An Evaluat ion of Digital

An Evaluat ion of Digital Image Correlat ion Cr iter iafor St rain Mapping Applicat ions

W. TongDepartment of Mechanical Engineering, Yale University, New Haven, CT 06520-8284, USA

A BST RA CT : The performance of four digital image correlation criteriawidely used in strain mappingapplications has been critically examined using three sets of digital images with various whole-�elddeformation characteristics. The deformed images in these image sets are digitally modi�ed tosimulate the less-than-ideal image acquisition conditions in an actual experiment, such as variablebrightness, contrast, uneven local lighting and blurring. The relative robustness, computational costand reliability of each criterion are assessed for precision strain mapping applications. Recommen-dations are given for selecting a proper image correlation criterion to ef�ciently extract reliabledeformation data from a given set of digital images.

KEY W O RDS: digital image correlation, image processing, in-plane strain measurement

N O T A T IO N

B Th e en t ire region of a sam plesurface com m on in both refer-en ce an d curren t im ages

C, CCC, CSSD Th e gen eric, cross-correlat ion ,an d sum -squared differen ceim age correlat ion coef�cien ts

C1, C2, C3, C4 Four speci�c sum -squared dif-feren ce (SSD) im age correlat ioncoef�cien ts

E or E11, E22, E12 Th e 2-D Lagran gian strain ten -sor an d its th ree com pon en ts

F Th e deform at ion gradien t ten sor

G(X, t0) Th e brigh tn ess or greyscale valueof each pixel of th e referen ceim age

g(x , t) Th e brigh tn ess or greyscale valueof each pixel of th e curren t im age

N Th e total n um ber of valid pixelsin an im age subset S

P or P1, P2,� ,P6, P7 A set of param eters th at de�n e apossible local deform at ionm appin g fun ct ion for an im agesubset S (usually cen tred at X0)

P* or P�1, P�2,� ,P�6, P�7 Th e param eters th at de�n e th eopt im ised local deform at ionm appin g fun ct ion of an im agesubset S

S an d s An im age subset of th e referen cean d curren t im ages respect ively

t0 Th e t im e wh en th e referen ceim age G(X, t0) is taken

t � t0 + Dt Th e t im e wh en th e curren tim age g(x , t) is taken after a t im ein crem en tDt

u � [u1, u2] Th e 2-D disp lacem en t vector(an d its h orizon tal an d vert icald isplacem en t com pon en ts)

up(X; P) Th e local d isplacem en t�eld ofan im age subset given by a set ofdeform at ion m appin g param e-ters P

w�~G; ~g� Th e p ixel-level weigh t fun ct ionde�n ed in term s of both refer-en ce an d curren t im ages

X � [X 1, X2] 2-D Cartesian coordin ates of am aterial poin t on th e referen ceim age G(X, t0)

X0 Th e cen tre or an ch orin g poin tof an im age subset S

x � [x1, x2] 2-D Cartesian coordin ates of am aterial poin t on th e curren tim age g(x , t).

Int roduct ion

Various correlat ion criteria h ave been used in th e

literatu re for strain m appin g m easurem en ts based on

digital im ages [1�17]. Assessm en t of each criterion is

often m ade in dividually based on th e digital im ages

acqu ired from n early ideal test con dit ion s an d freefrom m ajor im age degradat ion . In an actual experi-

m en tal en viron m en t, d igital im ages are som etim es

acqu ired with sign i�can t exposure an d/or ligh t in g

variat ion s over th e loadin g h istory. Overexposure or

un derexposure, un stable or n on -un iform ligh t in g,

un even con trast, an d blurrin g due to sam ple surface

deform at ion an d m otion are am on g th e m ost com -

m on ly en coun tered situat ion s. Th ere is a n eed toform ulate a robust im age correlat ion criterion an d

related im age processin g strategy for th ese im ages soth e result in g strain m appin g errors can be m in im ised

or even elim in ated.Th is paper presen ts a study on th e robustn ess,

com putat ion al cost, an d reliability of com m on ly

used im age correlat ion criteria by com parin g th eir

strain m appin g results of th ree sets of n um erically

m odi�ed digital im ages with various degrees of

average brigh tn ess, con trast an d sh arpn ess. In th e

followin g, a brief sum m ary of th e im age correlat ion

criteria evaluated in th is study is�rst given . Th e

procedure for select in g an d gen erat in g th e test im age

sets used in th e study is th en described. Th e strain

m appin g results obtain ed from correlat ion an alysesof th e test im age sets usin g th e various correlat ion

criteria are presen ted. Fin ally, th ese results are dis-cussed in term s of th e strain m appin g errors an d th e

relat ive perform an ce versus cost of th ese im age cor-relat ion criteria.

Criteria for Digital Image Correlat ion-Based St rain Mapping

For strain m appin g an d pattern recogn it ion applica-

t ion s based on digital im age correlat ion , several cri-

teria for m atch in g im age pairs h ave been reported in

th e literatu re [1�21]. Th ey will be sum m arised in th e

followin g usin g th e n otat ion s adapted by Ton g an d Li

[11] an d Li [12]. Con sider an object with a�at surface

B th at lies on X1�X2 plan e in th e initial or undeformedcon�gurat ion de�n ed by th e two-d im en sion al Car-

tesian coord in ates (X1, X2) at a certain start in g t im et0. Th e object is assum ed to un dergo on ly p lan ar

rigid-body m otion an d deform at ion so th e surface Brem ain s�at an d stays on X1�X2 plan e. Let a m aterial

poin t X � [X 1, X2] on th e �at surface of th e object

displace to th e locat ion x� [x1, x2] at th e t im e t�

t0 + Dt. Th e deform at ion equat ion s in th e Lagran gian

form ulat ion are de�n ed as:

x � x�X ; t�� X � u�X ; t�; t � t 0; (1)

wh ere u� [u1, u2] is th e disp lacem en t vector in th e

X1�X2 plan e. Th e deform ation gradien t ten sor F of an

in�n itesim al m aterial vector at X is given as:

F �@x@X

� I �@u@X

; X 2 B an d t� t 0: (2)

Th e deform at ion gradien t ten sor F is used exten -

sively in form ulat ion s of elast icity an d plast icity

deform ation th eories of m aterials. For exam ple,

th e Lagran gian strain ten sor E is com puted via E�

(FTF ) I)/2.Durin g an experim en t, th e object surface B at th e

t im es t0 an d t are recorded respect ively as referencean d current im ages. In prin cip le, th e on e-to-on e

correspon den ce between th e ph ysical an d im agecoordin ates of th e object h as to be establish ed via a

cam era/ len s calibrat ion procedure. For sim plicity, it

is assum ed th at such a correspon den ce is iden t ity so

th e spat ial coordin ates of th e object in term s of p ixels

of digital im ages will be used in th e rem ain der of th e

paper. Let G(X, t0) an d g(x , t) represen t th e brigh tn ess

distribut ion fun ct ion s of th ese two im ages in th e

initia l (referen ce) an d deformed (curren t) con�gura-

t ion s respect ively. Th e prin cip le of im age correlat ion

for deform at ion m easurem en t is to�n d a local

displacem en t m appin g u th at can m atch th e twoin ten sity d istribut ion fun ct ion s over a sm all but

�n ite area S in th e in it ial con�gurat ion (S is called asubset of B h ere) or over th e correspon din g area s

in th e curren t con�gurat ion . Th e displacem en tm appin g u over S is often param eterised by a

vector P:

u�X ; t�� u p�X; P�; P � P�X 0; t�� P1; P2; P3; . . .�;

(3)

wh ere X0 is a referen ce m aterial poin t usually cen tredwith in th e subset S. Th e quality of m atch in g between

th e two im ages over th e subset region for an assum ed

param eter set P can be m easured by a correlat ion

coef�cien t:

C�X ; t ; P�� Cf G�X ; t0�; g�X � u p ; t�g; X 2 S; (4)

wh ere both th e speci�c correlat ion criterion an d th eassum ed displacem en t vector�eld up(X; P) are def-

in ed over th e subset S. Both cross-correlat ion (CC)criteria an d sum -squared differen ce (SSD) correlat ion

criteria h ave been proposed in th e n um erical an dim age an alysis literature [18�21]. Con sequen t ly, th e

problem of deform at ion m appin g by im age correla-

t ion at an arbit rary poin t X0 located with in th e subset

S can be stated as to�n d a set of local m appin g

param eters P*� [P1*, P2*, P3*,� ] th at m axim ises

a CC coef�cien t or m in im ises an SSD correlat ion

coef�cien t:

C�P�� m ax

P2PD

CCC�P�f g

or

C�P�� m in

P2PD

CSSD�P�f g; (5)

Evaluation of Digital Image Correlation Criteria : W. Tong

wh ere PD con tain s all rat ion al param eter sets of an

assum ed 2-D in -plan e m appin g fun ct ion [e.g.

def(F) > 0 is usually required]. To lim it th e scope ofth is study, on ly a h om ogen ous deform at ion or af�n e

m appin g fun ct ion will be con sidered, i.e. P� [P1, P2,P3, P4, P5, P6], with P1 an d P2 as th e two disp lacem en t

com pon en ts an d P2�P6 as th e four d isp lacem en tgradien t com pon en ts. As CC coef�cien ts are actually

related to th e SSD correlat ion coef�cien ts [18�24],

on ly th e lat ter on es will be evaluated in th is study. A

gen eric SSD correlat ion coef�cien t can be de�n ed as:

CSSD�

Z

Sw ~G; ~g� �

~g�x ; t�� ~G�X ; t0�

h i 2dS; (6)

wh ere ~G�X ; t0� an d ~g�x ; t� are certain n orm alised

form s of brigh tn ess distribut ion fun ct ion s of th e

im age pair G(X, t0) an d g(x , t) respect ively, an dw�~G; ~g� is a p ixel-level weigh t fun ct ion . Four com -

m on ly used SSD criteria are sum m arised in th e fol-

lowin g [assum in g w�~G; ~g�� 1 an d th e total n um ber

of pixels with in th e subset is N]:

C1 �XN

i�1

gi � Gi� �2

�G2� 1 �

�g2

�G2�

2P N

i�1 giGi

�G2; (7a)

C2 �XN

i�1

gi � P7 � Gi� �2

�G2� 1 �

�g� P7� �2

�G2

�2

P Ni�1 gi � P7� �Gi

�G2; (7b)

C3 �XN

i�1

gi

�g�

Gi

�G

� �2

� 2 1�

P Ni�1 giGi

�g�G

!

; (7c)

C4 �XN

i�1

gi � g0

Dg�

Gi � G0

DG

� �2

� 2 1�

P Ni�1 gi � g0� �Gi � G0� �

DgDG

" #

; (7d)

wh ere P7 is an extra param eter accoun t in g for vari-

at ion of th e local average brigh tn ess of th e subset

between th e two im ages [8], an d

�g �

��

XN

i�1

g2i

vuut ; g0 �

XN

i�1

gi

N; Dg �

��

XN

i�1

gi � g0� �2

vuut ; (8a)

�G �

��

XN

i�1

G2i

vuut ; G0 �

XN

i�1

Gi

N; DG �

��

XN

i�1

Gi � G0� �2

vuut :

(8b)

As sh own above, th e CC coef�cien ts are in deed

related to th e SSD coef�cien ts (with addit ion al

assum ption s of�g � �G an d�g � P7 ��G for C1 an d C2

respect ively). In th is study, th e m in im isat ion prob-

lem of th e SSD correlat ion coef�cien ts with respect toeith er 6 or 7 local m appin g param eters P was solved

by th e n on lin ear Newton�Raph son m eth od for least-squares regression [18]. Bicubic splin e in terpolat ion

of th e curren t im age was used to com pute th e�rst-

order spat ial derivat ives of th e im age brigh tn ess an d

subsequen t ly to obtain th e Jacobian (gradien t) an d

approxim ate Hessian m atrices for each correlat ion

coef�cien t [8, 12]. Th e con vergen ce con dit ion s in th e

Newton�Raph son iterat ive rou t in e were set to en sure

th at variat ion s in d isp lacem en ts P1 an d P2 were equal

to or less th an 10)4 pixel an d variat ion s in d isp lace-

m en t gradien ts P2�P6 were equal to or less th an

0.5 �10)6 (i.e. 0.5lstrain s). A subset of 41�41pixels an d grids with 10�10 pixels were used in all

an alyses reported in th e followin g.

The Test Image Sets

Th ree sets of test im ages (640�480 pixels, 8-bit

greyscale) were used to evaluate th e perform an ce of

various SSD correlat ion criteria. To sim ulate th e dif-

feren t ligh t in g con dit ion s, th e origin al im ages arem odi�ed usin g a com m ercial d igital im age-processin g

program (Pain t Sh op Pro, Eden Prairie, MN, USA). Th e

�rst set of test im ages con sists of a st ill d igital im age of

an alum in ium plate (about 25 m m wide) as sh own

in Figure 1A an d its eigh t variat ion s due to ch an ges in

ligh tn in g con dit ion s, in cludin g 20% in crease in

brigh tn ess, 30% decrease in con trast, equalised h isto-

gram , blu rrin g (Gaussian�lterin g), etc. Th e st ill d igi-

tal im ages are used to assess th e level of possible

m in im um errors in troduced by th e variable ligh t in g

con dit ion s for sm all-strain m appin g applicat ion s. Th e

im age pairs in th e secon d set of test im ages areselected from a un iaxial ten sile test of a com pact dog

bon e-sh aped steel sh eet sam ple (Figure 1B). Th ewidth of th e ten sile sam ple is about 4.6 m m . Th e

im age of th e deform ed steel sh eet sam ple is m odi�eddigitally to accoun t for seven differen t ligh t in g con -

dit ion s. Th e secon d set of test im ages is used to

evaluate th e effects of variable ligh t in g con dit ion s on

th e local strain s in a m oderately an d n om in ally h o-

m ogen ously deform ed�eld. Fin ally, a pair of digital

im ages from a ten sile test of a tapered t itan ium sh eet

sam ple [14] is used as th e basis for th e th ird set of test

im ages. Th e origin al width of th e sam ple is about

4 m m an d th e im age resolut ion is 9.7 m icron s per

pixel (Figure 1C). Th e im age of th e deform ed

W. Tong : Evaluation of Digital Image Correlation Criteria

t itan ium sam ple is again m odi�ed digitally to accoun tfor seven differen t ligh t in g con dit ion s. Th e th ird set

of im ages is used to evaluate th e effects of variableligh t in g con dit ion s on th e overall strain d istribu t ion s

in a n on -h om ogen ous deform at ion�eld. Th e strain

level is also m uch h igh er th an th at in th e secon d

im age set. For a com plete list of th e digitally m odi�ed

test im ages, see tables given in Appen dix A.

Evaluat ion Results

The still image set

Th e eigh t im age pairs (TST0 versus TST1-8) were

evaluated accordin g to th e four SSD correlat ion cri-teria discussed above. Im age correlat ion an alysis

resu lts of th e st ill im age set are given in Tables 1�3 interm s of robustn ess, speed an d reliability respect-

ively. Table 1 sh ows th e pass/ fail resu lts of d igitalim age correlat ion calcu lat ion s (h ere f̀ail' m ean s th at

th e m axim um iterat ion step of 80 was reach ed

(A)

(B)

(C)

Figure 1: Th e referen ce im ages used in th e curren t evaluat ion :(A) a stat ion ary alum in um plate; (B) a steel sh eet sam ple un derun iaxial ten sion ; (C) a tapered titan ium sh eet sam ple un derun iaxial ten sion . Th e sam ple surfaces were decorated with sprayblack pain t speckles to en h an ce th e im age con trast for im agecorrelat ion processin g

Table 1: Sum m ary of th e robustn ess an d speed (un it : m in utes)for th e�rst im age set

L̀ighting' conditions C1 C2 C3 C4

+20% intensity (I) 1.5 1.1 3.2 2.6

)30% contrast (C) 2.5 2.4 3.5 2.8

)30% C and blurred Fail* Fail* Fail* 6.1

Equalised histogram 2.5 2.3 2.0 4.1

Equalised and blurred 1.8 1.7 2.0 5.2

+20% I and )30% C 3.3 2.4 Fail* 2.9

+20% I, )30% C and blurred Fail* Fail* Fail* 5.3

+20% I, )30% C and both 1.9 1.8 Fail* 4.9

*Convergence conditions could not be reached at the end of 80 iteration steps.

Table 2: Global overall strain s m easured for th e�rst im age set(th e un it for m axim um global strain is m icro-st rain s, i.e. 10)6)


+20% intensity (I) 81 <1 6.5 <1

)30% contrast (C) 17 7.1 7.3 1.1

)30% C and blurred � � � 3�6

Equalised histogram 90 15 5 1.9

Equalised and blurred 104 13 4.5 3.6

+20% I and )30% C 87 8.7 � <1

+20% I, )30% C and blurred � � � 3�6

+20% I, )30% C and both 93 29 � 10�16

Table 3: Local strain m appin g results for th e�rst im age set(th e un it for th e average an d stan dard deviat ion in strain s ism icro-st rain s, i.e. 10)6)


+20% intensity (I) 488 � 2140 <1 � 3 22 � 172 <1 � 3

)30% contrast (C) 70 � 403 11 � 144 22 � 181 <1 � 8

)30% C and blurred � � � 32 � 147

Equalised histogram 375 � 1485 49 � 329 34 � 161 3 � 40

Equalised and blurred 582 � 1801 85 � 402 27 � 166 23 � 158

+20% I and )30% C 405 � 1875 13 � 142 � <1 � 9

+20% I, )30% Cand blurred

� � � 37 � 149

+20% I, )30% Cand both

667 � 2386 157 � 812 � 96 � 743


with out m eet in g th e con vergen ce con dit ion s in th e

n on lin ear Newton�Raph son solut ion procedure des-

cribed above). Th e correlat ion criterion C4 was m ostrobust as n o failu re to con verge was en coun tered

wh ile th e criteria C1 an d C2 were th e secon d best(failed to process on ly two cases wh en both th e im age

con trast was reduced an d th e im age blu rrin g wasin troduced). Th e correlat ion criterion C3 was th e

least robust for th is set of im ages (h alf of th e eigh t

im age pairs failed to m eet th e con vergen ce con di-

t ion s). Th e total com putat ion al t im es th at took to

com plete th e successfu l im age correlat ion calcu la-

t ion s on a Pen t ium desktop com puter are listed in

Table 1 as well. Th e criteria C1 an d C2 were am on g

th e fastest on es wh ile th e correlat ion criterion C4 was

th e m ost com putat ion ally in ten sive (about h alf or

less th e speed of th e fast on es on average).

Th e reliability an d accuracy of th e strain m appin gresu lts by digital im age correlat ion were assessed in

term s of both th e global overall strain levels an d th elocal strain variat ion s (th e average an d th e stan dard

deviat ion of local strain s). Tables 2 an d 3 list th em axim um values am on g th e th ree in -plan e strain

com pon en ts E11, E22 an d E12. Clearly, th e criterion C4gave th e m ost reliable an d accurate resu lts (i.e. closest

to zero strain s) for all cases com pared. Un iform

ch an ges in in ten sity an d/or con trast as well as

equalisat ion in h istogram in duced lit t le errors in

both global an d local strain s usin g th is criterion .

Blurrin g of im ages in duced about 30lstrain s an d

150 lstrain s in th e average an d stan dard deviat ion

values of local strain levels. Lit t le global strain s (3�6

l strain s at m ost) were detected. However, sign i�can t

errors in local strain s existed for th e im ages with n on -un iform ligh t in g ch an ges (th e last case in wh ich

th ree differen t region s in th e im age were m odi�ed

separately (see Appen dix A for details). As expected,

th e criterion C2 was as accurate as th e criterion C4 in

dealin g with un iform sh ift in th e im age brigh tn ess.

Th ey were far less accurate in processin g im ages

un dergoin g a con trast sh ift or im age blu rrin g. Sur-

prisin gly, it also gave poor resu lts for im ages with th e

h istogram s equalised. However, th e criterion C3 gave

better resu lts th an th e criterion C2 for th e h istogram -

equalised im ages (th ey are less robust th ough ). For allth e cases th at could be processed successfu lly usin g

th e criterion C1, th e results were th e worst in term s ofboth global an d local strain s.

The homogenous deformation image set

Th e robustn ess an d speed perform an ce of th e four

correlation criteria are sum m arised in Table 4 for th e

secon d test im age set. Both criteria C2 an d C4 were

equally robust in successfu lly processin g all of eigh t

pairs of im ages wh ile th e criterion C2 was about th ree

t im es faster th an th e criterion C4. Usin g th e origi-

n al im age pair, th e th ree in -p lan e global straincom pon en ts (E11, E22 an d E12) over th e en t ire

region of th e strain m aps h ave been com putedas 0.04224�0.000628, )0.02235�0.000339 an d

0.000431�0.000331 respectively. For all th e cases

th at cou ld be processed successfu lly, lit t le differen ce

was detected in th e global overall strain s. In m an y

strain -m appin g applicat ion s, details in th e local strain

distribu t ion s are of in terest [5, 7, 12, 16]. Th e resu lts

obtain ed from usin g th e origin al im age pair based on

th e criterion C4 m ay be used as th e referen ce strain

m aps wh ile all oth er results are treated as test strain

m aps. Th e differen ces between th e test strain m ap-

pin g results an d th e referen ce on es can th en becom puted. Th e stan dard deviat ion s of th e local poin t-

to-poin t differen ce in th ree in -plan e strain com po-n en ts can th en be used as a m easure of strain m appin g

errors (Table 5). Th e n um bers in th e brackets inTable 5 are th e poin t-to-poin t CC coef�cien ts [18]

between th e referen ce an d test strain m aps.

As expected, th e criterion C4 gave th e best results for

each case in term s of reliability. A sim ple sh ift in th e

im age brigh tn ess or con trast in duced lit t le error usin g

th e criterion C4. A soften ed im age would h ave a local

strain error of <100lstrain s. Gam m a correct ion s could

cause th e errors up to about 100�200 lstrain s or so.

Equalised h istogram an d blurrin g would add th e errors

up to about 250�500 lstrain s an d gave th e sm allest CC

coef�cien t (m uch less th an 0.9). Th e strain m appin g

resu lts usin g th e origin al pair of im ages an d oth er th reecriteria sh ow lit t le errors (in dicat in g th e h igh quality

of th e origin al im ages; see case n o.1 in Table 5). Again ,

th e criterion C2 was very effect ive in processin g im ages

with th e sim ple sh ift in im age brigh tn ess an d som e-

wh at effect ive with im ages subjected to gam m a cor-

rect ion s. It was in effect ive with im ages un dergoin g

con trast ch an ges, soften in g/blu rrin g, an d h istogram

equalisat ion . Th e criterion C1 was least reliable as th e

Table 4: Sum m ary of th e robustn ess an d speed (un it : m in utes)for th e secon d im age set


The original deformed image 3.1 3.1 3.4 11.5

+20% brightness Fail* 3.1 6.5 11.5

)30% contrast and softened 7.0 6.7 8.6 10.8

Equalised histogram Fail* 9.3 7.8 14.4

Equalised and blurred Fail* 7.7 6.4 14.1

+20% I and )30% C (softened) Fail* 6.7 Fail* 10.8

Gamma correction (0.65) Fail* 3.4 5.6 13.0

Gamma correction (1.50) Fail* 3.4 6.3 12.0



CC coef�cien ts between th e test an d referen ce strain

m aps for cases oth er th an n o.1 were all far less th an 0.9.

The non-homogenous deformation image set

Th e origin al im age pair in th is im age set was selected

from th e load steps n os. 22 an d 38 of a con secu t ive

im age set th at was an alysed in an experim en tal study

on diffuse n eckin g in sh eet m etals [14]. As sh own in

Figure 2, axial strain d istribut ion s alon g th e cen tre-

lin e of th e tapered ten sile sh eet sam ple h ave been

foun d to be h igh ly n on -h om ogen ous with in creasin gload steps. Th e robustn ess an d speed perform an ce

of th e four correlat ion criteria are sum m arised in

Table 6 for th e th ird test im age set. Th e th ree in -

plan e global true strain com pon en ts (E11, E22 an d E12)

over th e en t ire region of th e strain m aps h ave been

com puted usin g th e origin al im age pair an d th e cri-

terion C4 to be 0.2438�0.06049,)0.1197�0.03212

an d 0.001950�0.007358 respect ively. Large stan d-

ard deviat ion s in strain s are related to th e h igh ly

n on -h om ogen ous deform at ion�eld itself (Figure 3).

Th e correlat ion criterion C4 was th e m ost robust in

successfu lly processin g all eigh t pairs of im ages wh ileth e criterion C2 was th e secon d best (failed to process

200 300 400 500

X1 (pixels)

�1 0 1 2Horizontal distance (mm)

0

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.8

Er1

42

41

40

39

38

37

36

34

32

30

25

20

15

10

28

Figure 2: Axial st rain d istribu t ion pro�les at selected loadsteps for th e tapered titan ium sh eet sam ple un der un iaxialten sion [14]. Th e n um bers overlaid th e curves are load stepswh en a digital im age was taken durin g th e test. Th e im age pairsused h ere (TIP0 an d TIP1) correspon d to th e on e at load stepsn os. 22 an d 38 respectively

Table 5: Stan dard deviat ion s of localstrain d ifferen ces dE11, dE22, dE12

between test an d referen ce strain m apsfor th e secon d im age set (th e un it forlocal strain differen ces is m icro-strain s,i.e. 10)6 an d th e n um ber in a bracket isth e cross-correlat ion coef�cien ts betweenth e strain m aps)


Original deformed image 23 (0.93) 32 (1.00) 13 (1.00) Reference maps

19 (0.88) 35 (0.99) 35 (0.99)

10 (0.95) 11 (1.00) 7 (1.00)

+20% brightness � 32 (1.00) 1054 (0.67) <1 (1.000)

35 (0.99) 1187 (0.54) <1 (1.000)

11 (1.00) 368 (0.59) <1 (1.000)

)30% contrast andsoftened

1628 (0.57) 989 (0.64) 1184 (0.64) 85 (0.99)

1687 (0.44) 1191 (0.56) 1357 (0.53) 85 (0.97)

689 (0.42) 344 (0.59) 409 (0.55) 42 (0.99)

Equalised histogram � 3846 (0.01) 3616 ()0.13) 266 (0.93)

3934 ()0.28) 3138 ()0.14) 256 (0.73)

1188 (0.36) 1492 (0.16) 140 (0.90)

Equalised and blurred � 4833 (0.02) 4165 ()0.13) 513 (0.80)

4663 ()0.32) 3752 ()0.16) 429 (0.50)

1448 (0.34) 1745 (0.16) 236 (0.81)

+20% I and )30% C(then softened)

� 982 (0.65) � 91 (0.99)

1186 (0.56) 85 (0.97)

339 (0.60) 47 (0.99)

Gamma correction (0.65) � 251 (0.92) 1628 (0.03) 214 (0.94)

244 (0.88) 1633 ()0.05) 149 (0.94)

143 (0.91) 648 (0.49) 116 (0.94)

Gamma correction (1.50) � 378 (0.90) 1209 (0.65) 139 (0.98)

333 (0.80) 1256 (0.52) 102 (0.95)

129 (0.92) 429 (0.54) 75 (0.97)

Table 6: Sum m ary of th e robustn ess an d speed (un it : m in utes)for th e th ird im age set


The original deformed image 10.1 10.2 11.2 27.5

+20% brightness Fail* 9.8 9.8 32.6

+20% brightness and blurred 12.9 12.2 Fail* 22.3

Equalised histogram Fail* 10.9 16.1 35.7

Equalised and blurred Fail* Fail* Fail* 29.9

)30% contrast 9.2 9.3 9.5 27.7

)30% contrast and blurred Fail* Fail* Fail* 20.7

+20% I and )30% C 10.9 10.6 Fail* 32.6



on ly two of th e total eigh t cases). Th e criterion C1

was th e worst as it cou ld on ly process four cases of

th em . Again , th e criterion C2 was about th ree t im es

faster th an th e criterion C4. For all th e cases th at

could be processed successfu lly, lit t le d ifferen ce in

th e global strain s were detected as well.Sim ilarly, th e results obtain ed from th e criterion C4

an d th e origin al im age pair m ay be used as th e refer-

en ce strain m aps wh ile all oth er results are regarded as

test strain m aps. Th e stan dard deviat ion s of th e local

poin t-to-poin t d ifferen ce between th e test strainm appin g resu lts an d th e referen ce on es are sh own

in Table 7 for all th ree in -plan e strain com pon en ts.Th e n um bers in th e brackets in Table 7 are th e CC

coef�cien ts between th e referen ce an d test strainm aps. An oth er way of assessin g th e reliability of th e

strain m appin g results is to use th e n orm alised errors in

local strain s. Th e n orm alised local strain errors are

com puted by dividin g th e local strain differen ce

(between th e referen ce an d th e test m aps) by th e local

strain s of th e referen ce m aps. As an exam ple sh own in

Figure 4 for th e case n o. 5 usin g th e criterion C4, th e

n orm alised error in th e tran sverse strain com pon en t is

at m ost a few per cen t an d is about on ly 0.1% at th e

cen tre of th e tapered ten sile sam ple. Th e criterion C2

gave th e local strain m appin g reliability sim ilar to th atof th e criterion C4, except th e case n o. 6. Th e criterion

C1 h ad th e worst reliability for th is set of test im ages,especially for cases n os. 3 an d 8.

Discussions

Main tain in g steady an d un iform ligh t in g an d expo-

sure con dit ion s durin g im age acqu isit ion in a

m ech an ical test is crit ical in ach ievin g th e best strain

0.1290. 129

0 .12

9

0 .12

9

0 . 1290.129

0.152

0 .1 5

20.

152

0.15

2

0 .15

2

0.152

.176

0. 17 6

0.17

6

0.1 760.176

0.17

0.199

0.19

9

0.199

0.19

9

0.1990.199

0. 22 30.2 23

0.223

0.2230.22 3

0.223

0.24

7

0. 2470.247

0.247

0.247

0.2 470.24 7

0.27

0

0.270

0.270

0 .2 7

0

0.270

0.270

0 .2 9 4

0.294

0.2 9

4

0.294

Figure 3: Th e axial true strain distribu t ion of th e taperedt itan ium sam ple between th e origin al im age pair of TIP0 an dTIP1 (criterion C4)

Table 7: Stan dard deviat ion s of localstrain differen ces dE11, dE22, dE12

between test an d referen ce strain m apsfor th e th ird im age set (th e un it for localstrain differen ces is m icro-st rain s, i.e.10)6 an d th e n um ber in a bracket is th ecross-correlat ion coef�cien ts between th estrain m aps)


The original deformed image 972 (1.00) 272 (1.00) 500 (1.00) Reference maps

539 (1.00) 139 (1.00) 285 (1.00)

403 (1.00) 70 (1.00) 232 (1.00)

+20% brightness � 2109 (1.00) 3882 (1.00) 2280 (1.00)

1062 (1.00) 1852 (1.00) 1158 (1.00)

790 (0.99) 1575 (0.98) 751 (1.00)

+20% brightness and blurred 16880 (0.96) 2310 (1.00) � 2530 (1.00)

8940 (0.96) 1392 (1.00) 1481 (1.00)

6249 (0.80) 959 (0.99) 867 (0.99)

Equalised histogram � 3795 (1.00) 6192 (1.00) 3684 (1.00)

1906 (1.00) 3087 (1.00) 1821 (1.00)

1309 (0.98) 2382 (0.95) 1077 (0.99)

Equalised and blurred � � � 4396 (1.00)

2322 (1.00)

1321 (0.98)

)30% contrast 1523 (1.00) 1517 (1.00) 1312 (1.00) 12 (1.00)

986 (1.00) 838 (1.00) 771 (1.00) 8 (1.00)

599 (1.00) 440 (1.00) 647 (1.00) 8 (1.00)

)30% contrast and blurred � � � 314 (1.00)

557 (1.00)

215 (1.00)

+20% I and )30% C 10664 (0.98) 1993 (1.00) � 2231 (1.00)

5220 (0.99) 1192 (1.00) 1135 (1.00)

3930 (0.90) 985 (0.99) 737 (1.00)


m appin g resu lts by digital im age correlat ion . Wh en

such con dit ion s can n ot be fu lly realised in an actualexperim en t due to eith er th e lim itation of th e im a-

gin g h ardware (especially th e scan n in g electron an datom ic force m icroscopes) or th e surface degradat ion

of deform in g objects, th e effect of variable ligh t in gsan d exposures on th e perform an ce of digital im age

correlat ion -based strain m appin g sh ould be assessed

an d th e m ost robust an d reliable im age processin g

strategy m ay be developed to m in im ise th e m eas-

urem en t errors.

In all th e successfu lly processed cases for th e th ree

sets of test im ages in vest igated h ere, th e global average

strain levels d iffer at m ost by 50�100 lstrain s. If such a

level of errors is acceptable for a given applicat ion ,

th en th e variable ligh tn in g an d exposure con dit ion s

sim u lated in th is study is of lit t le con cern . Th e selec-

t ion of th e correlat ion criteria can be based on eith er

th e robustn ess (th e criterion C4) or th e com putat ion alcost (th e criterion C2). However, wh en on e is con -

cern ed with relat ively sm all deform at ion s of th e order

of a few h un dreds m icro-strain s (�0.01�0.1%) in eith er

th e overall average levels (th e�rst test im age set) or

detailed local strain variat ion s (th e secon d test im age

set), th e m ost robust an d reliable correlat ion criterion

C4 sh ould always be used. Un less th e im age pair to be

processed is of h igh est quality, th e strain m appin g re-

su lts obtain ed by oth er correlat ion criteria are always

less reliable. Measurem en ts of deform at ion�elds with

large overall strain s an d h igh strain gradien ts ten d to

be far less sen sit ive to th e kin ds of degradat ion ofim ages con sidered h ere, alth ough on ly th e criterion

C4 is successfu l in processin g all cases. Th e localm easurem en ts of th e strain com pon en t E12 (about

2700 lstrain s) in th e th ird test im age set becom e

n everth eless un reliable.

Surprisin gly, both criteria C1 an d C3 h ave been usedwidely in a sign i�can t n um ber of strain m appin g

applicat ion s [1�4, 6, 13, 16, 17]. Wh ile th ey m ay beadequate for pattern m atch in g [20, 21] in gen eral an d

wh ole-�eld strain m appin g usin g im ages of very h ighquality, th e results in th is study h ave sh own th at both

criteria are far less robust an d reliable th an eith er cri-

terion C2 or C4. Th e correlat ion criterion C2 was ori-

gin ally in troduced in th e con text of processin g 3-D

surface pro�lin g data acquired by scan n in g probe

m icroscopy [8] an d it h as been used for m an y in -plan e

strain m appin g applicat ion s [5, 7, 9, 11, 12, 14, 15].

Th e extra param eter P7 effect ively acts as a correct ion

term on th e average brigh tn ess of th e im age pair.

Wh ile it is n ot as robust an d reliable as th e correlat ion

C4, its reduced com putat ion al cost m akes it at t ract ivefor m an y pract ical applicat ion s. Th is study sh ows th at

th e n orm alised correlat ion criterion C4 is m ost robustan d reliable for processin g im ages with variable ligh t-

n in g or exposure problem s con sidered h ere. Its on lydrawback is th e th reefold in crease in th e com puta-

t ion al cost wh en com pared with th e criterion C2.

Conclusions

Based on th e n um erical evaluat ion s of th ree sets of

test im ages presen ted h ere, th e perform an ce of th e

four correlat ion criteria con sidered in th is study can

be sum m arised as followin g:

� Robustn ess ran kin g: C4, C2, C3 an d C1;

� Reliability ran kin g: C4, C2, C3 an d C1;

� Speed ran kin g: C1, C2, C3 an d C4.

To ach ieve th e m ost reliable results in strain m appin g

applicat ion s, th e m ost robust correlat ion criterion C4

sh ould always be used. If on ly th e average deform a-

t ion levels of a deform ed im age are required or wh enim ages of good stability an d m in im al degradation

can be assured, th e fast correlat ion criterion C2 m aybe used to reduce th e com putat ion al cost. Oth er two

correlat ion criteria evaluated h ere are n ot recom -m en ded at all for wh ole-�eld strain m appin g m eas-

urem en ts except for very h igh quality im ages.

REFEREN CES

1. Sutton , M. A., McNeill, S. R., Jan g, J. an d Babai, M. (1988)Effect of subpixel im age restorat ion on digital correlat ionerror estim ates. Opt. Eng. 27, 870�877.

2. Bruck, H. A., McNeill, S. R., Sutton , M. A. an d Peters, W. H.(1989) Digital im age correlat ion usin g Newton -Raph sonm eth od of part ial d ifferen tial correct ions. Exp. Mech. 29,261�267.

-0.0

81

0.061

-0.061

- 0.0

40

-0.040

-0.020-0.020

0.020

-0.020

0.001

0.001 0 .00

1

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0. 0 0

0.001

0.001

0.021

0.021

0.021

0.021

0.041

0.0620.10

3

Figure 4: Th e n orm alised errors in tran sverse true straindist ribu t ion of th e tapered t itan ium sam ple (case n o. 5 listedin Table A3, criterion C4)


3. Jam es, M. R., Morris, W. L. an d Cox, B. N. (1990) A h ighaccuracy autom ated strain -�eld m apper. Exp. Mech. 30,60�67.

4. Fran ke, E. A., Wen zel, D. J. an d Davison , D. L. (1991)Measurem en t of m icrodisplacem en ts by m ach in e visionph otogram m etry. Rev. Sci. Instrum. 62, 1270�1279.

5. Ton g, W. (1997) Detect ion of p last ic deform at ion patternsin a bin ary alum in um alloy. Exp. Mech. 37, 452�459.

6. Ch oi, S. an d Sh ah , S. P. (1997) Measurem en t of deform a-t ion on con crete subjected to com pression usin g im agecorrelat ion . Exp. Mech. 37, 307�313.

7. Ton g, W. (1998) Strain ch aracterization of propagat ivedeform at ion ban ds. J. Mech. Phys. Solids 46, 2087�2102.

8. Ven droux, G. an d Kn auss, W. G. (1998) Subm icrondeform at ion�eld m easurem en ts, Part 2. Im proved digitalim age correlat ion . Exp. Mech. 38, 86�91.

9. Sm ith , B. W., Li, X. an d Ton g, W. (1998) Error assessm en tfor strain m appin g by digital im age correlat ion . Exp. Tech.22, 19�21.

10. Zh an g, D., Zh an g, X. an d Ch en g, G. (1999) Com pres-sion strain m easurem en t by digital speckle correlation .Exp. Mech. 30, 62�65.

11. Ton g, W. an d Li, X. (1999) Evaluat ion of two p last icstrain m appin g m eth ods. Proc. SEM Annual Conference onTheoretical, Experimental and Computational Mechanics,Cin cin n at i, Oh io, 23�26.

12. Li, X. (2000) Spat ial ch aracterizat ion of un stable p last ic�ows in two alum in um alloys. Ph D Th esis, Departm en t ofMech an ical En gin eering, Yale Un iversity, New Haven , CT.

13. Sch reier, H. W., Braasch , J. R. an d Sutton , M. A. (2000)System at ic errors in d igital im age correlat ion caused byin ten sity in terpolation . Opt. Eng. 39, 2915�2921.

14. Ton g, W. an d Zh an g, N. (2001) An experim en tal in vest i-gat ion of n eckin g in th in sh eets. Proc. ASME ManufacturingEngineering Division MED, New York, Vol. 12.

15. Raiser, G. F., Em ery, R. D. an d Ton g, W. (2002) Digitalim age correlation� usin g deform at ion m easurem en tsto accelerate m icroprocessor product developm en t.Proc. SPIE�The International Society for Optical Engineering,Vol. 4217, 485�490.

16. Wan g, H. an d Kan g, Y. (2002) Im proved digital specklecorrelat ion m eth od an d its applicat ion in fractu re an aly-sis. Opt. Eng. 41, 2793�2798.

17. Yon eyam a, S. an d Morim oto, Y. (2003) Accurate dis-p lacem en t m easurem en t by correlat ion of colored ran -dom pattern s. JSME Int. A46, 178�184.

18. Press, W. H., Flan n ery, B. P., Teukolsky, S. A. an d Vetter-lin g, W. T. (1986) Numerical Recipes: the Art of Scienti�cComputing. Cam bridge Un iversity Press, Cam bridge, UK:484�485, 509�528.

19. Gon zalez, R. an d Win tz, P. (1987) Digita l Image Processing,2n d edn . Addison Wesley, New York: 90�92, 100�109,425�427.

20. Jain , R., Kasturi, R. an d Sch un ck, B. G. (1995) MachineVision. McGraw-Hill, New York: 482�483.

21. Rit ter, G. X. an d Wilson , J. N. (1996) Handbook of Com-puter Vision Algorithms in Image Algebra. CRC Press, BocaRaton , FL: 225�229.

22. Ton g, W. an d Zh an g, N. (2004) On th e crack in it iat ionan d growth in a stron gly dyn am ic strain agin g alum in umalloy sh eet. J. Mater. Sci. Technol. 20 (Suppl. 1), 23�26.

23. Ton g, W. (2004) An adapt ive backward im age correlat iontech n ique for deform at ion m app in g of a growin g crack inth in sh eets. Exp. Tech. 28, 63�67.

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A PPEN DIX A : SU MMA RY O F T EST IMA GE SET S

Table A1: Th e st ill im age set

Image �le Comments

TST0.RAW The reference image (of an aluminium plate)

TST1.RAW Brightness increased 20% from TST0

TST2.RAW Contrast reduced 30% from TST0

TST3.RAW Blurred TST2

TST4.RAW Histogram-equalised TST0


TST6.RAW Brightness: +20% and contrast: )30% (TST0)


TST8.RAW Strip 1: brightness: +20% and contrast: )30%

Strip 2: only brightness increased 20%

Strip 3: only contrast reduced 30% (TST0)

All digital image modi�cations mentioned above (and below) are carried out usingthe functions in the program Paint Shop Pro.

Table A2: Th e h om ogen ous deform at ion im age set


STL0.RAW The reference image (of a steel sample)

STL1.RAW The deformed image (of the steel sample)

STL2.RAW Brightness increased 20% from STL1

STL3.RAW Contrast reduced 30% from STL1 (softened)

STL4.RAW Histogram equalised (STL1)

STL5.RAW Blurred STL1

STL6.RAW Brightness: +20% and contrast: )30% from theimage STL1 and then softened once

STL7.RAW 0.65 gamma correction on STL1 (darker)

STL8.RAW 1.50 gamma correction on STL1 (brighter)

Table A3: Th e n on -h om ogen ous deform at ion im age set


TIP0.RAW The reference image (of a titanium sample)

TIP1.RAW The deformed image (of the titanium sample)

TIP2.RAW Brightness increased 20% from TIP1

TIP3.RAW Blurred TIP2

TIP4.RAW Histogram equalised (TIP1)


TIP6.RAW Contrast reduced 30% from TIP1


TIP8.RAW Brightness: +20% and contrast: )30% from the image TIP1


An Evaluat ion of Digital

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