Possibilistic C-shell clustering with inter-cluster constraints

2006IEEE International Conference on Systems, Man, and Cybernetics

October8-11, 2006,

Taipei,

Taiwan

Possibilistic C-Shell Clustering with Inter-Cluster Constraints

Tsaipei

Wang, Member,IEEE, andJames M. Keller, Fellow,IEEE

Abstract-This paper describes our analysis of using extra constrainttermsregarding relations between cluster prototypes in possibilistic c-shell clustering. The extra constraints are

implemented as additional terms in the cost function. This allows users of these algorithms to incorporate additional knowledge regarding properties cluster prototype into the clustering process. Ouranalysis here focusesontheuse ofone extra term for locating circles (shell clustering with circular

prototypes)with similar radii. An adjustable factor is used to

control the strength of this constraint. For possibilistic

clustering, this couples the update procedure of the otherwise independent prototypes. Our experiments, using both simulation and real image data, indicate that this is especially

useful in locating actual clusters when the available data are

noisy.

I. INTRODUCTION

THE ability to locate specific shapes in images is important formany image analysis problems. Fuzzyand possibilistic c-shell clustering algorithms have been demonstrated to performwell for the task oflocating lines, circles, and quadric curves in 2-D images (orhyperplanes, hyperspheres, and hyperquadric surfaces in higher dimension data) [1-4], rectangles [5], and template-based shapes [6]. Examples of their application to real-world image analysis are in [7,8]. Among the two (fuzzy and possibilistic) approaches, the

possibilistic

approach is more

immune tothe effect of noisepoints, and isthe basis ofour

algorithm discussed in this paper. Compared with the generalized Houghtransform [9] ortemplate

matching,

shell clustering hasmuchsmallermemoryandtime

requirements,

and the precision of prototype parameters is not limited to

the bin size in the Hough transform or the parameter resolution of thetemplates [2].

Existing c-shell clustering algorithms allow the specification of the number and type of prototypes, the initial conditions (membership values or prototypes), and a

distance or similaritymeasure between prototypes and data points. They do not allow the inclusion of additional constraintsonclustercharacteristics in the

clustering

process. However, insomeparticularapplicationstheremay be other desirablepropertiesfortheclusteringresults. Forthese cases,

we present here the idea of introducing

approximate

constraints on cluster prototype parameters that

correspond

tothedesired clusterproperties. Sincefuzzyand

possibilistic

Tsaipei Wang is with theDepartment ofComputerScience, National Chiao Tung University, Hsinchu, Taiwan (phone: +886-3-5712121

ext.56689;e-mail:wangts(cs.nctu.edu.tw).

J. M. Keller is with the Department of Electrical and Computer Engineering, University of Missouri-Columbia, Columbia, MO 65211,

USA.

clustering algorithms identify the clusters through minimization of cost functions, it seems natural to try to incorporate these additional constraints on cluster parameters in the cost functions, and this is the approach we describein this paper.

There are many different types of additional constraints, depending on the particular applications and the extra information we know about the desired clusters. Such constraints can also be applied to the relations between clusters. For example, in c-shell clustering, we may want linear/planar prototypes that are approximately parallel to each other, or circular prototypes with similar radii. Different constraints mean different constraint terms added to the cost function. Here we focus on one example, where

we look for circular prototypes that have similar radii, and

use thisexample to illustrate someproperties ofusing such additionalconstraints. In the following, section II describes theformulation of the original and our modified possibilistic c-shell clustering (PCSC) algorithms for circles. Section III contains simulation results, and section IV describes an application in a real world scenario. Section V is our conclusion and discussion.

IL.

DESCRIPTION OFALGORITHM A. The OriginalPCSCforCircles

The original PCSC algorithm, presented in [2], is an extension ofthepossibilisticc-means (PCM) algorithm[10] with shell-shapedprototypes. Here we limit our description to circle prototypes only. The following distance measure betweenapointxjandacircleprototype

{c1,

ri},

ct2~~ 2 2

dij2=(|Xi

C1 (1)

is used to allow an analytical form for updating the prototypes, avoidingthe numerical methodsused by [4] for the original fuzzy c-shell clustering. The objective function of PCM is

C N C N

J= z

uijmdy:

+zlz(1 -uq)m.

(2)

i=lj=1 i=1 j=1

Here

uj1

is the possibilistic membership of vector Xj in prototype

{ci, rj},

N is the number of vectors, C is the number of clusters, and

ql

and m are two parameters that determine thedependenceof

uij

on

dcj.

Theupdateequations for the possibilistic membership values and the prototypes

U"j = I+ i m and

2(ff)

1wi respectively, where -2cj Pi= T 2 , -Ci Ci-ri N m X

[

1 Hi

]1Y

Li

iXjI'

m2UJ

(XjX)

and

Xjm(XjTXj

j=1 j=1

These update equations are derived by

setting

to zero the derivatives of theobjective functionJrelativeto

u1j

and

pi.

We can perform the original PCSC

algorithm

using

the procedurebelow:

1.Initialize the prototypes.

2. Repeat until convergence or for a

predefined

number of iterations

2.1. Calculate cluster

membership

U=

{uij}

using (3).

2.2. Updatethe prototypesusing (4).

Convergence can be defined as when the

changes

of prototypesbetween iterations fall below a preset threshold.

B. The

Modified

PCSCfor Circles

Since we are considering circle prototypes with similar radii, weadd an extra termJctotheoriginal costfunction J in(2)totakeradiussimilarityintoaccount:

J'=

J+Jc

(5)

For the purpose of simplicity, let us consider only two

clusters now. Let

(cl,r1)

and (c2,r2) represent the respective prototypes. For the constraint of radius similarity, we can usethefollowingform for

J,:

2

_2

2

c=

-a(r2

-r2

)

(6)

Here a is zero or a positive number

indicating

how strong theconstraintis. The selection ofits value willbe discussed later.

The next step is to derive theupdate equations. With the added term Jc, we are not able to derive a

single

equation like (4) that simultaneously updates both the center and radius of a prototype. It is necessary to differentiate J relative to the center and radius ofa prototype separately, and minimizeJbysetting these derivativestozero:

al N (

2_2

=-2j(ui)m(Xj -ci)ixj-ci

ri2

)0

(7)

and

ail

a(ri2

) =-2L(Ur2J )rx2)-Ci| ri

+2a

(ri2

-

r3-i2

)=O.

Here i 1,2.

Equations (7) and (8) do not allow us to obtain simple closed-form update equations for the prototype parameters, c1, r1, c2 and r2. This is because both r1 and r2 appear together due to the added term

J,

Therefore, an iterative method isnecessary for finding the solutions. Similar to

[4],

we choose to solve (7) and (8) using Newton's method. There are 6 quantities to be optimized simultaneously (2 each forclandc2 for two-dimensional data, and one each for r,andr2).

The procedure and stopping criteria of themodified PCSC algorithm are similar to those of the original PCSC, except that we update the prototypes by solving (7) and (8) using Newton's method. To reduce the amount of computation time,we further limit the prototype parameterupdating step toonly one iteration of Newton's method. This was indicated in [11] to still give goodclustering results, which we find to bethecase.

In order to gain some insight into the effect of thenew

parameter a, let us first consider a "perfect" scenario where the data points fall exactly on two circles and are evenly spaced in each circle, with

(cl*,r

*)

and

(c2*,r2*)

being the centers and radii of the two underlying circles. In addition, these two circles are assumed to be well separated so their corresponding data points can be treated as two separate sets (calledX(1) andX(2)below), with themembership ofa data pointin one settothe cluster of theothersetassumed tobe

zero. This makes the clustering process more like two

separateclusteringprocessesofX(l) andX(2) coupled only by the constraint term

J,

For this special case, one solution to (7) is when

ci=ci,

regardless of the value of

ri.

This can be understood by consideringthe assumedsymmetry. When c,

ci,

all the

xj

in X(i)have thesame

dij:

dij2 k

[(i)2

-2]i

for

xj

in

X(i),

which is independent ofj. This makes the memberships

uj

independent of

j

as well. (We will use u(l) and U(2) to represent thej-independent memberships.) Combine these with theassumption ofevenly spaced

Xj,

and wecan seethat the summation in (7) becomes zero when

ci=ci*.

Since we have notmadeanyassumption regardingthe prototyperadii, this proves that

cl=cl*

isasolution.

Our next task is to solve for

r,

and r2 given

cl=cl*

and C2 C2. First, from (8) we can obtain the following relation betweenr1 andtheother parameters:

2 a2 +

E

(U(l))m

Xj

-c1l

2

Xjc (l, a X((U) )) XjeX(]) (9)

LetN1 andN2 are thenumber of data points in X(1) and X(2), (8) respectively, and

N1'=Nl(u(,))m

andN2'=N2(u(2))m be the total membership in the two individual clusters. By into (9) substituting

xj

cl|=

xj

-c |=- forxjinX(i), we get

a2

+

N,1(*)2

2 - r2

±iv1tr1}

(IOa)

and similarly 2

*2+

r2

arI

+N2

(r2)

(lOb)

a

±N2'

We can further derive separateexpressions

(rj)2

and(r2)2 as

weightedaverages of(rl )2and

(r2

*)2:

2

(aN2

')(r )2

+(a+N2

')N1'

*)

2 r,2 = 2 r2 2

r)

( a) (aN2')+(at+N2')Nl' and 2

(aN,')(r>

)2±(a

N1

')N2

(r2 (b

r2

=

.i2r

(I

lb)

(aN1

')+(a+N

')N2'

This implies that solutions of both ri and

r,

have values between

r1*

and

r2*.

When acO,wehave

rl=rl*

and

r2=r2*

as

expected. When ais much larger than N1 and N2 (andN1' andN2'as

well),

it turnsoutthatr,and r2 become identical:

r]2, r22

N1 (r1)i N2'(r2)2 (12)

N1 +

N21

Equations (1 1)and(12)are notsolutions of

r,

and r2. This isbecause N,' actually depends on r1 through d112 and

u().

A

special case whenwe can obtain

simple analytical

solutions for

r1

andr2 iswhen we set

71

J24oo,

and from(3)wehave

uij-1.

This in turn makes

Nj'

-N. This allowsus to compute analytical solutions of r1 and r2

directly using (11),

with

Nl'

and N2' replaced by N1 and N2. In

addition,

because membership values are constant, these solutions will not

change between iterations of the

clustering algorithm.

In

otherwords, theycorrespondtothe prototypes found

by

our

modifiedPCSCalgorithmunder theconditionsabove. Inthe followingwecontinueto assume

Ni'=N1.

The purpose of thederivationsabove istounderstandhow the relations between rl, r2,

rl*

and

r2*

are affected

by

the relations between a, N1 and N2. One

interesting

aspect to

look at isthe ratio between

[(ri)2_(r

*)2]

and

[(r2

*)2

-(rl

)2].

This isa measure of how much the radius ofaprototype is affected by the difference between the two

underlying

circles. We caneasily obtainthisratio

by

subtract

(r1

*)2

from bothsidesof( lla):

rl2

*)2

aN2 (ri ) -(ri) __13 *2

*2I

a

(r*)2_

(r1*)2

aN2 +(a-+N2)N and

similarly,

2 *)2

(r2)

(r2)

- aN1 (I3b) *2 *2

(r1

)

_

(r2

)

aN++(aNc+

N

)N2

We candrawafew observations for

special

casesof

(13):

(1) The ratio in(13a)is closeto onewhen

oN2

>>

(c-N2)NI

(i.e., afterdividedby

aNIN2,

Nl-'>>a-+N2-

').

Thisoccurs whenboth aandN2aremuchlargerthan

N1.

Inthis case,we have

rP:-r2

.Onthe other

hand,

theratio in

(3.33b)

is closeto

zero under the same condition because

aNV

<<

aN2

<

(a-+N1)N2,

resulting in

r2=r2*.

Oneobservation we can draw

is that, when a is large compared with the number of data points, and there are many more data points in one set than the other, both prototypes will have radius similar to that of the underlying circle of the larger data set.

(2) When

a>»N2=N1,

both ratios in (13) become 1/2, giving

2 2

ri r2

[(rl*)2+(r2*)2]/2.

This means that both

prototypes

end up with the same radius when oc is very large and both sets have the same number of data points.

(3) The ratio in (13a) is close to zero when cxN2<<

(oaIN2)N1

(i.e.,

after divided

by

aNIN2,

Nl-l<<

-

±+N2-

).

Thisoccurs

when either aorN2is much smaller thanN1. In this case, we have

rl-rl,

meaning that r1 is barely affected by the extra constraint.

Overall, for the extra constraint to significantly affect the clustering result, ashould be large compared with either N2 orN1.

For more thorough investigation of how the relations between a;

N,

and N2 affect the clustering result, we plot in Fig. 1 r1 (solid lines) and r2 (dashed lines) relative to

rl*

and

r2*

as functions of a for a few different combinations of

N1

and N2, computed from(11). The radii of underlying circles in this simulation are set to

r1

10 and

r2*

=20. We can see that

r1=rl

* and

r2=r2*

when a-0, and that both r1 and r2

appear to move toward the same value when ais increased. In addition, this value shifts toward

r1

* when

N1/N2

increases, and toward

r2*

when

NI/N2

increases.

III. SIMULATION RESULTS

Inthis sectionourmodified PCSC algorithm is appliedto two dimensional simulation data. The assumption of independent sets of data for individual clusters is used throughout this section just for illustration purpose. This assumption does not affect the actual usage of thealgorithm because in the possibilistic approach to clustering, each prototype is updated independently regardless of what happens with the other prototypes, except for our added inter-cluster constraints.

Forcomparisonwiththe derivation in theprevious section, we start with the "perfect" scenario. We plot in Fig. 1

r1

(squares) and r2 (circles) obtained by applyingourmodified PCSC algorithm,instead ofusingtheanalytical results, for a few different values of a. The simulation results are

obtained with perfect initialization (initial prototypes overlapping with underlying circles of the data points),

qV500000,

andm=1.5. The value ofq is made verylargeso

that allmembership values will be closetoone, asassumed for the analytical results. We can see the prototype radii obtainedthrough clusteringare consistentwiththeanalytical results.

Fig. 2 contains the datapoints and prototypes found with various combinations of a,

N1

and N2. Parameter values

qr500000

and

m=1.5

are used. The small dots are data points and the large circles are prototypes found. Fig. 2(a) and (b)have

cr0,

and the prototypes found matchperfectly

N2= 100 20 15 10 20 rIl -o c m -1 (a)

_le

.1

.1

,.."i 0'-.,',:

I (c) .

4''

.. .~ ~ ~ ~~ 04 Co 0 Tl-0 100 200 0 100 200 0 100 200

a

Fig. 1. Radii ofprototypes found by PCSCrelative totheradii of

underlying circles ofthe data points,asfunctions of a, N1 andN,. Eachindividualplotcorrespondstoadifferent combination ofN, and N,. Radii ofunderlying circlesarer, =10andr,=20.The solidand

dashedlines areanalytical resultsfor r1 and r,,respectively,and the

small squares and circles are simulated results for r1 and r,,

respectively.

with the underlying circles ofthe data points. In Fig. 2(c),

we have ao0 and

N,.N2,

but both prototypes match perfectly with the underlying circles and have the same

radius because we have set rl

*r2

Fig. 2(d)-(f) correspond

to a>0,

N1.N2,

and

rlJ*r2*.

InFig. 2(d),wehave

N2>N,,

and the resulting prototypes match better with X(2). This is reversedinFig. 2(e).Fig. 2(f)is thesame asFig.2(d)except that ais larger, causing the two prototypes to have even

closer radii. The changeis moreevident forX(1), which has lessdata points.

Itmay seem strangethattheextra constrainttermactually makes the prototypes different from the underlying circles with differentradii, suchas inFig. 2(e). The

analysis using

the perfect scenario is to help us understand the relations between ocandotherparameters. Amorepracticaluseofthe modified PCSC algorithm is to more accurately locate the prototypes that are expected to have similar radii when the data arenoisy. For example, assume that

X(l)

contains only datapointsalonga

circle;

we can find this

underlying

circle withPCSC easily. Intheotherhand, forexample, letX(2) be

so noisy that simply clustering its data points produces a

prototypeverydifferent from theunderlying cluster. Insuch cases,we canthink ofthe"coupled"

clustering

as

having

the prototype for X(l) pullingthe prototype forX(2) along inthe clusteringprocess.

The scenario in the last paragraph is illustrated with simulated data in Fig. 3. HereX(1) and X(2) have the same

number of data points

(N,=N2=48),

but X(2) contains only part ofa circle (16 points)and therest arenoise points (32 points). The underlying circles have the same radius,

r*

=r2 =8,and arecentered at (0,0). We applyourmodified

PCSC algorithm using different values of c, Fig. 3(a)-(d) correspond to cc = 0, 5, 20, and 400, respectively. Initial

(d)

(e)

(f)

(1II~~~~~~~~~~~~~~~~~4

Fig.2. Simulation exampleof PCSC withconstraintusingnoiseless

data.The smalldotsaredatapointstobeclustered,andthe circlesare

foundprototypesfoundby ouralgorithm. Both axesofeach small rectangle have ranges of -25 to 25. Parameters used are: (a) rj=r =10,N=N,=20, cr=0; (b)rl=10,r,*=20,N,=N,=20, a-0; (c)

r==r,=10,N1=20,N,=60, cr20; (d)rl =10, r,=20,N1=20,N,=60, c-20; (a) r*=10,r,*=20,N1=60, N,=20,a-100; (a) r'=10, r, =20,

N,=20, N,-60, cr100.Theleftandrightrectangles of each of(a)-(d) correspondtoparameters withsubscripts1and2,respectively.

prototypes are set to be the underlying circles so that no erroris causedby bad initialization. Thevalue ofq is set to

(r1

*)4/2;2000.

The solidcirclesare thefinalprototypesafter 10iterations of ouralgorithm, and the dashed circles arethe underlying circlesoverlapped withX(2) forcomparison. The actual centers and radii of the prototypes found are also listed in Table 1. It is evident that,by increasing oc, we are abletobetteridentifytheunderlying circlesinthenoisydata

set(X(2)) through clustering. Forthetwo larger oxvalues, the prototypesfoundforcluster #2 are so close totheunderlying circlesthattheyarealmostinseparable.

To more systematically investigate the power of the constraint term in locatingthe correct underlying circles for this scenario, we repeat for 100 times the sameexperiment

as in Fig. 3 with the same underlying circles but different, randomly generated noise points. Furthermore, the initial prototypes have randomly selected centers within r1

*14=2

from (0,0) and radii of between 8±2. We include the variation in initialization to make the experiment more

realistic. The statistics is listed in Table 2. The quantity "radius error" is the absolute radius difference between the prototypes found and the underlying circles. The quantity "deviation"represents themaximum separation between the final prototype and the underlying circle and is used as a measureofhow accurately theclustering algorithm finds the underlying circle. Cluster #1, which corresponds to the noiseless set of data points (X(1)), remains very close to the

underlying circle for all a values. On the other hand, for cluster #2,which corresponds to the noisy set of data points (X(2)), both deviation and radius error are significantly reduced when a is increased. Even a--1, which is much smaller than both N1 and N2, results in significantly more accurateprototypes than when a-0.

(b)

Fig. 3. Simulationexampleof PCSC withconstraintusingonesetof noiselessdata and one set ofnoisy data.Thesmall dotsaredatapoints

to be clustered, and the circlesare found prototypes found by our

algorithm. Both axesofeachsmallrectanglehave ranges of -20to

20. (a)-(d)correspondtoc-0, 5, 20,and400,respectively. Theleft

andrightrectanglesof eachof(a)-(d)correspondtoparameters with

subscriptsI and2,respectively.

TABLE I

PROTOTYPESFOUND IN FIG. 3

Cluster#1(left) Cluster #2(right) Center Radius Center Radius (a) 0 (0.00,0.00) 8.01 (2.10,0.12) 9.85 (b) 5 (0.00, 0.00) 8.09 (0.15,0.05) 8.79 (c) 20 (0.00,0.00) 8.07 (0.33, 0.04) 8.20 (d) 400 (0.00,0.00) 8.05 (0.21,0.02) 8.05 TABLE 2 PROTOTYPEACCURACY Cluster#1 Cluster#2

a Deviation RadiusError Deviation RadiusError 0 0.01±0.00 0.01±0.00 3.38±9.95 1.41±4.35 1 0.02±0.02 0.02+0.02 2.16+1.40 0.88+1.05 5 0.06±0.04 0.06+0.06 1.51+0.92 0.55+0.67 20 0.07+0.06 0.07±0.09 0.92±0.49 0.25+0.30 100 0.08±0.06 0.08+0.09 0.71+0.36 0.13+0.15 400 0.08±0.06 0.08+0.10 0.65+0.33 0.09±0.11

Accuracy ofprototypes foundbyourPCSC algorithm usingoneset

ofnoiseless dataandone setofnoisydata. TABLE 3 PROTOTYPEACCURACY

ar Deviation Radius Error 0 1.60+2.66 0.59+0.88 1 1.32+1.44 0.48±0.60 5 1.12+0.82 0.42±0.49 20 1.04+0.77 0.39+0.45 100 1.03+0.69 0.40+0.45 400 1.03+0.69 0.39+0.45

Accuracy ofprototypes foundbyourPCSC

algorithmusingtwosetsofnoisydata.

Fig. 4. Processing ofeyeimages in with the PCSC algorithm with constraint. (a) The eye images. (b) The detected potential pupil boundary pixels of (a). (c) and (d)arethesame as(a) and (b), with solid anddashed circlesindicating pupil boundary locations obtained withouralgorithm and by manuallabeling,respectively. (e) and (f

arethesameas(c) and(d),exceptthattheoriginal PCSC is used.

Itis also interestingtoinvestigate the performance ofour

algorithm whenbothX(1) andX(2)arenoisy. Forthispurpose

we repeatthe experiment with both X(1) andX(2) consisting of24 points along apartial circle and 24 noise points. The

statistics of 100randomized trials are listed in Table 3; the

parameters and initialization conditions are the same as

those used in generating Table 2. While there is still some

improvement in accuracy here with increased a, this

improvement is evidently less dramatic as compared to the results in Table 2 even with verylarge a. Thereduction in

deviation and radius error seems to level offat larger a,

indicating that we can not achieve better accuracy by increasing a.

IV. EXAMPLEFORREAL IMAGE DATA

Afterall theexperiments with simulated data,weshowan

example of the application ofourmodified PCSCalgorithm on real data. Here the goal is to simultaneously find the

pupils, whicharecircles,from apairofeyeimages obtained in a screening test [12]. The procedure to extract the pupil boundary pixelsfromtheimagesisgiven in[13].

Fig. 4showsa casewhere the contrastbetween thepupils

andsurrounding regions is verylow in one eye. Fig. 4(a)is

the original eye images and Fig. 4(b) shows the extracted

boundary pixels, which serve as the data points in shell

clustering. InFig. 4(c) and4(d)thecircles(pupil boundaries)

obtainedthroughthe originalPCSC areoverlappedwith the

images in Fig. 4(a) and4(b).Thesolid circles indicatingthe

pupil boundaries found by our algorithm and the dashed circles indicating manually labeled pupil boundaries. The

boundaryof the darkerpupilisnotcorrectlylocated because

(a)

(c)

(b)

(d)

thealgorithm is affected by data pointsthat donotfallon the [12] G. W. Cibis, "Video Vision Development Assessment(VVDA): pupil*bundary.The parameters used for PCSC here are Combiningthe

Bruckner

Testwith EccentricPhotorefraction for

DynamicIdentificationofAmblyogenicFactors",Trans. Am.

m=1.5 and

qr2OOO.

In Fig. 4(e) and

4(f)

we plot the Ophthalmol.Soc., vol. 84, pp.643-685, 1994.

clustering resultswith themodifiedPCSC using a-100. We [13] T. Wang, "Eye Location and Fixation Estimation Techniquesfor

Automated Video Vision Development Assessment", Master'sThesis, can see that the new algorithm gives much more accurate Dept.Comp. Eng.Comp. Sci., University ofMissouri-Columbia,

pupilboundary locations. Columbia, MO,2002.

V. CONCLUSION

We present in this paper the use ofextra cost function terms as a method for incorporating additional desired properties ofclusterprototypes. The focus of our analysisis on the locating circular prototypes with similar radii, although the generalapproach certainlyisnotlimitedtothis case. The effect of the constraint

strength

factor

ix,

as

analyzed insections IIandIII, should be

applicable

toother cases with constraints on relations among clusters. We believe that this approach can enhance the

applicability

of clustering in various applications. Possible directions of future research include the investigation regarding other types of constraints, the effect of adjusting oa between iterations, and the effectonoverlappedclusters.

ACKNOWLEDGMENT

The authors would like to thank Dr. Gerhard Cibis of Children'sMercyHospital, KansasCity, Missouri, USA, for fruitfulcollaboration andthe use ofimage data.

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