Derivations for maximum likelihood estimation of particle size distribution using in situ video imaging

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1 2 TWMCC Texs-Wisconsin Modeling nd Control Consortium 1 Technicl report numer 27-1 Derivtions for mximum likelihood estimtion of prticle size distriution using in situ video imging Pul A. Lrsen nd Jmes B. Rwlings Mrch 16, 27 Astrct This technicl report contins supplementl mteril for the mnuscript Mximum likelihood estimtion of prticle size distriution for high-spect-rtio prticles using in situ video imging [2]. This report gives the derivtion of the proility densities used to formulte the likelihood function of the MLE. 1 Mximum likelihood estimtion of PSD Let X k = (X 1k,..., X T k ) e T -dimensionl rndom vector in which X ik gives the numer of non-order prticles of size clss i oserved in imge k. A non-order prticle is prticle tht is completely enclosed within the imging volume. A order prticle, on the other hnd, is only prtilly enclosed within the imging volume such tht only portion of the prticle is oservle. For order prticles, only the oserved length (i.e. the length of the portion of the prticle tht is inside the imging volume) cn e mesured. Accordingly, we let Y k = (Y 1k,..., Y T k ) e T -dimensionl rndom vector in which Y jk gives the numer of order prticles with oserved lengths in size clss j tht re oserved in imge k. We denote the oserved dt, or the reliztions of the rndom vectors X k nd Y k, s x k nd y k, respectively. The prticle popultion is represented completely y the vectors ρ = (ρ i,..., ρ T ) nd S = (S 1,..., S T +1 ) in which ρ i represents the numer of prticles of size clss i per unit volume nd S i is the lower ound of size clss i. Given the dt x nd y (the suscript k denoting the imge index is removed here for simplicity), the mximum likelihood estimtor of ρ is defined s ˆρ = rg mx p XY (x 1, y 1, x 2, y 2,..., x T, y T ρ) (1) ρ 1

2 TWMCC Technicl Report in which in which the suscript indictes the use of order prticle mesurements nd p XY is the joint proility density for X nd Y. In other words, we wnt to determine the vlue of ρ tht mximizes the proility of oserving exctly x 1 non-order prticles of size clss 1, y 1 order prticles of size clss 1, x 2 non-order prticles of size clss 2, y 2 order prticles of size clss 2, nd so on. A simplified expression for p XY cn e otined y noting tht, t lest t low solids concentrtions, the oservtions X 1, Y 1,... X T, Y T cn e ssumed to e independent. This ssumption mens tht the oserved numer of prticles of given size clss depends only on the density of prticles in tht sme size clss. At high solids concentrtions, this ssumption seems unresonle ecuse the numer of prticle oservtions in given size clss is reduced due to occlusions y prticles in other size clsses. At low concentrtions, however, the likelihood of occlusion is low. The independence ssumption does not imply tht the oservtions re not correlted. Rther, the ssumption implies tht ny correltion etween oservtions is due to their dependence on common set of prmeters. As n exmple, if we oserve lrge numer of non-order prticles, we would expect to lso oserve lrge numer of order prticles. This correltion cn e explined y noting tht the proility densities for oth order nd non-order oservtions depend on common prmeter, nmely, the density of prticles. Given the independence ssumption, we express p XY s T T p XY = p Xi (x i ρ) p Yj (y j ρ) (2) i=1 in which p Xi nd p Yj re the proility densities for the rndom vriles X i nd Y j. Using Eqution (2), the estimtor in Eqution (1) cn e reformulted s ˆρ = rg min ρ j=1 T log p Xi (x i ρ) i=1 T log p Yj (y j ρ) (3) The proility densities p Xi nd p Yj re derived in the following sections. These derivtions show tht X i P(m Xi ), or tht X i hs Poisson distriution with prmeter m Xi = ρ i α i, in which α i is function of the field of view, depth of field, nd the lower nd upper ounds of size clss i. Furthermore, Y j P(m Yj ), in which m Yj = T i=1 ρ iβ ij 2 Derivtion of proility densities The proility densities p Xi nd p Yi in Eqution (3) cn e derived given the prticle geometry nd the sptil nd orienttionl proility distriutions. Here, we derive p Xi nd p Yi for needle-like prticles ssuming the prticles re rndomly, uniformly distriuted, oth in their 3-dimensionl sptil loction nd in their orienttion in the plne perpendiculr to the opticl xis. To simplify the discussion, we initilly present the derivtion ssuming 2-dimensionl system with monodisperse, verticlly-oriented prticles. Lter, we relx these ssumptions nd present the derivtion for rndomly-oriented, polydisperse prticles in 3-dimensionl spce. j=1

3 TWMCC Technicl Report l l/2 A n l l A l/2 I I () S () S Figure 1: Depiction of hypotheticl system of verticlly-oriented prticles rndomly nd uniformly distriuted in spce. 2.1 Non-order prticles Let S e squre domin in R 2 with dimension B nd re A S = B 2. Let I e rectngulr domin in R 2 with horizontl nd verticl dimensions nd, respectively, nd re A I =. Assume A I A S nd I S. Let n tot e the totl numer of verticllyoriented prticles with midpoints rndomly nd uniformly distriuted in S, nd define ρ = n tot /A S s the density of prticles per unit re. Let the length of ll prticles e l, with l < min (, ), nd define A n s the re of the domin in which prticles re inside I ut do not touch the order of I. Becuse the prticles re oriented verticlly, it is esy to show tht A n = ( l), s depicted in Figure 1(). Finlly, let X e rndom vrile denoting the numer of non-order prticles ppering in I. Assuming the loction of ech prticle in S is independent of the remining prticles loctions, the proility tht specific prticle will pper entirely within I is given y p = A n /A S. Given the ove ssumptions, this proility is constnt for ll prticles. The proility of oserving x non-order prticles in I is nlogous to the proility of oserving x successes in n tot Bernoulli trils in which the proility of success in ech tril is p. Thus, X is inomil rndom vrile with proility distriution p X (x) = ( ntot x ) p x (1 p) ntot x Now, ssume B while keeping ρ constnt. Then n tot nd p = A n /A S = A n ρ/n tot while Np = ρa n remins constnt. The limiting distriution of X is therefore Poisson p X (x) = e m X m x X, m X = ρa n x! To extend the nlysis to polydisperse, rndomly-oriented needles, we discretize the length scle into T size clsses nd let X = (X 1,..., X T ) e T -dimensionl rndom vector

4 TWMCC Technicl Report l l/2 sin θ A n (l, θ) θ l/2 sin θ l/2 cos θ l/2 cos θ Figure 2: Depiction of geometricl properties used to derive the non-order re function A n (l, θ). in which X i gives the numer of non-order prticles of size clss i oserved in single imge. An orienttion Θ nd length L re ssigned to ech prticle, where Θ 1, Θ 2,..., Θ ntot re i.i.d. with density function p Θ (θ), θ [ π/2, π/2) nd L 1, L 2,..., L ntot re i.i.d. with density function p L (l), l (, inf). Θ nd L re independent of ech other nd independent of the prticle s sptil loction. We define S s the T+1-dimensionl vector of reks etween size clsses. A prticle of length l elongs to size clss i if S i l < S i+1. Let i = S i+1 S i. Our gol is to determine the proility tht prticle of size clss i will pper entirely inside the imge I, given its density ρ i. Following the pproch used to solve the Buffon-Lplce needle prolem [3, p. 4], Figure 2 shows geometriclly tht for given orienttion θ nd length l, A n (l, θ) cn e clculted s A n (l, θ) = { ( l cos θ)( l sin θ) θ π/2 ( l cos θ)( + l sin θ) π/2 θ (4) The proility tht given prticle in size clss i will pper entirely within I is given y p i = Si+1 π 2 S i π 2 Si+1 π 2 S i A n (l, θ)p Θ (θ)p L (l)dθdl π 2 A S p Θ (θ)p L (l)dθdl (5) Thus, the proility tht specific prticle of size clss i will pper entirely within the imge is given y p i = α i /A S, where α i is the numertor in Eqution (5). Following the sme rguments s ove, we cn show tht for n infinitely lrge system, X i is Poisson rndom vrile with prmeter m Xi = ρ i α i. Extending the nlysis to three-dimensionl spce is trivil ecuse we ssume the prticles re oriented in the plne perpendiculr to the cmer s opticl xis nd ssume no interction etween prticles. Thus, for three-dimensionl system, X i is Poisson rndom vrile with prmeter m Xi = ρ i α i, with α i = α i d, in which d is the depth of field. Assuming Θ is distriuted uniformly nd L is distriuted uniformly cross ech size clss, α i cn e clculted s follows. Let S i = S i+1 S i, S mx = 2 + 2, S i,mx =

5 TWMCC Technicl Report S mx S i, nd ssume >. For S i+1, α i = d ( S 3 i+1 S 3 ) i + ( S 2 i+1 S 2 ) i + 3π S i π S i For < S i, S i+1, α i = d ( ( [S i+1 Si+1 2 π S sin i S i+1 ( ) ) S i ( S 2i sin + Si + 2 log S i+1 + Si ( S S i + Si 2 i+1 2 Si 2 2 ) ) For S i, S i+1 S mx, α i = d [ ( ( ) ( {S i+1 Si+1 2 π S 2 + Si sin cos i S i+1 ( ) ( ))] S i [ Si Si (sin cos Si Si + 2 log S i+1 + Si log S i+1 + Si S i + Si 2 2 S i + Si 2 2 ( ) S i 1 ( S 3 3 i+1 Si 3 ) } ) S i+1 For S i S mx nd S i+1 > S mx, [ d α i = {S mx Smx π S ( ( ) ( Smx sin cos i,mx S mx ( ) ( ))] S i [ Si Si (sin cos Si Si + 2 log S mx + Smx log S mx + Smx 2 2 S i + Si 2 2 S i + Si 2 2 ( ) S i,mx 1 ( S 3 3 mx Si 3 ) } ))] S mx ))] 2.2 Border prticles As efore, we simplify the discussion y first presenting the derivtion of p Yi for monodisperse, verticlly-oriented prticles. Let Y e rndom vrile denoting the totl numer of order prticles ppering in I. Define A s the re of the domin in which prticles

6 TWMCC Technicl Report l l A I l A l I A A A S () () (c) S S Figure 3: Depiction of hypotheticl system of verticlly-oriented prticles rndomly nd uniformly distriuted in spce. touch the order of I, s depicted in Figure 3(). For the present system, A = 2l. The proility tht specific prticle will touch the order of I is given y p = A /A S. Following the sme rguments s ove, we cn show tht for n infinitely lrge system, Y is Poisson rndom vrile with prmeter m Y = ρa. Now, ssume we would like to incorporte dditionl informtion into our estimtion y tking into ccount not only the numer of order prticles, ut lso their oserved lengths. For monodisperse popultion, these oserved lengths cn tke on vlues nywhere etween nd l. We therefore discretize the length scle on [ l] nd let j denote the size clss corresponding to the oserved length. We define Y j s rndom vrile denoting the numer of order prticles ppering in I with oserved length in size clss j. Figure 3() illustrtes this pproch for two size clsses. In this figure, A 1 is the re of the region in which prticles produce oserved lengths from to l/2, corresponding to size clss 1, while A 2 is the re of the region in which prticles produce oserved lengths from l/2 to l, corresponding to size clss 2. The proility tht specific prticle will touch the order of I nd produce n oserved length in size clss j is p = A j /A S. Thus, Y j is Poisson rndom vrile with prmeter m Yj = ρa j. In Figure 3(), A 1 = A 2. This equlity etween the res of different oserved length size clsses does not hold in generl, however, s illustrted in Figure 3(c). In this figure, we ssume ll prticles re oriented digonlly, t 45 degrees from the horizontl, nd the figure illustrtes tht A 1 > A 2. Hence, in generl, order prticles re more likely to result in oserved lengths in the lower size clsses. To extend the nlysis to polydisperse systems with rndom orienttion, we define new rndom vrile Y ij tht gives the numer of prticles in size clss i tht intersect the imge order, producing n oserved length in size clss j. Given tht the size clss of ech order prticle is unknown, we define the rndom vrile Y j s the totl numer of order prticles producing oserved lengths in size clss j, noting tht Y j = i Y ij. Our pproch is to determine the proility density for Y ij for ll i nd to use these densities to derive the proility density for Y j.

7 TWMCC Technicl Report l cos θ lθ l l (l l) sin θ θ (l l) cos θ l sin θ l θ Figure 4: Depiction of non-order re for ritrry length nd orienttion. We define the function A j (l, θ) s the re of the region in which prticle of length l nd orienttion θ produces n oserved length corresponding to size clss j. To clculte A j (l, θ), it is convenient to define n re function Ã(l, θ, l) s the re of the region in which prticles of length l nd orienttion θ either intersect or re enclosed within the imge oundry nd produce n oserved length greter thn or equl to l. Ã(l, θ, l) cn e clculted using the geometric reltionships shown in Figure 4: In this figure, the thick-lined, outer rectngle is the imge region, nd the inner rectngle is the region inside which prticle with length l nd orienttion θ will e entirely enclosed within the imge oundry, thus producing n oserved length of exctly l. A prticle with its midpoint long the perimeter of the outermost hexgon would touch the imge oundry ut give n oserved length of. A prticle with its midpoint nywhere inside the innermost hexgon will produce n oserved length greter thn or equl to l. Using the reltionships indicted in this figure, nd ssuming l l, Ã(l, θ, l) cn e clculted s { Ã(l, θ, l) ( + (l 2 l) cos θ)( + (l 2 l) sin θ) (l l) = 2 sin θ cos θ θ π/2 ( + (l 2 l) cos θ)( (l 2 l) sin θ) + (l l) 2 sin θ cos θ π/2 θ (6) If l <, Eqution (6) is vlid only for θ on ( sin (/ l), sin (/ l)). If, l < ( ) 1/2, Eqution (6) is vlid only for θ on ( sin (/ l), cos (/ l)) nd (cos (/ l), sin (/ l)). A j (l, θ) is given y Ã(l, θ, S j ) Ã(l, θ, S j+1) l S j+1 A j (l, θ) = Ã(l, θ, S j ) A n (l, θ) S j l < S j+1 (7) l < S j

8 TWMCC Technicl Report The proility tht given prticle in size clss i will pper within I nd produce n oserved length in size clss j is given y p ij = Si+1 π 2 S i π 2 Si+1 π 2 S i A j (l, θ)p Θ (θ)p L (l)dθdl π 2 A S p Θ (θ)p L (l)dθdl (8) The proility tht specific prticle in size clss i will touch the order of I nd produce n oserved length in size clss j is p ij = β ij /A S, with β ij eing the numertor in Eqution (8). Thus, for n infinitely lrge system, Y ij is Poisson rndom vrile with prmeter m Yij = ρ i βij. Assuming Y 1j, Y 2j,..., Y T j re independent, then Y j = i Y ij is lso Poisson rndom vrile with prmeter m Yj = i ρ i β ij [1, p.44]. As in the nonorder cse, the nlysis is extended to three-dimensionl spce ssuming the prticles re oriented in the plne perpendiculr to the cmer s opticl xis nd tht the prticles do not interct. Thus, for three-dimensionl system, Y j is Poisson rndom vrile with prmeter m Yj = i ρ iβ ij, with β ij = β ij d. Assuming Θ is distriuted uniformly nd L is distriuted uniformly cross ech size clss, β ij is clculted s follows. Let the length scle discretiztion e the sme for oth order nd non-order prticles. As efore, let S i = S i+1 S i, S mx = 2 + 2, S i,mx = S mx S i, nd ssume >. Then β ij is given y Ā(i, S j ) Ā(i, S j+1) i > j β ij = Ā(i, S j ) α i i = j (9) i < j in which Ā(i, S) is clculted s Ā(i, S) = d [ ( Si 2γ1 4Sγ 2 4Sγ 3 + 3S 2 ) ( γ 4 + S 2 π S i+1 S 2 ) i (γ2 + γ 3 Sγ 4 ) ] i π/2 S < sin (/S) < S < γ 1 = sin (/S) cos (/S) < S < S mx sin (/S mx ) cos (/S mx ) S > S mx 1 S < /S < S < γ 2 = ( ) S 2 2 /S < S < S mx ( ) Smx 2 2 /S mx S > S mx 1 S < 1 S 2 2 /S < S < γ 3 = ( ) S 2 2 /S < S < S mx ( ) Smx 2 2 /S mx S > S mx

9 TWMCC Technicl Report S < γ 4 = 2 /S 2 < S < ( S 2 )/S 2 < S < S mx ( Smx)/S 2 mx 2 S > S mx 3 Vlidtion of Mrginl Densities To ensure the correctness of the proility densities derived in the previous section, four different Monte Crlo simultions were crried out in which rtificil imges of prticulte popultions were generted. Figure 5 shows exmple imges generted for ech simultion. Ech of these imges hs horizontl imge dimension of =48 pixels nd verticl dimension of =48 pixels. The first row displys four simulted imges for monodisperse prticles of length.5 with N c =25 crystls per imge. The second row shows imges of prticles uniformly distriuted on [.1.9] with N c =25. The third row shows imges of prticles normlly-distriuted with µ =.5 nd σ =.4/3 with N c =25, nd the fourth row shows exmple imges for simultions of prticles uniformly-distriuted on [.1 2.] with N c =15. For ech simultion, 2, rtificil imges were generted. Bsed on the oservtions in these 2, imges, histogrm ws generted for ech size clss giving the frequency of oservtions for oth order nd non-order prticles. These histogrms re compred with the theoreticl mrginl densities in Figures 6 13.

10 TWMCC Technicl Report Figure 5: Exmple imges for simultions of vrious prticle popultions. Row 1: monodisperse prticles of length.5, Nc =25. Row 2: prticles uniformly distriuted on [.1.9]. Row 3: prticles normlly-distriuted with µ =.5 nd σ =.4/3 with Nc =25. Row 4: prticles uniformly-distriuted on [.1 2.], Nc =15.

11 TWMCC Technicl Report Figure 6: Comprison of theoreticl nd simulted mrginl densities for rndomlyoriented, monodisperse prticles of length.5 nd mesured y prtitioning [.1.9] into ten ins. Results re for non-order prticles.

12 TWMCC Technicl Report () () (c) (d) Figure 7: Comprison of theoreticl nd simulted mrginl densities for rndomlyoriented, monodisperse prticles of length.5 nd mesured y prtitioning [.1.9] into ten ins. Results re for order prticles.

13 TWMCC Technicl Report Figure 8: Comprison of theoreticl nd simulted mrginl densities for rndomly-oriented prticles distriuted uniformly on [.1.9] nd mesured y prtitioning [.1.9] into ten ins. Results re for non-order prticles.

14 TWMCC Technicl Report Figure 9: Comprison of theoreticl nd simulted mrginl densities for rndomly-oriented prticles distriuted uniformly on [.1.9] nd mesured y prtitioning [.1.9] into ten ins. Results re for order prticles.

15 TWMCC Technicl Report Figure 1: Comprison of theoreticl nd simulted mrginl densities for rndomlyoriented prticles distriuted normlly nd mesured y prtitioning [.1.9] into 1 ins. Results re for non-order prticles.

16 TWMCC Technicl Report Figure 11: Comprison of theoreticl nd simulted mrginl densities for rndomlyoriented prticles distriuted normlly nd mesured y prtitioning [.1.9] into 1 ins. Results re for order prticles.

17 TWMCC Technicl Report Figure 12: Comprison of theoreticl nd simulted mrginl densities for rndomlyoriented prticles distriuted uniformly on [.4 2.] nd mesured y prtitioning [.4 1.] into 9 ins with 1th in spnning [1. 2]. Results re for non-order prticles.

18 TWMCC Technicl Report Figure 13: Comprison of theoreticl nd simulted mrginl densities for rndomlyoriented prticles distriuted uniformly on [.4 2.] nd mesured y prtitioning [.4 1.] into 9 ins with 1th in spnning [1. 2]. Results re for order prticles.

19 TWMCC Technicl Report References [1] Yvonne M. M. Bishop, Stephen E. Fienerg, nd Pul W. Hollnd. Discrete Multivrite Anlysis: nd Prctice. The MIT Press, Cmridge, Msschusetts, [2] Pul A. Lrsen nd Jmes B. Rwlings. Mximum likelihood estimtion of prticle size distriution for high-spect-rtio prticles using in situ video imging. In preprtion. [3] Herert Solomon. Geometric Proility. SIAM Pulictions, Phildelphi, PA, 1978.