Approximate Likelihood Procedures for the Boolean Model Using Linear Transects John C. Handley Xerox Corporation jhandley@crt.xerox.com 1
Boolean Random Set Model ξ ξ R ρ n Poisson process { , , } in , intensity K 1 2 R n Random compact set { , S S , } subsets of K 1 2 Boolean model: = + ξ U ( S ) B n n n lognormally distributed disks ρ = ρ = 0.1 1.0 2
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. 3
Applications • Materials science: aggregates • Cellular telecommunications • Ecology • Biomedical imaging • Food science 4
Background • Least squares and methods-of-moments based on capacity functional: = + ξ U ( S ) B n n n ( = ∩ ≠ ∅ = − − ρ µ ⊕ T ( K ) Pr( K ) 1 exp( E ( S K ) ) B 0 B compact K ( K is K reflected about origin Estimate LHS from realization, RHS is parameterized for some models Limited inference for these estimation methods Maximum likelihood methods difficult to formulate in continuous setting 5
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 6
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) 7 20
1D continuous Boolean model Poisson arrivals with rate λ Random segment lengths X with dist. C clump gap 8
Laplace-Stieltjes transform of clump-length density of 1D continuous Boolean model (see Hall (1988): • K = clump-length • C = segment length distribution • z - g ( ) = £ sx s e dP K ( x ) 0 F I L O z z - 1 • t M P G J = + l - l - - l - 1 ( / s ) exp st { 1 C x ( )} dx dt H K N Q 0 0 9
1D discrete Boolean model (binomial germ-grain model) Binomial arrivals p = marking probability Random segment length X with dist. C clump gap 10
Self-covering event E (1, m ) for the Boolean model m k = = « « + U U E ( , ) 1 m ( X k ) E '( , ) 2 j E ( j 1 , ) m 1 = = k 1 j 1 - j 1 m  ’ = - - - + P ( ( , )) E 1 m F m ( ) F j ( 1 ) F i ( 1 ) ( ( P E j 1 , )) m = = j 1 i 1 = - P ( ( , )) E 11 F ( ) 1 F ( ) 0 11
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. - j 1 m  ’ = = - - - = - P K ( m ) F m ( ) F j ( 1 ) F i ( 1 ) ( P K m j ) = = j 1 i 1 = = - + - P K ( 1 ) ( F ( ) 1 1 p )( 1 p ) / p 12
Approximate the continuous 1D model 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. − − − [ nx ] j 1 j 1 i 1 ∑ ∏ = = − = − > P K ( x ) F x ( ) F ( ) F ( ) ( P K x j n / ), x 0 n n = = j 1 i 1 Computationally efficient 13
Approximation of clump-length density of 1D model with constant segment length a = 1 using n = 1000 For continuous solution see P. Hall, Introduction to the Theory of Coverage Processes − 1 − − [ x 1] 1 ∑ { } { } − 1 = λ λ − − + λ − + − λ λ − + + 1 j f x ( ) ( e 1) 1 ( x ( j 1) e ( x ( j 1)) j j = j 1 14
Discrete likelihood (approximating a continuous likelihood) = = Let S {( B W , ), i 1, , N } be a sample of complete K i i clumps (black runlengths) and gaps (white runlengths) { } are i.i.d. ( B P K ; ) ? i λ { W } are i.i.d. Geometric( / ; ) n n i = B and W are independent for all i 1, , N K i i N ∏ − λ = λ − λ = W 1 L S ( ; , ) ? ( / )(1 n / ) n P K ( B ; ) ? i i = i 1 Take equispaced transects horizontally and vertically, treat the samples as independent. 15
Disks With Lognormally-Distributed Radii 1 1 [ ] 2 φ = − β ( ) r exp log( / r R ) / 0 β π 2 r 2 Segment-length distribution: “Classic” Boolean Model ∞ 1 ∫ = − − φ 2 2 C x ( ) 1 r x / 4 ( ) r dr M 1 x / 2 where ∞ ∫ = φ = β i 2 M r ( ) r dr R exp{ / 2} 1 0 0 λ = ρ Also, 2 M 1 20 = ρ = β = R 1.0, 0.4, 0.5 16 0
Estimation results ˆ ρ β = ρ β = ˆ MLE : ( , ) (0.36,0.51) where ( , ) (0.4,0.5) Likelihood ratio-based 95% confidence interval: ˆ ρ β − ρ β − ρ β ≤ χ = ˆ 2 {( , ) : 2( ( , l ) l ( , )) (0.05) 5.99} 2 17
Union of Two Randomly-Oriented Square Boolean Models ≤ x / 2 D , x D i i = − − − ≤ ≤ 2 2 2 C x ( ) 1 ( x 2 D x D ) / 2 xD , D x D 2 i i i i i i 1, otherwise. λ = ρ π 4 D / i i i Boolean model 1 has fixed diameter 1 Boolean model 2 has fixed diameter 4 18
= D 1.0, 1 ρ = 0.4, 1 = D 4.0, 2 ρ = 0.3 2 19
Estimation Results ρ ρ = ρ ρ = ˆ ˆ MLE : ( , ) (0.38,0.29) where ( , ) (0.4,0.3) 1 2 1 2 Likelihood ratio-based 95% confidence region: ρ ρ − ρ ρ − ρ ρ ≤ χ = ˆ ˆ 2 {( , ) : 2( ( l , ) l ( , )) (0.05) 5.99} 1 2 1 2 1 2 2 20
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 21
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