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