-free estimation of sphere size distributions from
slabs showing over- projection and truncation, with a review of
previous methods.
J. Microsc., 131:265–290, 1983.
D. Hlubinka and S. Kotz.
The generalized fgm distribution and its application to
stereology of extremes.
Applications of Mathematics, 55:495–512, 2010.
A.J. Jakeman and R.S. Anderssen.
Abel type integral equations in stereology: I. general discussion.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Journal of Microscopy, 105(2):121–133, 1975.
M.G. Kendall and P.A.P. Moran.
Geometrical probability.
Hafner Pub. Co, 1st edition, 1963.
S. Kötzer and I. Molchanov.
On the domain of attraction for the lower tail in wicksell’s on
the domain of attraction for the lower tail in wicksell’s
corpuscle problem.
In R.Lechnerova, I.Saxl, and V.Benes, editors, Proceedings
S4G. International Conference on Stereology, Spatial Statistics
and Stochastic Geometry. Prague, June 26-29, 2006., pages
91–96, 2006.
J. Mecke and D. Stoyan.
Stereological problems for spherical particles.
Math. Nachr., 96:311–317, 1980.
G.M. Tallis.
Stereology
Spheres
Other shapes
Local Wicksell pb.
Estimating the distribution of spherical and elliptical bodies in
conglomerates from plane sections.
Biometrics, 26:87–103, 1970.
R. Takahashi and M. Sibuya.
Prediction of the maximum size in wicksell’s corpuscle problem.
Annals of the Institute of Statistical Mathematics, 50:361–377,
1998.
10.1023/A:1003451417655.
S.D. Wicksell.
The corpuscle problem I.
Biometrika, 17:84–99, 1925.
S.D. Wicksell.
The corpuscle problem II.
Biometrika, 18:152–172, 1926.