Information Technology Reference
In-Depth Information
deformation. They extend the parametrization to the third dimension using stereo
endoscopy. A computationally efficient technique is used to compensate for illumi-
nation variations.
In the remainder of this section, we will detail one of the most efficient algorithm
for heart surface tracking, the ESM method. This algorithm searches the object po-
sition and orientation that best matches the object view. It is robust against illumi-
nation variation, partial occlusion and specular highlights. Also it is natively robust
against small deformations of the object. Furthermore, it can be easily extended to
account explicitly for deformations. And, last but not least, it can be implemented
to run in real-time at a high sampling rate.
6.4.1
Problem Formulation
Suppose that the object we want to track is a plane and has sufficiently rich texture
on the surface. Let the image of the object we capture be
I
, and the brightness of
the point
p
=(
u
1) in the image be
I
(
p
). Then the pattern we want to track is
expressed by
I
∗
(
p
∗
) where
p
∗
=(
u
∗
,
,
v
,
v
∗
,
1) and (
u
∗
,
v
∗
)
is
the index to identify the point in the tracking area. To simplify the expression we
write the point as
p
∗
i
for
i
= 1
∈{
1
,...
n
}×{
1
,...
m
}
q
and
q
=
mn
. The ordering can be either row-
major or column-major. Then the brightness of the point is
I
∗
(
p
∗
i
).
The problem can be formalized as a nonlinear estimation problem to find the
optimal displacement
G
that transforms the point
p
∗
i
to the point
p
i
in the current
image so that the brightness of the template and the current image becomes the
same:
,...,
I
(
p
i
)=
I
∗
(
p
∗
i
)
(6.9)
where
p
i
=
w
(
Gp
∗
i
) and with
w
the perspective transformation.
If the target is planar the transformation is expressed by an homography matrix
G
. Suppose that there are two camera positions and two images taken from these
two positions. The matrix
G
and the perspective transformation
w
transforms the
plane texture from one viewpoint to the other viewpoint:
p
i
=
w
(
Gp
∗
i
)
(6.10)
where
⎡
⎣
⎤
⎦
.
⎡
⎤
g
11
u
∗
+
g
12
v
∗
+
g
13
g
31
u
∗
+
g
32
v
∗
+
g
33
g
21
u
∗
+
g
22
v
∗
+
g
23
g
31
u
∗
+
g
32
v
∗
+
g
33
1
g
11
g
12
g
13
g
21
g
22
g
23
g
31
g
32
g
33
⎣
⎦
,
w
(
Gp
∗
i
)=
G
=
(6.11)
Estimation of the homography matrix becomes a nonlinear least-square mini-
mization problem: find the matrix
G
that minimizes the sum of square difference
(SSD) error defined by
q
i
=1
(
I
(
w
(
G
)
p
∗
i
)
−
I
∗
(
p
∗
i
))
2
.
(6.12)
Search WWH ::
Custom Search