Sutton, 1983, Determination of Displacements Using an Improved Digital Correlation Method

7/24/2019 Sutton, 1983, Determination of Displacements Using an Improved Digital Correlation Method

1/7

Determination of

displacements using an

improved digital correlation

method

M A Sutton, W J Walters, W H Peters, W F Ranson and S R McNeil1

An improwed digital correlation method is presented f or

obtaini ng the ful l-f ield in-plane deformations of an object.

The deformations are determined by numeri cally correla-

ti ng a selected subset

rom

he digi tized intensity

p ttern

of

the ~~defo~ed object. T& e mproved n~rn~~a~ ~o~e~at~o~

scheme is discussed in detail - The d~~~ac~~ ts of a simple

object, as computed by the correlation rou tine,

are shown CO

agree with theoretical calculati ons.

Keywords: digital, processirtg displacement;s

It has been stated many times that the computer will alter

the way we live. Without question, this simple statement

continues to be proven true every day. However, the full

potential of the computer has not been realized in many

areas. In particular, those people who wish to contribute to

a better understanding of our world through innovative

measurements have seen little change. The basic reason

for this is that older measurement techniques were

simply not developed for use with the computer. Recently9

however, researchers have developed a novel measure-

ment scheme which employs a digital imaging system. In

this scheme, a video camera observes an object and the

image is digitized and sent to a computer. Within the

computer, numerical schemes utilize the basic theory of

deformatian as a mapping.

In this paper we present a synopsis of the basic

theory of digital correlation as used in the analysis of

object deformation. An improved numerical scheme for

computing the deformation of an object is discussed in

detail. Finally, the technique is successfully employed to

analyse the displacements of a simple object, a cantilever

beam with an end load.

-

College of Engineering University of South Carolina Columbia SC

29208 USA

BASIC THEORY OF DIGITAL

CORRELATION AS APPLIED TO

SURFACE DEFORMATION

MEASUREMENTS

Suppose an object is viewed with a stationary video

camera as shown in Figure 1. The intensity distribution of

light reffected by the specimen can be stored as a set of

numbers or grey levels in a computer via an appropriate

information transfer. Usually, the continuously varying

intensity pattern is discretely sampled with an array of

sensors that

records

and stores an array of intensity

values. A typical size for such an array is 512

X 512.

Each

sensor converts the intensity to a number. For typical

scanners, the number will range from 0 to 255 (0

represents zero light intensity). The conversion of light

intensity to a digital number is controlled by the digit-

izer. In many cases the digitizer is controlled by

a

m~i~omputer, the PDP/gE in Figure 1. The PIXf8E

transfers the digital information into an array in

White light sowe

x

Specimen

White liqht source

I

~--_--

5-~,

Figur e 1. Schematic ofthe experimental confi guration for

correlation analysis

wol f no 3 arigust 1983

0252-88551831030133-579B83.40 0 I983 Butterworth & Co (Publishers) Ltd.

I33


2/7

Figure 2. D igi t al int ensit ies or a 10 X 10 subset

t

intensity. IQ EP)

Figure 3. Bi li near r econst rzl cti on of t he in tensit y surface

for four data poi nts.


3/7

To obtain the displacement and deformation gradient

terms for a local subset, we minimize the square of the

difference between the chosen subset in array A and all

other subsets of the same size in array B. Thus, ifwe define

a correlation coefficient C by the equation

.

C(z

[A(x) - B(x')J2 dx

i,j=lt2

(2)

I

then the analytical task we must perform is to minimize

the coefficient C with respect to the six mapping

parameters (ui and dui/dxj) for two-dimensional deform-

ation Physically, this analytical task may be understood

by reference to Figure 4. The analyst chooses a lagrangian

reference frame attached to the undefo~ed object. A

small subset centred at PO is chosen. The defo~ation of

this small subset due to applied loads is required. This

subset is moved and distorted homogeneously as shown.

it is then compared with the stored values of intensity

within the deformed array B x). The deformations which

minimize the difference in intensity, as given in equation

(2), are defined as the local mapping of the actual object

surface. It should be noted that a basic tenet of elasticity is

that there exists a subset within the body such that the

deformation in this small region may be expressed as a

homogeneous deformation. Therefore, if the subsets are

chosen sufftciently small and the various assumptions

noted previously are generally valid, then the method des-

cribed is useful for both large and small deformation

processes.

Relative to the above discussion, two remarks must

be included. First, the procedure for determining the

minimum in a function ofseveral variables is currently an

area of active research in applied mathematics. The

improved scheme used in this work will most certainly be

updated as more efficient methods are developed.

Secondly, it may seem that all this analysis is based on a

rather tenuous assumption: that the intensity pattern

deforms in a one-to-one correspondence with the object

surface. This assumption has been partially verified in

recent research-. This work indicated that large uniaxial

deformations (ie &i/&iin the range from 0.01 to 0.04)

were determined successfully, and rigid body displace-

ments of various ma~itudes have been measured quite

accurately. The use of the method for computing

displacements in a varying strain field is considered

here.

IXSCUSSION OF THE STRAIN ROUTINE

The subroutine Strain inputs light intensity data from an

undeformed surface and intensity data from the same

surface after it has been deformed. Strain examines the

points in the undeformed image and locates their new

positions on the deformed surface. Values for the variables

ult u2, dut:&r, duzidxz, dut/d~~ and duzidxl are also

obtained. As noted previously, it has been assumed that for

sufficiently small regions the deformation of straight lines

on the original surface yields straight lines. Therefore

Strain is limited to the processing of very small surface

areas. When a much Iarger surface area is to be examined,

it must be broken into many smaller areas that can be

handled individualiy by separate calls to this routine.

The deformation process can move points on the

original surface relatively large distances in any direction.

For this reason, the deformed surface data that is

examined represent a relatively large surface that is

centred about the undeformed surface.

Figure 6 is a simplified flowchart for this routine.

There are two major stages to the routine. These are called

the starting value procedure and the iteration pro-

cedure.

When the routine is entered, there are no available

values for the six variables of interest. The starting value

procedure examines the variables two at a time in order to

determine a good first estimate for each variable. Once the

starting values have been obtained, the iteration procedure

searches for the best set of values for the variables by

combining the effects of all six variables. The best set of

values minimizes (in the least-squares sense) the correl-

ation coefficient defined by equation (2).

Program

initiadon

At program initiation, information that must be input

includes

an m

X m

array of intensity data from the undeformed

image, where

m

is an integer from 1 to 50

four 100 x 100 data arrays for determination of

intensity data for the deformed image

upper and lower bounds on the six variables

end points for the start-up procedure

a minimum acceptable correlation coefftcient for

program termination

r____--L -__._- _

j~d_eslmo:es: or us and_ +;

______t______-

i--F?nGistmotei

or a

ax,

ond

a4ax -y

L--_______,______L_*_


4/7

Figure 7. Examina t ion of t he var iabl es u1 and ~42. gives

t he size and i ni ti alp osit io n of he undefomzed square subset ;

b is

an

array

of 121 cambinat io~ ~ of u1 and u2 val ues used o

shif t the or& ina~ subset a.

For example, a 15

X 5

intensity array represents a very

small subset of the intensity data from a much larger

undeformed object. This was the array size chosen for

most of the analysis to follow. The goal of Strain is to

determine the location of the 225 surface points after

deformation. The routine may be called many times for a

single set of undeformed and deformed data, and it

requires thousands of bilinear inte~la~ons to determine

the intensity data for deformed surface locations. There-

fore-, to minimize the interpolation calculations performed

within Strain, the deformed surface intensity array B)

must be converted to four arrays before any Strain call.

The new arrays FA, Fg, Fc and FD) are obtained from

FA I,J) = B f + l,J)-B i,J)

FB I,J) = B i,J + 1) - W43)

Fc Z, J) = B Z + 1,J + 1) + B Z, r>

(3)

F,(I, 3) = B(I, 3)

The upper and lower bounds for a variable define

the limits of values that will be examined. The range for a

variable is the difference between the limits, and the

increment size is the range divided by ten. This provides

11 acceptable values for a variable at any given time.

When a very small section of a surface is examined,

deformation can produce three types of change. The

surface section can be translated along the xi and x2 axes,

the x1 and x2 dimensions of the section can be expanded or

contracted, and the sides of the section can be rotated

about their vertices. These changes represent the respec-

tive effects of nonzero values for the three sets of

variables

Starting value procedure

Thestarting value procedure works as follows. First, the u1

and 24 values are examined to determine the translations

that occurred. The original surface data is translated to

each position (xi, x2) within the rectangle defined by the u1

and u2 ranges, as shown in Figure 7. At each

of

these 121

positions, the original 15 X 15 array ofundeformed data is

compared with a 15 X 15 array of deformed surface data.

For each of the 225 deformed surface points, the light

intensity is calculated by bilinear interpolation. A correl-

ation coefficient is calculated for the two intensity arrays.

The position of the translated square that yields the lowest

correlation coefficient produces the best values for u1 and

242. 1 is the difference between the x1 coordinate of the

position of this square and that of the centre of the original

square. 242 s found in a similar manner.

The end points mentioned previously are minimum

acceptable values for the increment sizes. If the variable

increments are greater than the end values for ZL~nd u2,

the ranges are reduced and centred about the position with

the lowest correlation coefficient. Again, the increment

size is the range divided by ten. The above procedure is

repeated. This process continues until the best values ofui

and ~2 are found such that they are incremented by

acceptably small amounts.

After good first estimates for ul and u2 have been

obtained the variables dullax1 and du2/axz are examined

to determine if the surface expanded or contracted in the

xi and x2 dimensions. The area of the deformed surface

that is to be searched is centred about the position

(xi0 + ui, xzO+ uZ). Position (xlO, x2,) is the centre of the

undeformed surface. For all variables, the increment size

is one-tenth of the range. Therefore the 121 possible

combinations of dullax and du2dx2 within their ranges

are examined. This amounts to finding the rectangle

which best correlates with the undeformed data. Some of

the possible rectangles are indicated in Figure 8.

Ic

L

Figure 8. Examinati on of dull & xl and du2/dx2. a gives

t he size of an undeformed square; b-f are examples of

rect angles t hat are correlat ed w it h he undeformed square.

136

image and vi sion computing


5/7

After the 121 combinations of dutldxt and

duZldxZ have been examined, the combination pro-

ducing the best correlation is retained. The ranges are

reduced and centred about this combination of values and

a new smaller set of increment sizes is obtained. The

process for finding dutldxt and duJdxZ is repeated in a

manner similar to that for ut and 2~2.After all combinations

have been examined, the ranges and increment sizes for

these variables are reduced. The search continues until the

increment size becomes sufficiently small.

The third set of variables examined are dul/dx2 and

du211dx1. in this process we examine the parallelograms

that are centred about (xl0 + ul, x20 + u2) and that have

sides of the lengths found from &t/&t and duddx2

(Figure 9). All 121 combinations of du11dx2 and &Z/&i

are examined to find the parallelogram that produces the

lowest correlation coefficient. This process is repeated for

reduced ranges as the two previous processes were

repeated.

While two variables are being examined, there is

considerable data processing at each of the 121 combin-

ations of values. First, locations must be determined for

each of the 225 points to be used for the deformed surface.

The locations are calculated from equation (3). During

much of the starting value procedure, several of the

variables in equation (3) are zero. This causes the

calculations to be relatively quick and inaccurate.

Once the location of a point has been determined,

the light intensity at that point must be determined using

bilinear interpolation. For a point at (I , J + G),

where Z and J are integers and


6/7

Top steel ptate

Sted rod

4 8 mm diameter

Dial

indicator

Plexiglas

block

I

25.4 mm

11

i j Floating table

bolts

/I

,

101.6 mm

,

F&we 10. F ixed end co~d~t~on or a cant~lev~ beam

Figur e 11. Typical random pattern

Digitalired imoge

I

I

I

I

I

Centre

-.

-1.

line

S et of i~estigatlon

F t;eUr e 12. Measuring beam defl ections using digital

imaging techniques

correlation method. The beam was machined from a

Plexiglas sheet to the dimensions shown in Figure 10. The

estimated material properties for the Plexiglas were

E = 34.48 GPa and v = 0.37.

The experimental procedure was as follows. First

the surface of the beam was coated with black rubberized

paint. It was then lightly spray painted white to obtain a

random pattern similar to that shown in Figure 11. This

surface was viewed with an Eyecom digitizing camera and

its associated equipment as shown in Figure 1. Next, the

cameras image magnification was set to 8.3 lines mm-

for all cases. However, because of equipment limitations,

only arrays of 100

x

100 could be processed. For each load

or noload condition, six separate camera positions were

required for the entire beam to be analysed. The

horizontal movements were accomplished by the use of a

specially designed horizontal translation stage on which

the Eyecom camera was mounted. The horizontal position

along the beam for each of the six views was obtained by

loosely attaching a ruler to the top of the beam. To ensure

correct positioning along the beam, the ruler information

was entered into each data file. Then the load was applied

to the beam. However, instead of measuring the applied

load, the vertical free end deflection (Se) was the measured

quantity. This procedure was used to simplify interpre-

tation of the results. Two values of I?* were analysed,

S* = 0.2616 mm and 6* = 0.503 mm.

For each value of s*, 12 100 X 100 arrays of ideas

were stored in the computer. Each of the 12 files was then

displayed on the Comtal image processor screen. The ruler

was disolaved on the screen to indicate the actual position

.

of a file along the beam.

70-

Y

60-

Q

x 50-

a/*

0-y

o-----I---I-----L-_

3 i f

+-

25.4 50.8 R

Distance, mm

Figure 13.

Verti cal displacement (S/S*) as determined

porn theory (-), fi ni te element analysis (-I -) and cross

co~e~ation (* )

(PexP = 0,262 mm)

Distance, mm

F igur e 14. Verti cat displacement as determined from

theory (-) and cross correlation (0) (c?*~~~ 0.503

mm)

138

image and vision computes


7/7

Top of beam

26 -

3 -

1:

/

36 ~

__-----

--o-

.

__-----

.4I -

,D------

C-

A/

/

_---

/

___--

46cf

__---

51 cv

/

N

.

_---

__--

__--

,, -

_---

76,_

/

/

. I/

rr

.--

.w

P

81 -

/

L_ -/I

pttm

of beam

, r(/ ,

,

I

I I

I _1._. ~_ . .L/ J

-0 II -0 IO -0.09 -0.08

-0.07 -0.06 -0.05 -0.04 -003 -002 -001 001 002

003 004 005 0.06 007 008 009 010 Ol

Pxes

F igur e 15. Hori zontal dtiplacement in pixels as determinedfrom cross correlation: 0, at 19.1 mm fr om the ree end;

0, at 25.4 mm fr om the fr ee end

Finally the Strain subroutine calculated the dis-

placements of the 15

x

15 subsets which were centred in

the front surface. Figure 12 shows the approximate

location of each subset.

The vertical deflection results are given in Figures

13 and 14. The theoretical values ofthe vertical deflections

were those obtained from simple beam theory with

In addition, a linear, elastic, finite element analysis of the

member was performed for the case where 6* = 0.2616

mm. The triangular plate bending element was used and

40 elements, four through the thickness and ten along the

length, were used to represent the structure.

Relative to these results, the following comments

can be made. First, the results of the finite element

analysis indicate that the cantilever member does behave

like a beam. Secondly, the experimental results for all

cases are quite good with a maximum error of 5 . Finally,

the results appear to have the largest errors in the regions

near the fixed support. This is to be expected since

elasticity theory is most accurate in regions far from the

imposed boundary condition.

The experimental results for the horizontal deflec-

tions of the beam are shown in Figure 15. These results

were obtained at two horizontal locations removed from

support uncertainties. The 15

x

15 subsets were selected

along a vertical line to detect the horizontal displacements

along the line. The results are quite poor and bear little

resemblance to the theoretical profile of displacements. As

the computed displacements in Figure 15 are all less than

0.12 pixels, it is clear that these results indicate that a

lower threshold for accurate measurement of displace-

ments is above 0.10 pixels.

CONCLUSIONS

An improved two-dimensional digital c:orrelation algor-

ithm has been developed and specially adapted to the

processing of digitized video signals so that surface

deformations can be obtained. The computation sub

routine Strain was found to require half the computer

execution time with minimal changes in computing

precision. This routine was used to deduce the two-

dimensional displacements of the centreline of a cantilever

beam. Comparisons of these results with known theoret-

ical results indicate that the algorithm can successfully

compute displacements larger than approximately 0.10

pixels by using a bilinear interpolation procedure.

ACKNOWLEDGEMENTS

We wish to acknowledge the encouragement of C J Astill

and the support ofthe National Science Foundation and of

the College of Engineering, University ofSouth Carolina.

REFERENCES

1

Peters, W H and Ranson, W F

Digital imaging

techniques on experimental stress analysis

Opt. Eng.

Vol21 No 3 (1982) pp 427-431

2

Chu, T C, Peters, W H, Ranson, W F and Sutton,

M A

Application of digital correlation methods to

rigid body mechanics

Proc. 1982 F all Meet. of SESA

pp 73-77

3

McNeill, S R, Peters, W H, Ranson, W

F

and Sutton, M A

A study of fracture parameters by

digital image processing

Proc. 1983 Spring Meet. of

SESA

to be published

vol 1 no

3

august 1983

139

Documents

Sutton, 1983, Determination of Displacements Using an Improved Digital Correlation Method