Approximate Likelihood Procedures for the Boolean Model Using Linear - - PowerPoint PPT Presentation

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Approximate Likelihood Procedures for the Boolean Model Using Linear Transects

John C. Handley Xerox Corporation jhandley@crt.xerox.com

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Boolean Random Set Model

1 2 1 2

Poisson process { , , } in , intensity Random compact set { , , } subsets of Boolean model: ( )

n n n n n

S S S ξ ξ ρ ξ = + K K

U

R R B 0.1 ρ = 1.0 ρ =

lognormally distributed disks

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Some References

• Matheron, G., Random Sets and Integral Geometry, 1975.
• Serra, J., “The Boolean model and random sets,” Computer

Graphics and Image Processing,1980.

• Cressie, N. and G. M. Laslett, “Random set theory and

problems of modeling,” SIAM Review, 1987.

• Hall, P., Introduction to the Theory of Coverage Processes,

1988.

• Stoyan, D., W.S. Kendall and J. Mecke, Stochastic

Geometry and Its Applications, 1995.

• Molchanov, I. S., Statistics of the Boolean Models for

Practitioners and Mathematicians, 1996.

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Applications

• Materials science: aggregates
• Cellular telecommunications
• Ecology
• Biomedical imaging
• Food science

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Background

• Least squares and methods-of-moments

based on capacity functional:

( ) ( ) Pr( ) 1 exp( ( ) ) compact is reflected about origin Estimate LHS from realization, RHS is parameterized for some models

n n n

S T K K E S K K K K ξ ρ µ = +   = ∩ ≠ ∅ = − − ⊕   ( (

U

B

B B

Limited inference for these estimation methods Maximum likelihood methods difficult to formulate in continuous setting

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Strategy

• Approximate 1D continuous model with 1D

discrete model

• Develop probability models for 1D discrete

model

• Take 1D samples of 2D model
• Form approximate likelihood for estimation
• Bonus: 1D methods are computationally

efficient

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Boolean model sampled by linear transects to produce linear Boolean models

An intersection of a line with a 2D Boolean model is a 1D Boolean model (consequence of Poisson mapping theorem)

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1D continuous Boolean model

clump gap Poisson arrivals with rate λ Random segment lengths X with dist. C

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Laplace-Stieltjes transform of clump-length density of 1D continuous Boolean model (see Hall (1988):

• K = clump-length
• C = segment length distribution

g ( ) ( ) s e dP K x

sx

= £

• z

= +

• L

N M O Q P F H G I K J

z z

• 1

1

1

( / ) exp { ( )} s st C x dx dt

t

l l l

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1D discrete Boolean model (binomial germ-grain model)

clump gap Binomial arrivals p = marking probability Random segment length X with dist. C

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Self-covering event E(1,m) for the Boolean model

E E E E E E ( , ) ( ) '( , ) ( , ) ( ( , )) ( ) ( ) ( ) ( ( , )) ( ( , )) ( ) ( ) 1 2 1 1 1 1 1 11 1

1 1 1 1 1 1

m X k j j m P m F m F j F i P j m P F F

j k k m i j j m

= = « « + =

• +

=

• =

= =

• =

’ Â U U

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Clump-length distribution of linear discrete Boolean model

• K = clump length
• C = segment length distribution
• p = marking probability 1-F(0)
• F(x) = 1 - p + pC(x), C(0) = 0.

P K m F m F j F i P K m j

j m i j

( ) ( ) ( ) ( ) ( ) = =

• =
• =

=

• Â

’

1 1

1 1 1

P K F p p p ( ) ( ( ) )( ) / = =

• +
• 1

1 1 1

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Approximate the continuous 1D model

1 [ ] 1 1

1 1 ( ) ( ) ( ) ( ) ( / ),

j nx j i

j i P K x F x F F P K x j n x n n

− = =

− −   = = − = − >    

∑ ∏

Sample a unit interval into n equal subintervals Use the Binomial approximation to the Poisson: p=λ/n where λ is the linear Poisson intensity. Computationally efficient

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Approximation of clump-length density of 1D model with constant segment length a = 1 using n = 1000

{ } { }

1 [ 1] 1 1 1

1 ( ) ( 1) 1 ( ( 1) ( ( 1))

x j j

f x e x j e x j j j

λ λ

λ λ λ

− − − − − =

    − = − + − + − + +          

∑

For continuous solution see P. Hall, Introduction to the Theory of Coverage Processes

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Discrete likelihood (approximating a continuous likelihood)

1 1

( ; , ) ( / )(1 / ) ( ; )

i

N W i i

L S n n P K B λ λ λ

− =

= − =

∏

? ? Let {( , ), 1, , } be a sample of complete clumps (black runlengths) and gaps (white runlengths) { } are i.i.d. ( ; ) { } are i.i.d. Geometric( / ; ) and are independent for all 1, ,

i i i i i i

S B W i N B P K W n n B W i N λ = = = ? K K

Take equispaced transects horizontally and vertically, treat the samples as independent.

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Disks With Lognormally-Distributed Radii

[ ]

2

1 1 ( ) exp log( / ) / 2 2 r r R r φ β β π   = −    

2 2 1 / 2 2 1 1

Segment-length distribution: 1 ( ) 1 / 4 ( ) where ( ) exp{ / 2} Also, 2

x i

C x r x r dr M M r r dr R M φ φ β λ ρ

∞ ∞

= − − = = =

∫ ∫

1.0, 0.4, 0.5 R ρ β = = =

“Classic” Boolean Model

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Estimation results

2 2

ˆ ˆ : ( , ) (0.36,0.51) where ( , ) (0.4,0.5) Likelihood ratio-based 95% confidence interval: ˆ ˆ {( , ) : 2( ( , ) ( , )) (0.05) 5.99} MLE l l ρ β ρ β ρ β ρ β ρ β χ = = − − ≤ =

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Union of Two Randomly-Oriented Square Boolean Models

2 2 2

/ 2 , ( ) 1 ( 2 ) / 2 , 2 1,

• therwise.

4 /

i i i i i i i i i i i

x D x D C x x D x D xD D x D D λ ρ π ≤    = − − − ≤ ≤     =

Boolean model 1 has fixed diameter 1 Boolean model 2 has fixed diameter 4

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1 1 2 2

1.0, 0.4, 4.0, 0.3 D D ρ ρ = = = =

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Estimation Results

1 2 1 2 2 1 2 1 2 1 2 2

ˆ ˆ : ( , ) (0.38,0.29) where ( , ) (0.4,0.3) Likelihood ratio-based 95% confidence region: ˆ ˆ {( , ) : 2( ( , ) ( , )) (0.05) 5.99} MLE l l ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ χ = = − − ≤ =

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Summary

• MLE for continuous Boolean model

difficult to formulate

• Take 1D samples from 2D model
• Approximate 1D BM with discrete model
• Express likelikood interms of discrete

model and maximimize for approximate MLE