Wicksell's Corpuscle Problem in Local Stereology - …pure.au.dk/portal/files/45769572/wicksell.pdf · The classical Wicksell pb.Local Wicksell pb.Uniqueness & Reconstruction Wicksell’s

The classical Wicksell pb. Local Wicksell pb. Uniqueness & Reconstruction

Wicksell’s Corpuscle Problem in Local Stereology

Markus Kiderlen, (joint work with Ólöf Thórisdóttir)

CSGB, University of Aarhus

Workshop on Convexity and Geometric Tomography

Aarhus, June 7, 2012


Motivation

General goal: Reconstruction ofparticles from planar sections.

• Geometric tomography

deterministic particles, known section planes,

• Stochastic geometry (’model based approach’)

random particle system (stationary RACS), arbitrary sections

• Stereology (’design based approach’)

particle system intersected with randomized planes• classical: IUR sections (isotropic uniform random)• local: central sections (individual sections through a reference

point)


Motivation


• Geometric tomographydeterministic particles, known section planes,

• Stochastic geometry (’model based approach’)

random particle system (stationary RACS), arbitrary sections



point)


Motivation



• Stochastic geometry (’model based approach’)random particle system (stationary RACS), arbitrary sections



point)


Motivation



• Stochastic geometry (’model based approach’)random particle system (stationary RACS), arbitrary sections

• Stereology (’design based approach’)particle system intersected with randomized planes

• classical: IUR sections (isotropic uniform random)• local: central sections (individual sections through a reference

point)


I. The classical Wicksell problem

for spheres


The size distribution of particles

Wicksell’s corpuscle problem:Determine the distribution of sphericalparticles from planar sections.

Assume: X random collection of spheres, stationary.

• Wanted:NV = mean number of spheres per unit volume in R3,

FR= distribution function of spheres’ radii R in R3.• We can estimate:NA mean number of spheres per unit volume in E ,

Fr= distribution function of profiles’ radii r in E .

Densities: fR and fr .


Relations between R and r

Two effects:1. Given a sphere hits E , its radius Rw has

size weighted distribution: fRw (y) = 12ER 2yfR(y).

2. Given sphere with radius Rw = y hits E ,

fr (x |Rw = y) =x

y√

y2 − x2, 0 6 x 6 y .

=⇒ fr (x) = xER

∫∞x

1√y2−x2

dFR(y)






fr (x |Rw = y) =x

y√

y2 − x2, 0 6 x 6 y .

=⇒ fr (x) = xER

∫∞x

1√y2−x2

dFR(y)






fr (x |Rw = y) =x

y√

y2 − x2, 0 6 x 6 y .

=⇒ fr (x) = xER

∫∞x

1√y2−x2

dFR(y)


Put another way: Relations between R and r

If Λ ∈ [0, 1] is independent of R with density s 7→ s/√1− s2, then

r = ΛRw

• Consequence: Moment relations

Erk = (EΛk)(ERkw ) = ck+1

ERk+1

ER, k = −1, 0, 1, . . . .

with explicitly known ck ’s.• Mean diameter: 2ER = π

2 (Er−1)−1, (k = −1)

• Specific total number: NV = (2ER)−1NA = 2π (Er−1)NA

NV ≈ 2nπ

∑ni=1 r

−1i “Fullman’s formula”




r = ΛRw



ERk+1

ER, k = −1, 0, 1, . . . .


2 (Er−1)−1, (k = −1)


NV ≈ 2nπ

∑ni=1 r





r = ΛRw



ERk+1

ER, k = −1, 0, 1, . . . .


2 (Er−1)−1, (k = −1)


NV ≈ 2nπ

∑ni=1 r



Reproducing distribution

• There are populations for which Er > ER (!)

• Z is Rayleigh(σ)-distributed⇐⇒ Z =

√X 2 + Y 2, where (X ,Y ) ∼ N(o, σ2)

Density: fZ (t) = tσ2 e− t2

2σ2 , 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.


Reproducing distribution

• There are populations for which Er > ER (!)

• Z is Rayleigh(σ)-distributed⇐⇒ Z =

√X 2 + Y 2, where (X ,Y ) ∼ N(o, σ2)

Density: fZ (t) = tσ2 e− t2

2σ2 , 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.


The integral equation and analytical unfolding

fr (x) = xER

∫∞x

1√y2−x2

dFR(y)

• 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

1− FR(y) = 2π (2ER)

∫∞y

fr (x)√x2−y2

dx .

• the Abel integral is smoothing,(corresponds to “1/2 integration”)

• the unfolding problem is (moderately) ill posed


The integral equation and analytical unfolding

fr (x) = xER

∫∞x

1√y2−x2

dFR(y)

• 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

1− FR(y) = 2π (2ER)

∫∞y

fr (x)√x2−y2

dx .

• the Abel integral is smoothing,(corresponds to “1/2 integration”)

• the unfolding problem is (moderately) ill posed


Numerical unfolding I

Shoals of algorithms have been suggested.

1. Inverse integral equation with empirical distribution“naive estimation”

-

r

61

0 1 2 3 4 5 6 7

−0.

50.

00.

51.

0

Naive estimator

R

R ∼ Rayleigh(2) ⇒ r ∼ Rayleigh(2)left: n = 7, F̂R compared with FR .


Numerical unfolding I

Shoals of algorithms have been suggested.

1. Inverse integral equation with empirical distribution“naive estimation”

-

r

61

0 1 2 3 4 5 6 7

−0.

50.

00.

51.

0

Naive estimator

R

R ∼ Rayleigh(2) ⇒ r ∼ Rayleigh(2)left: n = 7, F̂R compared with FR .


Numerical unfolding II

3. Discretization of the direct integral equatione.g. Scheil-Schwartz-Saltykov method S3M

- -q q q q qr R

0 2 4 6 8

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Scheil−Schwartz−Saltikov, 20 bins

R

0 2 4 6 8

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35


R

R, r ∼ Rayleigh(2), n = 1000 with 20 (left) and 8 (right) bins.


Numerical unfolding II

3. Discretization of the direct integral equatione.g. Scheil-Schwartz-Saltykov method S3M

- -q q q q qr R

0 2 4 6 8

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35


R

0 2 4 6 8

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35


R

R, r ∼ Rayleigh(2), n = 1000 with 20 (left) and 8 (right) bins.


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 methodproper study with a fitted parametric model (max. likelihood)


Variants• Thin sections of opaque spheres: “tomato salad problem”[Bach 1959, 1967,. . .]: Fr FR ;

(ER + δ)Erk = ckERk+1 + δERk ,

where 2δ = thickness of the slab.• Thin sections of transparent spheres “swiss cheese”[Coleman 1981,1982,1983]

• Truncated Fr and measurement errorse.g. small sphere radii unobservable or inexact due topreparation [Cruz-Orive 1983], [Coleman 1980], . . .

• Sections with lines:[Spektor 1950, Lord & Willis 1951](more unstable than 2D sections)

• Design based formulation [Jensen 1984]


II. The Wicksell problem in local stereology


Wicksell’s corpuscle problem in local stereology

Local stereology: Particle propertiesfrom 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 profileq = relative distance of P fromprofile circle center

Problem:Find distributions of R and Qfrom those of r and q.


Wicksell’s corpuscle problem in local stereology

Local stereology: Particle propertiesfrom 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 profileq = relative distance of P fromprofile circle center

Problem:Find distributions of R and Qfrom those of r and q.


Integral relations

Throughout: P(R > 0) = 1 and (for the talk) P(Q > 0) = 1.

o ∈ random ball, intersected by an indep. isotropic hyperplane

For 0 6 x , 0 6 y < 1 we have

F(r ,q)(x , y) = 1− σn−1

σnE

[B

(1− (1− Z )+;

12,n − 12

)],

with Z = 1Q2 max

{(R2−x2)+

R2 , (Q2−y2)+

1−y2

}.

Notation used:• constants σk = surface area of Euclidean ball in Rk ,• incomplete beta function B(z ;α, β) =

∫ z0 tα−1(1− t)β−1 dt

• positive part x+ = max{x , 0},


Consequences and alternative formulation

Consequences:• Fr and Fq always have densities,• F(r ,q) generally does not have a density.

Alternative formulation:

There is a stoch. variable Γ (whose density is a fct. of FQ) s.th.

r = ΓR.

If R and Q are independent, so are R and Γ.


Consequences and alternative formulation

Consequences:• Fr and Fq always have densities,• F(r ,q) generally does not have a density.

Alternative formulation:

There is a stoch. variable Γ (whose density is a fct. of FQ) s.th.

r = ΓR.

If R and Q are independent, so are R and Γ.


Moment relations

Assume that Q and R are independent:

• Moment relations: Erk = ck(Q)ERk k = 0, 1, 2, . . .,

ck(Q) = EΓk n=3= 1

2E[Q−1

(σk+3σk+2− B(1− Q2; k+2

2 , 12))]

.

• As c1(Q) = EΓ < 1 there is no reproducing distribution for R .

• For n = 3, ck(Q) = 1− Ef (q) with

f (y) = kπ

∫ y0

1−s2s

∫ s2

0

√t(1−t)(k−2)/2√s2−t dt ds.

• In particular 3c2(Q) = 2 + E [(1− q2)3/2], and12π 1

N

∑Ni=1 r

2i

2+ 1N

∑Ni=1(1−q2

i )3/2

estimates the average surface area 4πER2.


Moment relations



ck(Q) = EΓk n=3= 1

2E[Q−1

(σk+3σk+2− B(1− Q2; k+2

2 , 12))]

.



f (y) = kπ

∫ y0

1−s2s

∫ s2

0

√t(1−t)(k−2)/2√s2−t dt ds.


N

∑Ni=1 r

2i

2+ 1N

∑Ni=1(1−q2

i )3/2



Moment relations



ck(Q) = EΓk n=3= 1

2E[Q−1

(σk+3σk+2− B(1− Q2; k+2

2 , 12))]

.



f (y) = kπ

∫ y0

1−s2s

∫ s2

0

√t(1−t)(k−2)/2√s2−t dt ds.


N

∑Ni=1 r

2i

2+ 1N

∑Ni=1(1−q2

i )3/2



Moment relations



ck(Q) = EΓk n=3= 1

2E[Q−1

(σk+3σk+2− B(1− Q2; k+2

2 , 12))]

.



f (y) = kπ

∫ y0

1−s2s

∫ s2

0

√t(1−t)(k−2)/2√s2−t dt ds.


N

∑Ni=1 r

2i

2+ 1N

∑Ni=1(1−q2

i )3/2



Uniqueness I

The relation between Q and q: For 0 < y < 1

fq(y) = 2σn−1σn

yn−2

(1−y2)n/2

∫ 1

y

1√s2−y2

(1−s2)(n−1)/2

s dFQ(s).

This is essentially an Abel integral ⇒ Fq determines FQ .

The relation between R and r : For y > 0, n = 3, R,Q indep.

fr (y) = y

∫ ∞y

1√t2−y2

1t

∫ 1√1−(y/t)2

dFQ(s)s dFR(t).

This is a generalized Abel integral.


Uniqueness I

The relation between Q and q: For 0 < y < 1

fq(y) = 2σn−1σn

yn−2

(1−y2)n/2

∫ 1

y

1√s2−y2

(1−s2)(n−1)/2

s dFQ(s).

This is essentially an Abel integral ⇒ Fq determines FQ .

The relation between R and r : For y > 0, n = 3, R,Q indep.

fr (y) = y

∫ ∞y

1√t2−y2

1t

∫ 1√1−(y/t)2

dFQ(s)s dFR(t).

This is a generalized Abel integral.


Uniqueness II

Assume that Q and R are independent.

Then r and q are independent and the marginals of R and Q areuniquely determined by the distribution of (r , q).

Main idea: log r = log Γ + logR⇒ ϕlog r = ϕlog Γ · ϕlogR (characteristic functions)

Without independence assumption:• The marginals of r and Q do not determine F(R,Q),

• It is open whether F(r ,q) determines F(R,Q).


Uniqueness II

Assume that Q and R are independent.

Then r and q are independent and the marginals of R and Q areuniquely determined by the distribution of (r , q).

Main idea: log r = log Γ + logR⇒ ϕlog r = ϕlog Γ · ϕlogR (characteristic functions)

Without independence assumption:• The marginals of r and Q do not determine F(R,Q),

• It is open whether F(r ,q) determines F(R,Q).


Numerical Unfolding: Reconstruction of FR

S3M with 20 bins and N = 100 section profiles.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u

FR

(u)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

uF

R(u

)

Q ∼ unif(0, 1), R ∼exp(1) Q ∼ Beta(2, 5), R ∼unif(0, 1)


Concluding example

What if the reference set S (yellow) is not a point?

Assume: S is a sphere of fixed radius ρ > 0,concentric with the particle;the section plane is IUR hitting S

Alternative interpretation: Local Wicksell problem with Q = ρ a.s.

• Relation between r and R:

fr (y) = yρ

∫ y/√

1−ρ2

ydFR(t)

t√

t2−y2

• Moments: Erk = ck(ρ)ERk , k = 0, 1, . . .

where ck(ρ) = 12ρ

(σk+3σk+2− B(1− ρ2; k + 1, 1

2))


Concluding example

What if the reference set S (yellow) is not a point?

Assume: S is a sphere of fixed radius ρ > 0,concentric with the particle;the section plane is IUR hitting S

Alternative interpretation: Local Wicksell problem with Q = ρ a.s.

• Relation between r and R:

fr (y) = yρ

∫ y/√

1−ρ2

ydFR(t)

t√

t2−y2

• Moments: Erk = ck(ρ)ERk , k = 0, 1, . . .

where ck(ρ) = 12ρ

(σk+3σk+2− B(1− ρ2; k + 1, 1

2))


References.

R.S. Anderssen and A.J. Jakeman.Abel type integral equations in stereology. ii. computationalmethods of solution and the random spheres approximation.Journal of Microscopy, 105(2):135–153, 1975.

G. Bach.Über die Grössenverteilung von Kugelschnitten indurchsichtigen Schnitten endlicher Dicke.Zeitschrift wiss. Mikroskopie, 57:265–270, 1959.

G. Bach.Kugelgrößenverteilung und Verteilung der Schnittkreise; ihrewechselseitigen Beziehungen und Verfahren zur Bestimmungder 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.


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 fromslabs showing over- projection and truncation, with a review ofprevious methods.J. Microsc., 131:265–290, 1983.

D. Hlubinka and S. Kotz.The generalized fgm distribution and its application tostereology 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.


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 onthe domain of attraction for the lower tail in wicksell’scorpuscle problem.In R.Lechnerova, I.Saxl, and V.Benes, editors, ProceedingsS4G. International Conference on Stereology, Spatial Statisticsand Stochastic Geometry. Prague, June 26-29, 2006., pages91–96, 2006.

J. Mecke and D. Stoyan.Stereological problems for spherical particles.Math. Nachr., 96:311–317, 1980.

G.M. Tallis.


Estimating the distribution of spherical and elliptical bodies inconglomerates 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.

Documents

Wicksell's Corpuscle Problem in Local Stereology - …pure.au.dk/portal/files/45769572/wicksell.pdf · The classical Wicksell pb.Local Wicksell pb.Uniqueness & Reconstruction Wicksell’s