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Fig. 7.4 DIC: evolution of a
subset during deformation
from instant t n to instant
t n + 1
observed subset
t n+1
t n
where A(m, n) is the intensity at position (m,n) in undeformed conditions, B(m + i,
n + j) is the intensity at position (m + i, n + j) in deformed conditions, i and j are
the offsets, M a and N a are the numbers of horizontal and vertical pixels in
the subset. This definition of cross correlation is equivalent to that reported in the
second row of Table 1 in [ 1 ] and called NCC, in which the values are weighed by
their square means over the subset. It has the property of being unaffected by a
linear change in illumination lighting, but it is affected by a lighting offset;
however, since the measurements considered here were carried out consecutively,
under the same lighting conditions, this problem is not under discussion. The
maximization if the cross-correlation yields the position of the subset with the
resolution of one pixel, i.e. the obtained displacement is multiple of one pixel, and
this can be regarded as the first step of the tracking procedure.
Thus, a second step is required to obtain a finer, sub-pixel, resolution. An
effective and simple way to achieve this goal is given by the curve-fitting method,
reported in [ 24 ]. A fitting surface C(x, y) is defined based on a 3 9 3 grid, centred
in the pixel position given by maximization of ( 7.1 ):
Cx ; ðÞ ¼ a 0 þ a 1 x þ a 2 y þ a 3 x 2 þ a 4 xy þ a 5 y 2
ð 7 : 2 Þ
where coefficients a k (k = 0,…, 5) are found by least squares fitting the values
assumed by the cross-correlation r in the 9 points of the grid. The position of the
maximum is obtained by zeroing the derivatives of C:
o C
ox ¼ a 1 þ 2a 3 x þ a 4 y ¼ 0;
o C
oy ¼ a 2 þ 2a 5 y þ a 4 x ¼ 0
ð 7 : 3a ; b Þ
which gives:
x ¼ 2a 1 a 5 a 2 a 4
a 4 4a 3 a 5
y ¼ 2a 2 a 3 a 1 a 4
a 4 4a 3 a 5
;
ð 7 : 4a ; b Þ
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