Chemistry Reference
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where
y
ij
is the rotation required for the vector from point
i
to point
j
in the
theoretical shape in order to have the same attitude as the homologous vector in
the experimental profile. The translational parameters are calculated from the
simple (nonrepeated) medians as:
a
¼
x
i
t
u
i
cos
y
v
i
sin
y
med
i
f
½
g
(9a)
b
¼
y
i
t
u
i
sin
y
þ
v
i
cos
y
med
i
f
½
g
(9b)
The advantages of double median robust techniques over traditional least
squares regression methods have been discussed by Siegel et al. [
166
]. One
particular advantage specific to the shape comparison problem can be understood
by comparing (
6
) with (7) (9). The least squares minimization is sensitive to local
residuals between individual points, which are, however, only remotely related to
the overall shapes of the two profiles being compared. The robust method affects a
more global shape comparison, as can be seen from examining (7) and (
8
). Instead
of comparing individual points of the two curves, the method compares vectors or
line segments between all points
i
and
j
on the experimental profile with the
corresponding vectors on the theoretical profile. In addition, the values of
t
*
,
y
*
,
*
for each shape comparison (i.e., for each value of the shape parameter
B
)
are specified directly by the robust relationships (7) (9). Thus, the five-parameter
optimization is reduced to a single variable optimization of the shape parameter
B
[
155
].
The application of the robust shape analysis algorithm is illustrated in Fig.
2
for a
drop of polystyrene (PS,
M
n
¼
*
, and
b
a
10,200;
M
w
/
M
n
¼
1.07) in a poly(ethyl ethylene)
Fig. 2 Quality of the fit
obtained by the application of
the algorithm to an
experimental profile for a PS
10,200 drop in a PBDH 4080
matrix at 147
C.
Solid line
is
the theoretical profile, and the
data points
denote the
original segmented
experimental drop profile
[
20
]. The interfacial tension is
2.6 dyn/cm
Drop X Coordinate
