Geoscience Reference
In-Depth Information
Fig. 4 Examples of high (a), medium (b), and low (c) density of tracer particles images
inspection. In case of medium density, the images of individual particles can still
individuate, but it is not possible to identify image pairs. In this condition, the
velocity is evaluated considering the displacement of small group of particles
(patterns), assuming that they do not change their relative position in the group
during the time interval between images. This medium density mode is the one
properly named PIV.
In PIV mode, pattern displacements are evaluated by statistical means, comput-
ing two-dimensional correlations, autocorrelation or cross-correlation, on pairs of
images. Autocorrelation was traditionally used with the double exposure mode,
while cross-correlation is actually commonly used with double frames-single
exposure method.
The general concept of cross-correlation technique is to individuate the corre-
sponding particle patterns in the two images of the pair using a cross-correlation
coefficient. In digital images, a
value
which represents the gray-level intensity
is associated to each pixel. Considering the two images,
I
1
and
I
2
(Fig.
5
), taken
at time distance
D
t
, a grid of points at which the velocities are determined is defined.
Then an
interrogation area
,
A
1
, of size
l
w
l
z
, centered in
P
(
i, j
)
,
is considered.
This area contains a group of particles, forming a pattern in the flow, characterized
by a distribution of intensities. In the second images, a larger area, circular or
rectangular, of dimension
r
, containing
P
and named
research area
, is selected. In
this area, the interrogation area
A
2
of size
l
w
l
z
, corresponding, or in other words
containing the same particle pattern, to the one of the first image is se
arch
ed. In
particular, different areas
A
2
, offset from the point
P
by the amount
PP
2
, with
P
2
(
i
y
)
,
are considered and compared with
A
1
. The distance
PP
2
, which
furnishes the best correspondence, represents the particles movement in the time
interval
D
t
and is used for computing the velocity vector.
The best correspondence can be selected by evaluating the cross-correlation
coefficient:
þ
x
,
j
þ
P
P
l
w
l
z
ð
I
1
ð
i
j
Þm
1
Þ
ð
I
2
ð
i
þ
x
j
þ
y
Þm
2
Þ
;
;
i¼
1
j¼
1
C
ð
x
y
Þ¼
p
(3)
;
s
1
s
2
