Algorithms for thinning and rethickening binary digital pattern

Oewey

HD28

.M414

MIT LIBRARIES DUPL

39080 00932 7575

Algorithms

for

Thinning

and

Rethickening Binary

Digital

Pattern

M.V.

Nagendraprasad

Patrick

Wang

Amar

Gupta

WP#3764

1993

Productivity

From

Information

Technology

(PROFIT)

The

Productivity

From

Information

Technology

(PROFIT)

Initiative

was

established

on

October

23,

1992

by

MIT

President

Charles

Vest

and

Provost

Mark

Wrighton

"to

study

the

use

of

information

technology

in

both

the private

and

public

sectors

and

to

enhance

productivity

in

areas

ranging

from

finance

to

transportation,

and

from

manufacturing

to

telecommunications."

At

the

time

of

its

inception,

PROFIT

took

over

the

Composite

Information

Systems

Laboratory

and

Handwritten

Character

Recognition Laboratory.

These two

laboratories are

now

involved

in

research

re-lated

to

context

mediation

and

imaging

respectively.

./.as&achuseits institute

OF TECHNOLOGY

MAY

2

3 1995

LIBRARIES

In

addition,

PROFIT

has

undertaken

joint efforts

with

a

number

of

research

centers,

laboratories,

and programs

at

MIT,

and

the

results

of

these

efforts

are

documented

in

Discussion Papers published

by PROFIT

and/or

the

collaborating

MIT

entity.

Correspondence can be addressed

to:

The

"PROFIT"

Initiative

Room

E5

3-3

_10,

MIT

50 Memorial

Drive

Cambridge,

MA

02142-1247

Tel:

(617)

253-8584

Fax:

(617)

258-7579

E-Mail:

profit@mit.edu

EXECUTIVE

OVERVIEW

Financial enterprises

rely

heavily

on

paper-based

documents

to

conduct

various

operations;

this is

true

both

for

external

operations

involving

customers

and

other

financial institutions,

as

well

as

internal

operations involving various

departments.

Researchers

at

MIT

have

looked

at

the

possibility

of

taking information

directly

from

paper documents,

especially

handwritten documents,

to

computer-accessible

media.

Automated

reading involves

several steps as follows:

(i)

Scanning

of

document;

(ii)

Location

of

area

to

be

"read";

(iii)

Decomposing

the selected

area into

separate

characters;

(iv)

Adjusting

size

and

slant

of

each

character;

(v)

Recognizing each

character;

and

(vi)

Testing

whether

input

has

been

correctly read.

Based

on

several

years

of

sustained

research, the researchers

have

attained

very

high

"reading"

speed

and

accuracy,

even

in

situations

where

the quality of the

input

material

is

poor.

Patent

rights for

some

of the

new

techniques

have

been

applied

for.

Sponsor

companies

are

eligible to test

the

new

techniques

in their

respective

environments

at

no

charge.

The

work

performed

so

far is

described

in

a

number

of

published

paper

and

working

papers.

The

list

of

working

papers

is

as follows:

IFSRC

#

107-89 Optical

Image

Scanners

and

Character

Amar

Gupta

Recognition

Devices:

A

Survey

and

New

Sanjay

Hazarika

Taxonomy

Maher

Kallel

Pankaj Srivastava

IFSRC

#

123-90R

An

Improved

Structural

Technique

for PatrickS. P.

Wang

Automated

Recognition

of

Handprinted

Symbols

Amar

Gupta

Revised

October

1990

IFSRC

#

124-90 IntegrationofTraditional

Imaging, Expert

Evelyn

Roman

Systems,

and

Neural

Network

Techniques

for

Amar

Gupta

Enhanced

Recognition

of

Handwritten

John

Riordan

Information

IFSRC

#

151-91

Handwritten

Numeral

Recognition

Using

Ronjon

Nag

Dynamic Programming

Neural

Networks

on an

Alexis

Lui

Off-Line

Basis

Amar

Gupta

IFSRC

#

162-91R Algorithms

for

Thinning

and

Rethickening

M.

Nagendraprasad

PROFIT

93-03

Binary

DigitalPatterns Patricks.

Wang

Amar

Gupta

Handwritten

Characters

IFSRC

#

214-92

An

Algorithm

for

Segmenting Handwritten

Numeral

Strings

IFSRC

#

21 5-92

A New

Algorithm

forCorrecting

Slant in

Handwritten Numerals

TFSRC

#

218-92

A

System

for

Automatic

Recognition

ofTotally

Unconstrained

Handwritten Numerals

IFSRC

# 219-92

A

CoUection

of

Papers

on Handwritten

Numeral

Recognition

IFSRC

# 261-93

An

Adaptive

Modular

Neural

Network

with

Application

to

Unconstramed

Character

Recognition

IFSRC

# 287-94

An

Integrated Architecturefor

Recognition

of

PROFIT 93%4

TotaUy

Unconstrained

Handwritten Numerals

IFSRC

# 288-94

Detection

of

Courtesy

Amount

Block

on

Bank

PROFIT

93-09

Checks

IFSRC

# 289-94

A

Knowledge

Based

Segmentation Algorithm For

PROFIT

94

14

Enhanced R^ognition

of

Handwritten

Courtesy

Amounts

Amar

Gupta

Peter L.

Sparks

M.

V.

Nagendraprasad

Amar

Gupta

M.

V.

Nagendraprasad

Amar

Gupta

Vanessa

Feliberti

M.

V.

Nagendraprasad

Amar

Gupta

LikMui

Arun

Agarwal

P. S. P.

Wang

Amar

Gupta

M.

V.

Nagendraprasad

A.

Liu

Amar

Gupta

S.

Ayyadurai

Arun

Agarwal

Len M.

Granowetter

Amar

Gupta

P. S. P.

Wang

Karim

Hussein

Amar

Gupta

Arun

Agarwal

Patrick

Shen-Pei

Wang

the

papers,

and

the

software

developed

at

MIT.

The

Principal Investigator

for

the

imaging

area

is

Dr.

An>f

Gupta

Co-Directo'^-PROF,^-

IniHaavl

MIT

_^'-^^^3^

^-^^-^-^.^^^'^^^^^^^

^^^^;

fSf

-a^uTSt-r

S'c^e^^d

suggestions

are

DIGITALSIGNAL PROCESSING3,97-10211993)

Algorithms

for

Thinning

and

Rethickening

Binary

Digital

Patterns

M.

V.

Nagendraprasad,

Patrick

S. P.

Wang,* and

Amar

Gupta^

MassachusettsInstitute ofTechnology, Cambridge, Massachusetts

02139

1. INT

RODUCTION

Pattern recognition

and

image processing

applica-tions frequently deal with

raw

inputs that contain

linesofdifferentthickness. In

some

cases, this

varia-tioninthe thicknessis

an

asset,enablingquicker

rec-ognition of the featuresinthe input image. For

exam-ple, in processing aerial photographs, detection of

major

landmarks

canbeaidedbythe variationsinthe

thickness of the contours. In other cases, the

varia-tion can be a liability,

and

can cause degradation in

theaccuracy

and

the speedof recognition. For

exam-ple, in the case ofhandwrittencharacters, thedegree

of uniformity ofthe thickness of individual strokes

directlyimpactsthe probability of successful

recogni-tion, especially if neural

network

based recognition

techniques areemployed.

For the latter category of applications, uniform thickness can beattained, priorto recognition stage,

by firstthinningthe input pattern to a thickness of a

single pixel

and

then rethickening it to a constant thickness.

The

basic structure

and

the connectivity of the original pattern can be preserved irrespective of

the underlying complexity, through the stages of

thinning

and

rethickening.

Digitized

bitmap

patterns consist of

an

array of

pixels,

where

each pixel is either 1 ("on" pixel) or

("ofT' pixel). In thinning,alsocalledskeletonization, the

redundant "on"

pixels are eliminated

from

the

original pattern to yield its equivalent skeletonized pattern.

During

thesubsequentstage of rethickening,

•CollegeofComputerScience, NortheasternUniversity.

'_{Thisresearchwasfunded by}

the International Financial

Ser-vices Research Center at MIT's Sloan School of Management.

Comments andsuggestions should be addressedtothe principal

investigator for this project: Dr. Amar Gupta,

Room

E53-311,

MIT,Cambridge,

MA

02139,USA;telephone(617) 253-8906.

"on"

pixels are systematically

added

to reconstruct

an

equivalent of the original pattern. Because the

thinning process isusually considered

more

difficult

than the rethickening process, thebulkofthispaper dealswith thinningaspect.

Section 2 deals with basic notation.

The

thinning

stage is discussed in Section 3. Section 4 presents a theoreticalproofrelatedtoa

new

and

fasterthinning algorithm.

The

rethickening stageisdiscussedin

Sec-tion5. Results arepresentedinSection 6

and

conclu-sions in Section 7.

2.

BAS

IC

NOTATION

One

ofthe authors [11] has previously presented

definitions

and

notation relatedtothethinning

algo-rithms presented here. In order to facilitate a direct

comparison

of the

new

algorithm with a previousone

proposedin [11],the

same

notation isutilized in this

paper.

Definition1.

The

neighborsof apixel,p:[i,)],are

identifiedbythe eightdirections, [i

-

l,j], [i

—

l,j

+

1], [',;"

+

1], [i

+ hj

+

1], [i

+

IJ],[i

+

IJ

-

1], [ij

-1], [i

-

I,j

-

1).

The

directions are also assigned a

number

k taking values

from

0, . . ., 7 as

shown

in

Fig. 1.

Definition2.

The

contourpoints of adigital

pat-tern aredefined as thosepixels for

which

at leastone neighbor is off. InFig. 2,"a,""b," . . . , "k"

and

some

ofthepixels 'm'

and

'n'arecontourpoints while

none

ofthe"1"s isacontourpoint.

Definition3.

The

contourloopisasetofcontour

points

which

are connected into a loop.

More

for-mally,asetofcontourpointsc,,Cj,. . .,c„(for(n

>

1

)

form

a loopiffc,isaneighborofc,+,for

K

i

<

n and

c„

isaneighborofc,.

We

use L(l),. . . ,

L{m)

tolabelthe

1051-2004/93 $4.00 Copyright

©

1993byAcademicPress,Inc.

The

parallel thinning algorithm by

Wang

and

Zhang

in [11] performed thinningasfollows:

Algorithm

WZ:

initial;

g=

1;

repeat

<?

=

<?1;

if

(g==

1)

theng=

0:

elseg=

1;

for

;:

=

1

to

flido

if

73[;f] = 1

then

begin

_p

=

FIRST[;f]

;

x

= PREV[;c] ;

h[k]

=

_0;

repeat

z

=

successor

{x, _p) ;

x

=

p;

case

goi

0:

ifl<B(p)

<7

_{and (A(p)}

= 1

or

c(p)

=

₁₎

and

(

p[2] +

P[4])*p[0]*

P[6]

=0

then

begin

deletion

{Ql. p)

h[k]

=

1

end;

1:

if

1

<B(p)

<7and

(i4(p) = 1

or

c(P)

=

₁₎

and(

p(0] +

P[6])

*Pl2]* p[4]

=0

then

begin

deletion

(^1, p)

;

h[k]

=

1

end;

end

case;

P=

z;

until

loop-end-test

{k)

end

until h[l] +

• •

+ h[m]

= _0.

The

proposed

new

parallelalgorithmoperates as

fol-lows:

Algorithm

NWG:

g=

1;

repeat

h

=

0:

Q =

Q1:

if

g==

1

then

g=

0;

else

g=

1;

for each

_{array element p}

_of

_^

begin

if contour(

_p)

then

case

gof

0:

if

1

< S(p) <

7

and

(>l(p) = 1

or

c{p)

=

₁₎

and

_(p[2]

-I-P[4])*p[0]*

p[6]

=0

then

begin deletion

_{(^1, p)} ;

h[k]

=

1; A

=

_1;

_end;

1:

if

1

< 5(p) <

7

and

(>l(p)

=

i

or

c{p)

=

₁₎

and

{

p[0] +

P[Q])*p[2]*

Pl4]

=0

then

_{begin deletion}

(^1 , p) ;

h[k]

=

1;

A =

1;

end;

end

case;

end

for;

until

h

=

0;

In each iteration, the

new

algorithm uses only a subset of the operations involved in theearlier

algo-rithm.

During

asinglepass over the full array,

Algo-rithm

NWG

uses

two operations—

contour(p)

and

de-letion (Ql, p), whereas Algorithm

WZ

used initial,

loop-end-test(k), successor(x, p),

and

_{deletion (Q, p),}

FIRST[*],

PREV(/e)

and

_{contour(p) as a part of}

ini-tial, FIRST[/e]

and

_{PREV[)ij]. Further, the function}

contour(p)isusedat leastas

many

timesinFIRST[/e]

asitisinAlgorithm

NWG;

at least

one

scanisneeded

to_{detect the starting points ofallthecontour}

loopsin

the array.

The

speed ofthe

new

algorithm is

signifi-cantly higherthantheearlierone. Alsothe

new

algo-rithm can be

programmed

more

readily.

JUNCTIONAL

EQUIVALENCE

PROOF

This section provides a proofthat the

new

algo-rithm is _{operationally equivalent to the}

earlier

algo-rithm, that identical intermediate outputs occur at

the

end

ofeachiteration for bothcases,

and

that the

finaloutputs will alsobe identical.

Lemma

1. Ifthedata arrayatthebeginningofith

pass'is_{identical forboth the algorithms,}

_and

_further if

Algorithm

WZ

marks

a point

_p

for deletionduring the

pass in the data array, then Algorithm

NWG

also

marks

the

same

point

_p

for deletion.

Proof For Algorithm

WZ,

to

mark

point

p

for

de-letion, _{the following conditions}

_must

_besatisfied:

• point

_p

must

be

on one

of the

m

contour loops;

and

• thecasestatement

must

have been satisfied.

But

ifthe point

_p

isa part of

any

contourloop, then

contour(p)

must

besatisfied. Ifcontour(p)issatisfied

and

the case statement is satisfied, then Algorithm

NWG

will

mark

the point

p

for deletion.

Lemma

2. //the data arrayatthebeginningofith

passis_{identical forboth the algorithms,}

_and

_further if

Algorithm

NWG

marks

a point

_p

for deletion during

thepass in the data array, then Algorithm

WZ

also

marks

the

same

point

_p

for deletion.

Proof

For

Algorithm

NWG,

to

mark

point

p

for

deletion, _{the following conditions}

_must

_besatisfied:

'Here_{apassmeansthenumber}

of times

Q

= Ql sUtement is

executed. Ifthisalgorithm ismodifiedforexecutionona parallel

computer, thenthe step involving

Q

=Qlwillnotexist.Thenthe

numberofpasses over the arrayisthenumberoftimes

Q

=_Qlgets

Ifa point

p

is

marked

outfordeletion

by

Algorithm

NWG,

then the following conditions

must

be

satis-fied:

• contour (p)

must

be true;

and

• the case statement

must

have been satisfied.

But

ifcontourip) istrue,then

p

must

belongto

one

of

the

m

contours

and

hence

must

be

one

ofthe points

encountered while Algorithm

WZ

traverses the

con-tourloops.

When

thispointisencountered

and

ifthe

case statement is satisfied, then Algorithm

WZ

will

mark

the point

p

fordeletion.

Lemma

3. Algorithm

WZ

and

Algorithm

NWG

ex-ecute

an

identical

number

ofpasses over the dataarray.

Proof.

The

proofis obviousfromthe

way

the

two

algorithms are constructed.

During

apass. Algorithm

WZ

goes overallthecontoursinthe arrayonce

and

if

there isa deletion duringthe traversal of

any

of the

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contour points, then there is another iteration over the data array. Algorithm

NWG

goes over ail the pointsinthe array

and

deletesonly points thatreside

on

the contour. If there is

any

deletion during the

pass,thenit goesthrough another iteration.Sinceat

the

end

of

any

intermediate iteration, the

same

out-put is

produced by

bothofthem, they willterminate

after

an

identical

number

ofiterations.

Theorem.

Algorithm

NWA

and

Algorithm

WZ

are operationally equivalent, that is, given the

same

input, theyproduce the

same

output.

Proof. Follows

from

Lemmas

1, 2,

and

3 above.

5.

RETHICKENING

The

skeletonization algorithm above

was

devel-oped

as a part of a larger ongoing project

on

auto-mated

handwritten numerals.

The

numerals

inputto

a generalhandwriting recognition system could have

beenwritten using variouskindsofpenslikefeltpens,

ball pens, ormicrotippedpens.

Each

ofthese writing

instrumentscreatescharacters ofdifferentthickness.

In order toreducethese variations,theinput pattern

is scaled

down

to a single pixel thickness

and

then

rethickened to a uniform thickness.

The

algorithm

abovethins

down

the

numeral

patterntoa thickness

of

one

pixel. This"skeletal"

bitmap

ofthe

numeral

is

now

rethickened to a standard thickness, allowing

numerals

ofvaryingthickness to be rethickenedtoa

uniformthickness.

Even

withinthe

bitmap

array of a

numeral,the thickness of the

numeral

may

be

differ-entin different parts.

Thinning

and

rethickeningcan

also

smoothen

these variations to a large extent.

The

algorithmforrethickening looksata

pre-spec-ified neighborhood of each l_pixel

and

fills this

neighborhood with I's.

6.

RESULTS

The

National InstituteofStandards

and

Technol-ogy

(NIST) Handprinted

Character

Database

was

utilized as the basis for providing standardized

in-puts.

The

raw

dataina

bitmap

form,with

an

average

sizeofabout 50

X

350,were normalizedtoa 16

X

150

array.Inthelatterarray,thedata

were

typically4 to6

pixels in thickness. Figure 3

shows

a

sample

set of

numerals

before skeletonization

and

the

correspond-ingskeletonized

numerals

obtained as the output of

theproposedalgorithm.Figure4

shows

afewsamples

ofdigits

which were

skeletonized

and

then rethick-ened.

Note

that the digits inthe first column,

which

7.

CONCLUSION

MIT LIBRARIES

39080 00932 7575

Apart frompresenting themotivationfor thinning

and

rethickening,thispaper hasdiscussedalgorithms

for performingthese tasks efficiently.

The new

thin-ning algorithm has been

shown

tobefaster,but

func-tionally identicalto,one of the fastestthinning

algo-rithms in existence.

By

using the algorithms

pre-sented in this paper, raw inputs with underlying

deficiencies intermsofunequalthicknesscan be

nor-malized to yieldhigheroverall speed

and

accuracy.

REFER

ENCES

1. Adairid.M. R..Martin, G..andSuen,C. Y., Tracing center-linesofdigitalpatterns usingSquare and

Maximal-CircleAlgorithms. Proc. of VisionInterface' 90. Halifax, Can-ada, 1990, pp.67-79.

2. Arcelli,C,Cordelia, L.P.,andLevialdi,S..Fromlocalmaxima toconnectedskeletons.

IEEE

Trans. Pattern Anal. Mach. In-(e«. 3(1981), 134-143.

3. Arumugam,A., Radhakrishnan, T.,andSuen,C.Y.,

A

thin-ning algorithmbased ontheforcechargedparticleandits

ex-perimental comparison. Personalcommunication.

4. Davies,E.R.,Plummer,A. P. N.,Thinningalgorithms:

A

cri-tique and a new methodology. Pattern Recognition 14. 1

(1981),53-63.

5. Lu, H.E.,andWang,P. S.P.,

A

commenton"Afast parallel

algorithmfor thinningdigital patterns." Commun.

ACM

29

(1986),239-242.

6. Naccache,N.J.,andShinghal,R.,Aninvestigationinto

skele-tonization approach of Hilditch. Pattern Recognition 17. 3 (1984),274-284.

7. Pavalidis, T.,

A

flexible parallelthinning algorithm. In

Pro-ceedings of IEEE ComputerSociety Conference on Pattern Recognitionand Image Processing, 1981, pp. 162-167.

8. Sossa,J.H.,Animprovedparallelalgorithmforthinning digi-talpatterns.Pattern RecognitionLett.(1989),77-80. 9 Stefanelli,R.,andRosenfeld,A.,Someparallelthinning

algo-rithmsfor digital pictures.J.

ACM

18(1971),255-264.

10. Tamura, H.,

A

comparisonofline thinning algorithmsfrom

digitalgeometryviewpoint.Inthe Proceedingsof4th

Interna-tionalConference on Pattern Recognition, 1978, pp.715-719.

11. Wang,P. S.P.,and Zhang.Y.Y.,

A

fastandflexiblethinning algorithm.

IEEE

Trans. Comput.38.5 (1989).

M.V.

NAGENDRAPRASAD

obuinedhisB.Tech fromIndian

Institute ofTechnology, Madras, and M.S. degreesfrom Indian

InstituteofTechnology, Madras, and Massachusetts Institute of

Technology.Heiscurrentlyastudentatthe Universityof

Massa-chusetts,Amherst, working towardsadoctorateinDistributed

Ar-tificial Intelligence. His interesU include artificial intelligence,

connectionist systems,machinelearning,andcognitivescience.

PATRICK

S. P.

WANG

isaProfessorofComputerScienceat

NortheasternUniversity. He received his Ph.D. degree in

com-puter sciencefrom OregonState University,hisM.S.I.C.S. degree

from Georgia InstitueofTechnology, his M.S.E.E. degree from

National Taiwan (.'niversity,and his B.S.E.E. degree from

Na-tionalChiaoTungUniversity.

Before joining Northeastern University, he had been with

WANG

Laboratories,

GTE

Laboratories, BostonUniversity,and

the UniversityofOregonassoftware engineeringspecialist,senior

memberoftechnicalsUflF,adjunct AssociateProfessor,and

Assis-tantProfessor, respectively.From 1989to 1990,hewasavisiting scientistatthe

MIT

Artificial IntelligenceLaboratory.Heisnow

alsoteaching part-timeatHarvard University,andisa research consultantatthe

MIT

Sloan SchoolofManagement.

Professor

Wang

has published sixbooks andover 70technical

papers in imaging technology, pattern recognition,and artificial intelligence.Oneof hismostrecent inventions concerning pattern recognitionsystemshasbeengrantedapatentbythe U.S. Patent Bureau. He has attendedandorganized numerousinternational conferencesforIEEE,

ACM,

ICS,AAAI, CLCS,andthe

Interna-tional AssociationforPattern Recognition. Heisalso areviewer

forgrants proposalsofthe

NSF

IntelligentSystemsDivision,and

for several computerjournals including

lEEEPAMI.

Computer

Vision, Graphics,and ImageProcessing,InformationSciences,and

Information ProcessingLetters,and aneditor-in-chargeofthe In-ternationalJournal of Pattern Recognition and Artificial Intelli-gence.

Prof.

Wang

isaseniormemberofIEEE, andalso amemberof

ACM,

AAAI. andthe Pattern Recognition Society. Dr.

Wang

is

chairman oflAPR-SSPR(1992-1994) andhasbeena governing

boardmemberandthe newslettereditoroftheChinese Language

ComputerSociety(CLCS), Washington.

DC

(1983-1989). His

re-searchinterests include software engineering,programming lan-guages, pattern recognition,artificialintelligence,image

technol-ogy,andautomation.

Professor

Wang

hasalsowrittenseveral articlesonthe operasof Verdi, Puccini, and Wagner, and on Beethoven's, Mozart's and

Tsaikovsky'ssymphonies.

AMAR

GUPTA

(SM'85) is Senior Research Scientist at the

SloanSchool ofManagement.MassachusettsInstitute of

Technol-ogy,Cambridge.Heholdsabachelor's degreein electrical

engineer-ingfromthe Indian InstituteofTechnology, Kanpur, a master's degreein managementfromtheMassachusetUInstitute of

Tech-nology, anda doctorate incomputertechnologyfrom the Indian

Institute ofTechnology,

New

Delhi.Heconductedthe researchfor hisdoctoratedissertation atthreeprestigious universities in India,

England, andtheUnitedStates.

Dr.Guptahasbeeninvolvedinresearch,management,and

pub-lishing activities since 1974. Since joining

MIT

in 1979, he has

been active in the areas ofmultiprocessor architectures,

perfor-mancemeasurement,distributedhomogeneousandheterogeneous

data bases, expert systems, personal computers,and transfer of information technology. Heserves as aconsultantto anumberof corporations, government agencies, and international bodies on various aspects ofcomputertechnology,andisanactivetechnical advisorto severalinternational organizationsandcommittees.

Dr GuptaistheChairmanofthe TechnicalCommitteefor

Mi-croprocessor ApplicationsoftheIEEEIndustrialElectronics Soci-ety and assistant chairman for several annual

lECON

confer-ences.Hewasalsoinvolved with theVery Large Data Base Confer-ence

(VLDB

'87),heldinEnglandin 1987.Hereceived theRotary Fellowship for International Understanding in 1979 and the

BrooksPrize(Honorable Mention)in 1980).

Hehas writtenmorethan 50technicalarticlesandpapers,and producedseven books.

Dr.Guptaisa citizenof theUnitedStates.