Weakly Constrained Lucy–Richardson with Applications to Inversion of Light Scattering Data

(1)

Re eived:date/A epted:date

Abstra t Lu y-Ri hardson(LR)isa lassi aliterativeregularizationmethod

largelyusedfortherestorationofnonnegativesolutions.LRndsappli ations

inmanyphysi alproblems,su hasfortheinversionoflights atteringdata.In

theseproblems,thereisoftenadditionalinformationonthetruesolutionthat

isusuallyignoredbymanyrestorationmethodsbe ausetherelatedmeasurable

quantitiesarelikelyto beae tedbynon-negligiblenoise.

In this arti le we propose a novel Weakly Constrained Lu y-Ri hardson

(WCLR)method whi hadds aweak onstraintto the lassi alLR by

intro-du ing apenalization term, whose strength an be varied over avery large

A.Bu ini

DepartmentofMathemati alS ien es,

KentStateUniversity,

233MSB,1300LeftonEsplanade,

44242Kent,Ohio,

Tel.:+13306721481

E-mail:abu inikent.edu

M.Donatelli

DipartimentodiS ienzaeAltaTe nologia,

UniversitàdegliStudidell'Insubria,

ViaValleggio11, 22100Como,Italy, Tel.:+39-031-2386378 Fax:+39-031-2386209 E-mail:mar o.donatelliuninsubria.it F.Ferri

DipartimentodiS ienzaeAltaTe nologia,

To.S a.Lab,

UniversitàdegliStudidell'Insubria,

ViaValleggio11, 22100Como,Italy, Tel.:+39-031-2386251 Tel.:+39-031-2386294 Fax:+39-031-2386250 E-mail:fabio.ferriuninsubria.it

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range.The WCLR method is simpleandrobustasthestandardLR, but

of-fersthegreatadvantageofwidelystret hingthedomainrangeoverwhi hthe

solution an be reliably re overed.Some sele ted numeri al examples prove

theperforman esoftheproposed algorithm.

Keywords Lu y-Ri hardsonAlgorithm

·

LightS attering

·

Parti leSizing Mathemati sSubje t Classi ation (2000) 65R32

·

78A46

·

78A10

1Introdu tion

Inthisworkwe onsider the aseof aFredholmintegralequationoftherst

kind

I(θ) =

ˆ

K(θ, R) N (R)dR.

(1)

where

I(θ)

representsthedataevaluatedattheindependentvariable

θ

,

N (R)

is the unknown fun tion that depends on the variable

R

, and

K(θ, R)

is a knownintegralkernelwith ompa tsupport.Thelatter onditionimpliesthat

equation(1)isill-posed,meaningthatthesolutionisnotuniqueanddoesnot

depend ontinuouslyonthedata, i.e., smallperturbationsin

I

greatlyae t thesolution

N

.

Wearegoingto onsiderinverseproblemsoftheform(1) omingfromlight

s attering methods in opti s [2,6,1518℄, where the main goalof theopti al

te hnique is to re over the parti le-size distribution

N (R)

whi h hara ter-izes thesampleunder investigation.Thusthe solution

N (R)

isdened to be positive(

N > 0

)withapositivesupport(

R > 0

).Amongthevarious s atter-ing te hniques, themost popular ones are Dynami LightS attering(DLS),

Stati orElasti LightS attering(ELS) andMultiWavelengthTurbidimetry

(MWT). All of them an be easily implementedby using simple

experimen-tal setups(seeforexample[7℄andreferen estherein),with thepossibilityof

hara terizingsimultaneouslyaverylargenumberofparti les,the

hara ter-izationbeing arriedoutinsitu and almostreal time.Re entappli ationsof

these methods to parti le sizing an be found in Refs. [25,28℄ for the DLS

te hnique,in Refs.[21,30℄forELS,andRefs.[14,31℄forMWT.

Thegoalofane ient,wellperforminginversionalgorithmisthea urate

andfastre overyofthesampleparti le-sizedistributionoverthelargest

pos-siblerangeofparti leradii.Indeed,inversionalgorithmsareexpe tedtowork

prettywellonlywhentheparti lesizestobere overedliewithinagivenrange

[R

min

, R

max

]

,whi hdependsontherangeofthe

[θ

min

, θ

max

]

independent vari-ablebeingprobed.Althoughwhat followsisof generalvalidity,in thispaper

wewillfo usontheELSte hnique,where

I(θ)

representstheintensityofthe light s attered by the sample at various s attering angles

θ

. In the past we haveshown[12℄ thatthedynami alextensionof the

R

−

range(i.e.,theratio

[R

max

/R

min

]

)whi h an be probed,s alesproportionallyto

[θ

max

/θ

min

]

and, therefore anberatherlimitedifthelatterratioisnotsu ientlylarge.

(3)

avoidthepresen eofnoisein thedata,thedire tinversionof

A

,forinstan e be the MoorePenrose pseudoinverse, leads to very poor approximations of

theoriginaldesired

N

.Tore overameaningfulapproximationofthetrue so-lution, a ommon approa his to resortto regularization,i.e., use numeri al

methodsthat omputeasolutionofanearbywell-posedproblem,see[3,9℄for

lassi alregularizationmethodsforinverseproblems.

Forlargeproblems,a ommonapproa histoadoptiterativemethodsthat

require only matrix ve tor-produ ts. Among these iterative pro edures, one

ofthemostpopularonesis undoubtedlythe lassi alLu y-Ri hardson(LR),

known also as Ri hardson-Lu y method [26,29℄, whi h has the remarkable

feature ofensuring nonnegativity ofthesolutions.LR is alsoquite simpleto

implement,robustagainstnoiseand, providedthat theiterativepro edureis

stoppedafteraproperly hosennumberofsteps,doesnotrequireany

parame-tertobeoptimized.Furthermore,itisthespe i natureofourproblemthat

suggests the usage of the LR method, mainly be ause the noise present on

thedataisnotadditive,but multipli ativewhiteGaussiannoiseasdes ribed

in Se tion4.5,and theoperator

A

isseverelyill- onditionedand underdeter-mined. These features suggest that methods relying on the minimization of

the Eu lidean norm of the residual would not perform well on this spe i

problem. Moreover,in the onsidered setting, the knowledge of the norm of

thenoiseisnotavailableandonlysomestatisti alinformationonit anbe

re-trieved,implyingthatthe lassi aldis repan yprin iple annotbeemployed.

Thus,amethodlikeLRseemstobemoreee tivethanothermethods based

ontheminimization oftheEu lideannorm.

Ontheother side,theLRmethod isknownforbeingrelativelyslowand,

as mentioned above and reported in [12℄ 1

, for being not so e ient when

when therangeof re overableradiiis toolarge.Thelatterlimitation o urs

be ause,foranygivenangularrange

[θ

min

, θ

max

]

,asparti lesbe omesmaller andsmaller(

R

≪ λ/θ

min

),theangulardistributionofthes atteredlighttends tobemoreandmoreuniform[

I(θ

min

)/I(θ

max

)

∼ 1

℄,thus arryinglessandless informationon parti lesize.Conversely,forlargerand largerparti les(

R

≫

λ/θ

max

), the angular distribution varies widely with a high signal dynami range [

I(θ

min

)/I(θ

max

)

≫ 1

℄, but theshapeof

I(θ)

tendsto beindependent onparti lesize[23℄.

1

Afterthepubli ationof[12℄,itwasrealizedthatthemethod alledmodi ationofthe

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Inthis paperweproposeanewmethod alled WeaklyConstrained

Lu y-Ri hardson (WCLR), whi h improves the lassi al LR method by adding a

weak onstraint to the true solution asso iated to a physi al property (the

parti levolumefra tion on entration)thatisknownor anbemeasuredwith

ratherhigha ura y.Theintrodu tionofthisadditionalknowledgea elerates

remarkablythe onvergen erateofthemethodand,atthesametime,largely

improvesthe qualityof therestoration.In parti ularit allowstoenlargethe

R

−

rangebymorethanoneorder ofmagnitude,apropertythat is ru ial in manyrealappli ations.

Thepri etopayforaddingthe onstraintistheintrodu tionofadamping

parameter,

γ

, whi h has to be properly adjusted. However, this task is not riti alatallbe ausethestrengthof

γ

an bevariedoveraverybroadrange and its optimal value an be easily found by exploiting a se ond physi al

property(the parti le number on entration),whi hneedsto beknownonly

veryroughly.Notethatusingaweak ratherthananexa t onstraintisavital

feature of the WCLR method. Indeed, while imposing an exa t onstraint

has the advantage of requiring no damping parameter, its implementation

(by using lassi aloptimization algorithms for linear onstrained problems)

mightstillneedthetuningofotherparametersand,duetounavoidableerrors

alwayspresentintheestimateofthe onstraint,itdoesnotprovidene essarily

an improvement in the quality of the restoration. Conversely, our WCLR is

quite simple to implement, it is exible and robust against small (

.

±2%

) un ertaintiesasso iated tothe onstraintand, ex eptfor

γ

, doesnotrequire anyotherparametertobeset inthesamespiritofthe lassi alLR.

This work is stru tured as follows. In Se tion 2 we give some physi al

details abouttheproblem weare goingtoanalyze. Inparti ular,Se tion 2.1

des ribesthedis retizationpro ess,whileSe tion2.2 onsidersthe onstrains

wearegoingtouse.Se tion3isdevotedtotheformulationofthemathemati al

modelandtothedenitionofournumeri almethod.Thelatteroneistested

and omparedwithLRonsomenumeri alexamplesin Se tion4.Se tion5is

devotedto on ludingremarksandfuturework.

2Physi al details

Inthisse tionwegivesomeinsightintothephysi alproblemwearegoingto

onsider,i.e.,theproblemasso iatedtotheinversionofelasti lights attering

(ELS) data, where the main goal is to re over the size distribution of the

parti lespresentinthesample.

A ordingto ELStheory[23℄,whenasamplemadeofpolydisperse

parti- les hara terizedbyarefra tionindexdierentfromthatofthesurrounding

mediumisilluminatedbyalaserlightofwavelength

λ

,partoftheradiationis goingtobes atteredatanglesdierentfromthein identdire tion.Ifthe

par-ti les arehomogeneouslydispersed in themedium andtheir on entrationis

solowthatthey anbe onsideredasnon-intera ting,theangulardistribution

(5)

Fig.1 (a):normalizedbehaviorofthekernel

K(θ, R)

appearingin(1)asafun tionofthe s attering angle

θ

for veparti les withdiameters

d = 2R

ranging from

d = 0.01 µm

to

d = 100 µm

;the normalizationissu hthat

K(θ = 0, R) = 1

.(b):behavior ofthe kernel amplitude

K(θ = 0, R)

asafun tionoftheratio

R/λ

.Thestraightlineswithslopes

6

and

4

indi atethat

K(θ = 0, R)

growthsas

∼ R

6

or

∼ R

4

,the rossovero urringat

R ∼ λ

.

s attered bythesingleparti les [23℄.Thusthe systemis linearand

I(θ)

an bewritten as(1) where

θ

is the s attering angle (the angle between the in- identlaserbeamand thedire tionat whi hthes atteredlightisdete ted),

N (R)

istheunknownnumber- on entrationdensity[

cm

−3

_µm

−1

℄ofparti les

ofradius

R

,and

K(θ, R)

isthe(known)kernelofthesystem,representingthe intensitys atteredbyasingleparti leofradius

R

atangle

θ

.Typi ally,

I(θ)

is dete tedatanitenumberofangles

θ

i

(i = 1, .., n)

withinaboundedinterval

[θ

min

, θ

max

]

.

If the parti les are spheres, the kernel

K(θ, R)

is provided by the Mie theory [22℄,a ordingto whi h the angulardistribution of

I(θ)

s atteredby aparti le ofradius

R

ismostly onnedto thedira tion lobe

θ

diff

∼ λ/2R

and

I(θ)

amplitudestronglyin reaseswithparti leradiusas

I

∼ R

6

forsmall

parti les(

R

≪ λ

)and

I

∼ R

4

forlargeparti les(

R

≫ λ

).Figure1(a)reports anexampleof thebehaviorsof

I(θ)/I(0) = K(θ, R)/K(0, R)

versus

θ

overa rangeof

[2

− 180 deg]

forparti lesofdierentdiameters

d = 2R

from

d = 0.01

to

100 µm

.Asone annoti e, forsmallparti les

I(θ)

tendsto beratherat, whereasforlargeparti les

I(θ)

exhibitsmanyos illationsandde aysbymany orderofmagnitudeoverthereported

θ

range.Atthesametimethezero-angle amplitude

I(0)

varies widely, passing from

I(0)

∼ 10

−11

at

R/λ = 10

−2

to

I(0)

_{∼ 10}

8

at

R/λ = 10

2

,seeFigure 1(b).Thus, itis learthattheinversion

of (1) might be ome an unbearable task when the parti le size distribution

N (R)

tobere overed ontainsparti leswithverydierentradii.

2.1Dis retizationoftheFredholmIntegralEquation

We now des ribethe dis retization of (1). Let us onsider that only anite

numberof

θ

i

(

i = 1, ..., n

) anbea essedexperimentallyandwithinalimited range

[θ

1

, θ

n

]

. Thus, if the parti le size distribution

N (R)

is approximated by ahistogram onstituted by

m

bins(or lasses)delimited by theradii

r

j

,

(6)

j = 0, 1, ..., m

,theequation(1)be omes

I(θ

i

) =

m

X

j=1

A

i,j

N

j

,

i = 1, 2, . . . , n,

(3)

where

N

j

is the number on entration density[

cm

−3

_µm

−1

℄ of the parti les

belongingtothe

j

-th lassofwidth

∆r

j

= r

j

− r

j−1

and

A

i,j

=

ˆ

r

j

r

j−1

K(θ

i

, r) dr.

(4)

Whenthe lassesarenarrowenough,we anpinpointthem intermsoftheir

averageradius

R

j

= (r

j

+ r

j−1

)/2

and width

∆R

j

:=∆r

j

, where

∆r

j

= r

j

−

I(θ

i

)

arethedataprovidedby theexperiment,thematrix entries

A

i,j

are known and

N

j

aretheunknowns tobere overed.

Although somewhatarbitrary,itisoften onvenientto hoosethe

r

j

grid points sothat, within the range

[r

0

, r

= r

0

a

j

,

where

a = (R

m

/R

1

)

1/(m−1)

and

r

0

= 2R

1

/(1 + a)

. In this way theaverage radiusandthewidthofea h lasss aleas

R

j

= R

1

a

j−1

,

∆R

j

= ∆R

1

a

j−1

,

j = 1, ..., m,

(5) thus

α = 2[(a

− 1)/(a + 1)]

and for

a & 1

,

α

≈ a − 1

. A sket h layoutand dis retizations hemeofthe lassesisreportedinFigure2.Typi allyifwewant

to overthree ordersof magnitudein size,i.e.

R

m

/R

1

= 10

3

with

α = 0.02

, approximately

m = 350

lassesarene essary.

2.2Constraints

Theveryrst onstraintwe animposeisthat

N

j

> 0,

j = 1, . . . , m,

(6) This omesfromthesimpleobservationthatthenumberofparti le annotbe

negative.

Amoreinteresting onstraintisrelatedtotheintegralof

N (R)

obtaining

c

N

=

m

X

j=1

(7)

Fig.2 Dis retizations hemeofequation(1)

where

c

N

represents the parti le number on entration [

cm

−3

℄, i.e., the

to-tal number of parti les ontained in the sample divided by the sample

vol-ume.This onstraint anbeappliedwheneverthenumberofparti les anbe

ounted, a somewhatdi ult task that an be arried outonly under some

experimental onditions.

Thelast onstraintisrelatedtotheparti levolumefra tion on entration

c

V

, whi h is given by the totalvolume o upied by the parti les dividedby thesamplevolume

c

V

=

m

X

j=1

N

j

v

j

∆R

j

,

(8) where

v

j

=

1 ∆R

j

ˆ

R

j

+∆R

j

/2

R

j

−∆R

j

/2

(4/3)πR

3

_dR

(9)

is the averagevolume of oneparti le belonging to the

isofparti ularsigni an ebe auseinmost experimentsthevolume on entrationisaquantitythat anbemeasuredquite

easilyandwithhigh a ura y.

Inourmethodwewouldliketoexploit both on entration onstraints(7)

and(8)(the positiveness onstraint,(6) ,beingfullledautomati ally,see

be-low),buttheyarenotequivalentfromaphysi alpointofview.Asmentioned

out above, whereas an a urate value of

c

V

an be easily obtained experi-mentally,theestimateof

c

N

mightbesomewhattroublesomeandae tedby largeerrors. Thus, we propose aweakly onstrainedversionof theLR

algo-rithm basedon the

c

V

onstraintalone, and wewill use theestimate of

c

N

onlyfor ross- he kingtheself- onsisten yoftheinversionpro edure,i.e.,for

(8)

3The Weakly Constrained Lu y-Ri hardson method

We now des ribe how the physi al model translates into the linear algebra

language.Thelinearsystem(3)is ompa tlyrewrittenas

AN = I

(10) where

A

∈ R

n×m

,

N

∈ R

m

and

I

∈ R

n

.

Similarly,thethree onstraintsofSe tion2.2 anberewrittenas:

(i) From(6)

N

_{> 0}

,meaningthat

N

j

> 0

,for

j = 1, . . . , m

; (ii) From(7)

c

N

= N

t

∆R,

(11) where

0 < ∆R

∈ R

m

,with

(∆R)

j

= ∆R

j

denedin(5)for

j = 1, . . . , m

; (iii) From(8)

c

V

= N

t

V

=

m

X

j=1

V

j

N

j

.

(12) where

0 < V

∈ R

m

and

V

j

= v

j

∆R

j

,being

v

j

denedin (9).

Thisinformationwillbeusedin thefollowingtodeneasimpleandee tive

iterativepro edureto omputeasolutionof(10),where,ex eptfromthe

pos-itiveness(i) ,the onstraints(ii)-(iii)arenotallstri tlysatised,butareused

toimprovethe omputedapproximationortoestimatepossibleparameters.

Notethatequations(11)and(12) anbeseenasweighted

ℓ

1

normbe ause both

V

and

∆R

arenonnegative,sowe an dene

kNk

1,V

= N

t

_V

_,

kNk

1,∆R

= N

t

_∆R.

AsstatedattheendofSe tion2.2,wearegoingtousedire tlyonlyaweighted

versionofthe onstraint(iii) .

Inordertoinsertthe onstraint(iii)weopportunelypadthematrix

A

and theright-handside

j

V

_j

arethearithmeti averagesof

theentries ofthematrix

A

andtheve tor

V

,respe tively.Wedene

˜

A =

A

γϕV

t

and

˜I =

I

γϕc

V

.

(14)

Notethat,sin ethefa tor

γϕV

j

= γ < A

i,j

> (V

j

/ < V

j

>)

appearingin(14), musthavethesamedimensionalunitsas

< A

i,j

>

,

γ

is orre tlya dimension-lessparameter.Asa onsequen e,itsee tof (14)isindependentontheunits

(9)

theinitialguess

N

0

_{> 0}

,the

k + 1

approximatedsolutionofequation(15) an bere ursivelywrittenin termofthe

k

−

iterateas

N

k+1

₌

N

k

a

◦

A

˜

t

· _˜I

˜I

_k

!

,

k = 0, 1, . . .

where

•

and

• ◦ •

are respe tivelytheentry-wisedivision andmultipli ation and

• · •

is the usual matrix-ve tor multipli ation. The ve tor

a

_{∈ R}

m

is denedas

a

j

=

n+1

X

i=1

˜

_A

t

j,i

=

n

X

i=1

A

i,j

+ γϕV

j

,

j = 1, . . . , m,

(16) and

˜I

k

is

˜I

k

_{= ˜}

_AN

k

₌

AN

k

γϕ

N

k

_1,V

.

Letus onsiderthe

j

− th

omponentof

N

k+1

:

N

j

k+1

=

N

k

j

a

j

"

˜

A

t

· _˜I

˜I

_k

#

j

=

N

k

j

a

j

ξ

j

,

(17) thefa tor

ξ

j

is

ξ

j

=

n+1

X

i=1

˜

_A

t

j,i

˜I

i

˜I

k

i

=

n+1

X

i=1

˜

_A

i,j

˜I

i

˜I

k

i

=

n

X

i=1

A

i,j

I

i

I

k

i

+ γϕV

j

c

V

c

k

V

,

(18) with

c

k

V

=

N

k

1,V

.Combining(16)and(18)with(17),weobtain

N

j

k+1

= N

j

k

P

n

i=1

A

i,j

_I

I

k

i

+ γ

_hA

i,j

i

_hV

V

j

_j

_i

c

_c

V

k

V

P

n

i=1

A

i,j

+ γ

hA

i,j

i

V

j

hV

j

i

V

j

,

j = 1, . . . , m.

(19)

where wehaveexpli itlyreportedthe expressionfor

ϕ

givenin(13). We an seethat onstraint(iii)isnotblendedwiththeotherterm,itisde oupledfrom

thedatattingpartandisweightedby

γ

1

representsthe ve torwithentries allequalsto

1

.

(10)

3.1Heuristi interpretation

We now want to givean heuristi interpretation of theformulation of (15).

A standardapproa h forpassingfrom a onstrainedleast squareproblem to

anun onstrainedproblemisthewell-knownquadrati penalizationte hnique

[5℄. In our ase, onsidering the onstraint (iii) , we obtain the minimization

problem

min

N

kAN − Ik

2

+ (γϕ)

2

kNk

1,V

− c

V

Ψ (N) =

kAN − Ik

2

+ (γϕ)

2

kNk

1,V

− c

V

2

= N

t

_A

t

_AN

− 2N

t

_A

t

_I

_{+ I}

t

_I

_{+ (γϕ)}

2

_N

t

_VV

t

_N

− 2c

V

N

t

V

+ c

2 V

.

Thegradientof

Ψ (N)

is

∇Ψ(N) = 2[A

t

AN

_{− A}

t

I

_{+ (γϕ)}

2

_(VV

t

N

_{− c}

_V

V

_)].

Assumethat

V

j

6= 0

forall

j

.Then

Ψ (N)

is oer ive.Infa t

Ψ (N)

_{≥ γϕ}

min

j

{V

j

}N

t

1

− c

V

2

→ ∞

as

kNk

2

→ ∞.

Thustheminimumof

Ψ

exists,byWeierstrass theorem,andsatises

∇Ψ(N) = 0.

(21)

Thefun tional

Ψ

is onvex,butnotstri tly onvex,sin e,ingeneral,

N (A) ∩

N (

V

t

₎

6= {0}

,whereby

N (A)

wedenotethenullspa eof

A

.Thus, ondition (21) is only ne essary and the minimum is not unique. However, sin e the

fun tionalis oer ive,wehavethattheminimum

N

∗

of

Ψ

satises

kN

∗

_k

2

<

∞

. Inotherwordswehaveproventhefollowing

_{) = 0}

.

Condition(21) anberewrittenequivalentlyas

A

t

_γϕV

A

γϕV

t

N

_{= A}

t

_γϕV

_I

γϕc

V

,

(22)

Re allingthedenitionsof

A

˜

and

˜I

,we anseethat(22)issimplythenormal equations of (15). This, oupled with (i) , leadsto the idea of looking for a

(11)

Let us all

N

_γ

the nonnegativesolution of (22)obtained with a ertain

hoi e of

γ

and suppose to know exa tly

c

N

. Thus, we expe t that the best hoi e for

γ

istheonethat,besideprovidingthebestre onstru tion for

N

γ

, minimizesalsotheerroron

c

N

.Therefore,we hoose

γ = γ

opt

su hthat

γ

opt

= arg min

γ

{|c

N

− kN

γ

k

1,∆R

|}.

(23)

In pra ti e, as we will see in Se tion 4, the hoi e rule (23) is not so stri t

be ausethere is alargerange of

γ

-valuesaround

γ

opt

where the re onstru -tion is equallygood and even avalue of

γ

veryfar from

γ

opt

would provide a urateresults.Thisfeatureisoffundamentalimportan ebe ause,whenever

the onstraint

c

N

isnotknownand anbeonlyroughlyestimated(asitmight happen experimentally),

γ

opt

annot bedetermined with high a ura y and the ondition(23)would beinappli able.

Toobtainanestimationof

γ

opt

we ompute

N

_γ

forasmallsetofpossible

γ

andsele ttheonethat minimizes

|c

N

− kN

γ

k

1,∆R

|

.

Summarizing,ourweakly onstrainedLR(WCLR)algorithmisthe

follow-ing:

1. x

N

0

_{= 1}

andasmallsetofpossiblevaluesfor

γ

; 2. ompute

N

γ

forevery

our algorithm performs against the lassi al LR method, i.e., when

γ = 0

. Wewillalso showhowto ndtheoptimalvalue

γ

opt

, onsistentlywith what des ribedabove.

4.1GenerationofmatrixA

The

n

×m

matrix

A

was omputedbynumeri allyintegratingequation(4)with thekernel

K

providedbytheMietheory[22℄andillustratedinFigure1.The numberofangleswas

n = 100

,s aled a ordingto ageometri alprogression

(12)

with

θ

min

= 2 deg

and

θ

max

= 180 deg

.Thebinsforthere overeddistribution were also hosen a ordingly to a geometri al progression (see Se tion 2.1)

with

R

min

= 10

−3

_µm

and

R

max

= 10

3

_µm

andtheirnumberwas

m = 600

.In thiswayallthebinswere hara terizedbythesamerelativewidth

∆R

j

/R

j

≈

0.023

.

4.2Generationofarti ialtestdata

Thegenerationofarti ialtest datawas arriedoutbysupposingtoknowa

I

be omes

I

_{= I}

_true

_{◦ (1 + ǫe) ,}

(24) where

e

is ave torwhose entries are realizationsof a randomvariable su h that

(e)

j

∼ N (0, 1)

.Typi alvaluesfor

ǫ

arein theinterval

[10

−3

_{, 10}

−2

_]

.

4.3Inversionpro edure andstopping riteria

Thearti ialtestdatain(24)wereinvertedbyxingavalueoftheparameter

γ

andusingtheiterativealgorithmgivenby(19).Theiterativepro edurewas stoppeda ordingto thefollowing riteria:

(1) Firstofallweimposeaminimumnumberofiterations

k

min

= 10

5

,whi his

ne essaryfortheWCLR(ortheLR)algorithmtoworkproperly.The

min-imumnumberofiterations

k

min

wasestimatedbyinvertingnoiselessdata and he king that

10

5

iterations are enough for retrieving the expe ted

distributionwith high a ura y (

RRE . 10

−2

, see below, Se tion 4.4).

Wealso he kedthat fornoisydata withnoiselevelstypi alof our

prob-lem (

∼ 1%

) when the pro edure would prefer to stop at a number of iterations

k < k

min

(seebelowpoints3.and 4.),for ingit to ontinue up to

k

min

= 10

5

doesnotspoilsigni antlythequalityofthedistribution.

(2) Wealso imposea maximumnumberof iterations

k

max

= 10

6

, whi h

en-suresthat the inversionstops evenwhen the riteria belowreported are

notmet.

(3) Forany

k

min

< k < k

max

thepro edureisstoppeda ordinglytoarelative dis repan y prin iple,whi hisamodiedversionofthe lassi al

dis rep-an yprin iple, largelyusedwith iterative regularizationmethods [9℄.As

relativedis repan yparameterwedenethequantity

ǫ

k

=

1 √

_n

I

k

_{− I}

I

2

,

(25)

(13)

unknownandhen ethe lassi aldis repan yprin iple annotbeused.

A ordingto (25),thealgorithm wasthereforestoppedafter

k = k

∗

iter-ationssothat

ǫ

k

∗

≤ ǫ,

(26)

where

ǫ

isthefra tionalnoiselevelintrodu edin(24).

(4) Finallywhenever,for

k

min

< k < k

max

, ondition(26) isnotmet but

ǫ

k

attains aminimum and begins to in rease (the sequen e

{ǫ

k

}

de reases forsmallvaluesof

k

,butisnotalwaysmonotoni de reasingforlarge

k

), thepro edureisstoppedin orresponden eoftheminimum.

it

thenumberofiterationsatwhi hthe pro edureisa tuallystoppedand the orresponding

ǫ

K

it

willbe alledMean RelativeStandardDeviation(MRSD).

4.4Evaluationparameters

Thea ura y ofthe inversionalgorithm wasevaluatedby omparingthe

re-trieveddistributionwiththetrueone.However,sin efromaphysi alpointof

view,volume (ormass)distributions aremu hmoresigni antthan number

distributions, we ompared retrieved and true distributions on the basis of

volume-fra tiondensitydistributions,denedas

φ(R) = N (R)v(R),

here

φ(R)

hasthedimensionsof[

µm

−1

℄.

Forassessingthea ura yoftheinversionpro edure,wedenea

γ

-dependent RelativeRestorationError(RRE)as

RRE(γ) =

kφ

γ

− φ

true

k

2

kφ

true

k

2

(14)

whi h orrespondstotherelativeaveragerootmeansquare(

r.m.s.

)deviations betweentheretrievedandtruemassdistributions.

Similarly,for assessingthea ura y onthe re onstru tionof the two

pa-rameters

c

N

and

c

V

that hara terize the true distribution, we dene the quantities

D

N

(γ) =

k

N

_γ

_k

1,∆R

− c

N

c

N

and

D

V

(γ) =

k

N

_γ

_k

1,V

− c

V

c

V

,

whi hrepresenttherelativeerrorsbetween

c

N

and

c

V

andthe orresponding re overedparameters.

4.5Numeri alresults

Inthefollowingnumeri altests,theWCLRandLRalgorithmswere ompared

forseveralvaluesof

γ

inagivenrangeofre overableradii.Thelatteronewas hosento beasubset oftheoriginalrangedened in Subse tion4.1, so that

R

max

/R

min

= 100

.Forea h

γ

,thetestswererepeated

30

times,withdierent noiserealizations(ofthesame

ǫ

level).

Test1 Inthersttestweshowthat,whenthedistributiontobere overedis

hara terizedbyparti lesthatprodu esignalswhosefeaturesare:(a)

asymp-toti ally onstantatlowanglesand(b)exhibitadynami rangebetweenrst

and lastangle of severalorder of magnitudes(see

d = 0.1 µm

or

d = 1.0 µm

urves in Figure 1(a)),the original LR andour WCLR algorithms arequite

equivalent.Tothisaim,wesele tedastruenumberdistribution

N

true

a Gaus-sian enteredinthemiddleofthere overablerange

[R

min

, R

max

]

,i.e.,withan averagevalue

hRi = 1 µm

andstandarddeviation

σ

R

= 0.1 µm

.Theparti le number on entrationwas(arbitrarily) hosentobe

c

N

= 10

16

_cm

−3

andthe

r.m.s.

noiseleveladded to thedata was

ǫ = 0.01

.Theinversionwas arried outbytrimmingthere overableparti leradii intherange

[0.1 µm

− 10 µm]

, so that number of bins was

m = 200

. Correspondingly, the matrix used in thistestwasobtainedbysele tingapropersubset

(100

× 200)

oftheoriginal

(100

× 600)

matrix omputedat se tion4.1.Thevalueof

γ

rangesfromzero (originalLR)to

γ = 10

6

and

K

it

= 10

6

.

Thendingsofthistestshowthatouralgorithmperformsequallywell

in-dependentlyofthe

γ

−

value(

0 − 10

4

)anditsperforman eswerequitesimilar

totheones providedbytheoriginalLR algorithm.ThisisshowninFigure 3

wheredatare onstru tionsandaveragere overeddistributionsarehighly

a - urate and, asmatter of fa t, indistinguishable between our algorithm (run

with

γ = 1

)andthe lassi alLRalgorithm(

γ = 0

).

Test 2 Theee tive dieren ebetweenthe two algorithmsbe omes evident

onlywhentheparti lesare losetotheboundariesofthe

[R

min

, R

max

]

range. Forthisse ondtestwesele tedaGaussiandistribution hara terizedbylarge

(15)

Fig.3 Test1:ComparisonbetweentheoriginalLR(bluesymbols)andourWCLRalgorithm

(redsymbols)runwith

γ = 1

foraGaussiandistributionwithanaveragevalue

hRi = 1 µm

andstandarddeviation

σ

R

= 0.1 µm

.(a)Re onstru ted(symbols)andtrue(line)signals. (b) Re onstru ted (symbols)and true (line) distributions. The two algorithmsperforms

equallywelland,asmatteroffa t,bothresults(a)and(b)arealmostindistinguishable.

parti les,i.e., with

hRi = 100 µm

and

σ

R

= 10 µm

.Thenoiseleveladded to the data was

ǫ = 0.01

, the inversion was arriedout in the range

[10 µm

−

1000 µm]

(sothat

m = 200

),and

γ

wasvariedin therange

[10

−2

− 10

7

_]

.

Dierently from the rst test, the ee t of hanging

γ

is quite relevant, as shown in Figure 4 where the behaviors of the parameters

K

it

, whi h we re all that denotes the numberof iterations, (a), MRSD (b), RRE ( ),

D

N

(d), and

D

V

(e) arereported asafun tion of

γ

.First of allweobservethat thereisabroadrangeof

γ

∼ [10

−1

− 10

5

_]

wheretheinversionpro edurewas

stoppedatanumberofiterations

K

it

≤ 10

6

(seeFigure4(a)).Correspondingly,

withinthisrange,there onstru tionparameter

M RSD

(seeFigure4(b))was omparablewiththenoiseleveladdedtothedata,meaningthattheiterative

pro edurewasmu hfasterandstoppeda ordinglytotherelativedis repan y

prin iple(26).

Then,whereas

D

V

(seeFigure4(d))de reasesmonotoni allywith in reas-ing

γ

(whi h is onsistentwith thefa t that thestrongerthe onstraint,the higherthea ura yofitsre overedvalue),theparametersdes ribingthe

a u-ra yoftheretrieveddistribution,

RRE

(seeFigure4( ))andthea ura yon there overednumber on entration,

D

N

(seeFigure4(e)),exhibitverybroad valleys whose at regions over almost the same range of

γ

∼ [10

1

_{− 10}

5

_]

.

Thus,the hoi eofanoptimalvaluefor

γ

isnot riti alat allandanyvalue hoseninthe entralpartofthisinterval(forexample

γ

opt

∼ [10

2

−10

4

_]

)leads

bothto small errors in the re overyof

c

N

and to verya urate distribution re onstru tions,as shownin Figure 5(e-f-g). Conversely,for valuesof

γ

out-sidethisrange,there overyof

c

N

be omesin reasinglylessandlessa urate and,atthesametime,thedistributionsaremoreandmorepoorlyre overed,

asshownin all theotherpanelsof thegure.Inparti ular, wewould liketo

pointouttheremarkablemismat hingbetweenthetruedistribution andthe

onere overedinFigure5(a),showingthat the lassi alLR algorithm(

γ = 0

) istotallyunabletoperformsu hatask.

(16)

Fig. 4 Test 2: Behavior as a fun tion of

γ

of the average parameters

K

it

(number of iterations)(a),MRSD(b),RRE ( ),

D

N

(d),and

D

V

(e)foraGaussiandistributionwith

hRi = 100 µm

and

σ

R

= 10 µm

.Theerror bars arethe standard deviationsasso iatedto the various parametersderivingfromthe noise(

ǫ = 0.01

)presentonthedata (error bars largerthanthedatapointsarenotdisplayed).

Finally,wewouldliketopointoutthattherathersimilarbehaviorsbetween

RRE

and

D

N

guaranteesthat,inarealexperimentwheretheparameter

RRE

annotbemeasuredbe ause

φ

true

(R)

isnotknown,theoptimalrangefor

γ

opt

anbeinferredbylookingatthebehaviorof

D

N

(Figure4(d)).Thefa tthat su harangeisremarkablybroad(

∼ 2 − 3

ordersofmagnitude)ensuresthat evenahugeun ertaintyonthevalueof

c

N

wouldnotae t signi antlythe a ura ywithwhi h

φ

true

(R)

willbere onstru ted.

Test3 Inthistestwesele tedadistribution losetotheleftsideoftherange,

namely aGaussian with

hRi = 0.05 µm

and

σ

R

= 0.005 µm

.In this ase, as shown inFigure1,the

I(θ)

dataareratherat;therefore,fornot orrupting ompletelytheirbehavior,thenoiseleveladdedtothedatawas

ǫ = 0.001

.The inversionwas arriedoutintherange

[0.005 µm

− 0.5 µm]

(sothat

m = 200

), and

γ

wasvariedintherange

[10

−7

_{− 10}

2

_]

.

Figure6showsthebehavioroftheparameters

K

it

(Numberofiterations) (a), MRSD (b),RRE ( ),

D

N

(d), and

D

V

(e), whereasthe omparison be-tweenthere overeddistributionand

φ

true

(R)

isshown inFigure 7varying

γ

in therange

[10

−6

− 10

3

_]

. As for the large parti les, there is an even larger

optimalrange

γ

opt

∼ [10

−3

− 10

1

_]

,where onvergen erateisfasterand both

distributionre onstru tionand

c

N

re overyareverya urate.Were allthat

γ

opt

an be safely identied by the riterion (23) thanks to the very broad andshallowminimumregionof the urveof

D

N

inFigure 6(d).Finally,note

(17)

Fig.5 Test2:Comparisonbetween theaveragere overeddistributions(redsymbols)

ob-tainedwithouralgorithmatvarious

γ

-values(

γ = 0

isLR)andatrueGaussiandistribution

φ

true

(R)

withwith

hRi = 100 µm

and

σ

R

= 10 µm

.Theerrorbarsarethestandard devia-tionsasso iatedtothebinsofthere overedhistogramduetothenoise(

ǫ = 0.01

)presents inthedata.

that,also in this ase, theoriginal LR algorithmis totallyunable to re over

thetruedistribution(seeFigure7(a)).

4.6Errorsonthe

c

V

onstraint

Inthis se tionwereportanexampleof theee ts ofsmallsystemati errors

onthe onstraint

c

V

withrespe ttothequalityofthere overeddistributions. We investigated the spe i ase of the true distribution used in Figure 7,

i.e., aGaussian distributionwith

hRi = 0.05 µm

and

σ

R

= 0.005 µm

,andwe supposedtohaveerrors

δc

V

varyingwithinarangeof

±10%

.Thenforanyxed valueof

γ

andweinvestigatedhowthere overeddistributions deterioratesas

δc

V

isin reased. Thisanalysis was arried outfollowingthe samepro edure des ribed forthe previoustests, andwasrepeated for

8

dierent valuesof

γ

spanningtherange

γ

∼ [10

−5

_{− 10}

2

_]

(

1

pointperorderofmagnitude). Themain result is summarized in the rst panel of Figure 8, where the

behaviorof the parameter RRE is reported as a fun tion of the per entage

error

δc

V

. As expe ted, within the optimal range

γ

opt

∼ [10

−3

− 10

1

_]

, the

minimum of RRE o urs at zero error and in reasesas

δc

V

be omes larger andlarger,butas longas

δc

V

.

±2%

,itsvalueremains

.

2 − 3 × 10

−2

_.

This

trendis learlynotfollowedbyvaluesof

γ

thatareoutsidethisoptimalrange be ause,evenat zero errorasshown in Figure 6(a), theinversionalgorithm

(18)

Fig. 6 Test 3: Behavior as a fun tion of

γ

of the average parameters

K

it

(a), MRSD (b), RRE ( ),

D

N

(d), and

D

V

(e) fora Gaussian distributionwith

hRi = 0.05 µm

and

σ

R

= 0.005 µm

.with

hRi = 100 µm

and

σ

R

= 10 µm

.The error bars are the standard deviationsasso iatedtothevariousparametersderivingfromthenoise(

ǫ = 0.001

)present onthedata(errorbarslargerthanthedatapointsarenotdisplayed).

Fig.7 Test3:Comparisonbetween theaveragere overeddistributions(redsymbols)

ob-tainedwithouralgorithmatvarious

γ

-values(

γ = 0

isLR)andatrueGaussiandistribution

φ

true

(R)

withwith

hRi = 0.05 µm

and

σ

R

= 0.005 µm

.Theerrorbarsarethestandard de-viations asso iatedto thebinsofthe re overedhistogram dueto the noise(a

ǫ = 0.001

) presentonthedata.

;( )-(d)-(e): omparisonbetweenthe truedistribution(solidline)andtheaveragere overeddistributions(redsymbols)obtained

with

γ = 10

−

2

and

−5%

,

0%

and

+5%

per entageerrors.Thestandarddeviationsasso iated tothe binsofthere overedhistogramsareduetothe noise(a

ǫ = 0.001

)presentonthe data.

is not ableto work properly. The other importantparameter des ribingthe

performan esoftheinversionalgorithminpresen eofsystemati errorson

c

V

isthebehaviorofMRSD(seeFigure8(b)).Asshown,withintheoptimalrange

γ

opt

∼ [10

−3

− 10

1

]

andforerrors

.

±5%

,alltheMRSD valuesare ompatible with the fra tional statisti al errors on the data (

ǫ = 10

−3

). Outside these

ranges,MRSD tendstoin reasebe ause onvergen eissloweddownand

10

6

iteration are not su ient or be ause signal re onstru tions are very poor

whatsoever.

InthelastthreepanelsofFigure8wereportanexampleofthedistributions

re overed for

γ = 10

−2

at errors on

c

V

equal to

−5%

,

0%

and

+5%

. As expe ted,the entraldistribution(paneld,

0%

error)mat hesverya urately the input ones (RRE

≈ 8.7 × 10

−3

), whereas the other two are learly less

a urate, beingRRE

≈ 3.2 × 10

−1

for the

−5%

errorand RRE

≈ 8.8 × 10

−2

forthe

+5%

error,butstill verymeaningfulfromaphysi alpointofview.

5Con lusions

Inthispaperwehaveproposedasimpleandee tivevariantoftheLRmethod

by addinga weak onstraintto be imposed asa penalty term onthe

re ov-eredsolution.Thenewmethod, alledWeakly ConstrainedLu y-Ri hardson

(WCLR), wasapplied and tested on the inversionof simulated elasti light

(20)

volume-size,

φ

true

(R)

, distributions of the parti les omposing the sample over the largestpossibleintervalof parti lesizes.

anbeae ted by large errors.Thus in this paperweused asa onstraintonly

c

V

andexploited

c

N

fordeterminingaaposteriori riterionforadjusting thestrength,

γ

,of

c

V

.

By the use of extensive numeri al simulations, we have shown that the

optimal value

γ

opt

, i.e., the onethat provides the highesta ura ies in the re overyof both

c

N

and

φ

true

(R)

(or

N

true

(R)

),isnot riti al at allbe ause there isabroadrange(

∼ 2 − 4

ordersofmagnitude)of

γ

-valuesaround

γ

opt

where bothrestorations are equally good. Thus, in areal experiment where

φ

true

(R)

isnotknown,theoptimal

γ

opt

anbeinferredonlybylookingat

c

N

k

min

< K

it

5

was

areasonable ompromise.Intheextremelyrare asethatthebestrestoration

isobtainedataniterationlowerthan

k

min

,wedonotobservetheinsurgen e ofartifa ts, ortheyareat leastfairly weak, be auseWCLR algorithm hasa

verystable onvergen e,espe ially againstreasonable levels (

∼ 1%

). In any ase,forhighernoiselevels,aprepro essingdenoisingpro edure ouldalways

beapplied.In on lusiontherefore,although

k

min

ouldhavebeenoptimized from aseto ase,thesinglevalue

k

min

= 10

5

allowedustore overa urately

distributions spanning the entire

[R

min

, R

max

]

range, without introdu ing a furtheradjusting parameterinthepro edure.

Ournumeri alsimulationsshowthat,whentheparti lesizesto be

re ov-eredliewithinarangeextension omparablewiththat ofthemeasurements,

WCLR and LR are fairlyequivalent:they areboth simple,robustand,

pro-vided thatthe iterativepro edure isstopped after alargeenoughnumberof

steps, equallya urate. The only slightdieren e is that, whereas LR truly

requires no adjusting parameter, in the ase of WCLR, the only adjustable

parameterwould be

γ

, but asmentionedabove,it anbevariedoversu h a largerangeofvaluesthat,asamatteroffa t,doesnotneedtobeoptimized.

(21)

errors on the knowledge of the onstraint

c

V

with respe t to the quality of there onstru ted distributions. Ourndings showthat, aslongastheerror

on

c

V

is within

±2%

, the quality of the re onstru ted distributions is well preservedandtheuseofaweak onstraintisadvantageouswith respe tto a

strong onstraint.Thisfeatureisfundamentalinpra ti alappli ationswhere,

typi ally,therelativea ura yon

c

V

isoftheorderofafewper ents. AswestatedintheIntrodu tionthenatureoftheproblemitselfsuggested

that theLR algorithmwouldhavebeenagood hoi e, whileother methods,

inparti ulartheoneswhi hreliesontheminimizationoftheEu lideannorm,

would havebeeninee tive.

Themaindrawba koftheproposed approa hisstilltherequirementofa

fairlyhighnumberofiterations,eventhoughwehaveshownthatthenumber

ofiterationsrequiredbyWLCRismu hsmallerthantheonerequiredbythe

standardLRmethod. However,thedimensionoftheproblem athand,while

notsmall,isnottoolarge(asseenintheexamplesinSe tion4)andthusthe

overall omputationeortislimitedevenwhenahighnumberofiterationsis

required.

Itwillbesubje toffuturestudiesthe ombinationofthete hniques

pro-posedin this paperwithmoree ientmethods from theliterature,starting

from the work in [8,27℄. In parti ular, it would be interesting to insert the

methods from[13℄ in ourframework,aswell astheto ombine ourapproa h

withModulus-Methodforill-posedproblems, see[1℄andreferen estherein.

Finally, we would like to point out that variants of the lassi al

Lu y-Ri hardson,likethosein [4,24℄, ouldbeusefulforfurther improvingour

al-gorithm.Wewouldalsoliketostressthat,althoughinthisarti leourWCLR

methodhasbeentestedanditsperforman esas ertainedforthespe i

opti- alproblemrelatedtotheinversionoftheelasti lights atteringdata,the

pro-posemethod anbeinprin ipleappliedtoanyotherproblemwheresome

on-straintsasso iatedtointegralsofthedistribution tobere overedareknown.

Examplesofsu hproblemsareagainintheeldsofopti swiththete hnique

known asmulti-spe tral extin tion turbidity [10,11℄ or in the eld of

stere-ology,aninterdis iplinarymethodologythat is on ernedwiththere overing

of the 3D properties of asample from its 2D se tions (see for example [20℄

and referen es therein). Work is in progress for extending WCLR to these

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6A knowledgments

The author would like to thank the reviewers for their insightful omments

thatgreatlyimprovedthereadabilityandtheoverallqualityofthiswork.The

work of thersttwoauthorsis supported in partbyMIUR -PRIN2012 N.

2012MTE38NandbyagrantofthegroupGNCS ofINdAM.

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