Biomedical Engineering Reference
In-Depth Information
2
α
The Euler-Lagrange equation corresponding to (9.5) is
∇
=
δ
. The solution
for non-special
α
(read: non-integer) is of the form
i
=
1
λ
i
(
x
−
x
i
N
(
r
)
=
r
2
α
−
2
g
=
)
with
(9.6)
r
The polynomial kernel term does not appear here due to technical restrictions
of the Fourier domain definition of the criterion (9.5).
9.3.3.2
The Influence of
α
The coefficient
α
translates into the assumed smoothness of the deformation—
the higher the
α
, the smoother the deformation. For practical purposes, we
will use 0
.
5
<α
<
≈
5; the interpolation becomes point-wise unstable for smaller
α
and does not change much for the larger ones. For
α>
1
.
5 the prior can no
longer be interpreted as fractal Brownian motion, although the criterion remains
usable.
The choice of the order
α
obviously has an influence on the interpolation
results. In Fig. 9.6 we present an example of this effect for the landmark (bi-
variate) case, (see [86] for additional examples.) We have chosen two images
from a four-chamber ultrasound sequence of a heart
5
and declared one of them
reference and the other a test. We have manually identified six pairs of conform-
ing points in both the images and we have also put additional stable landmarks
in the corners of the image. Then we warped the test image onto the reference
image varying the parameter
α
. Note that only non-integer values of
α
were
used, for which the formula (9.6) remains valid; integer values need a special
treatment.
How do we choose the best
α
? As we have seen in the previous section,
the statistically optimal
α
can be determined directly, when the characteristics
of the stochastic process generating the deformation are known. However, in
practice, this is never the case . Therefore,
α
must be found experimentally. We
observe that small
α
yields more localized and abrupt changes in the deforma-
tion field, while higher
α
gives rise to smoother and more global changes, as
predicted.
5
Acknowledgements:
Images and landmark placement are the courtesy of
Marıa J. Ledesma, Universidad Polit ecnica de Madrid, and Laboratory of Echocardiog-
raphy, Hospital General Universitario Gregorio Mara n on, Madrid, Spain.
