Biomedical Engineering Reference
In-Depth Information
sinogram data. The incremental change in the likelihood is
N
M
N
M
dE
data
dt
=
dt
E
p
ij
,
p
i
,
j
d
x
=
d
E
p
ij
,
p
ij
dp
ij
dt
d
x
,
S
S
i
=
1
j
=
1
i
=
1
j
=
1
(8.38)
where
E
=
∂
E
/∂
p
, which, for Gaussian noise, is simply the difference between
p
and
p
. Next we must formulate
dp
/
dt
, which, by the transport equation, is
dp
ij
dt
=
[
β
1
−
β
0
]
d
δ
(
R
θ
i
x
−
s
j
)
d
x
dt
=
[
β
1
−
β
0
]
S
δ
(
R
θ
i
x
−
s
j
)
n
(
x
)
·
v
(
x
)
d
x
,
(8.39)
where
n
is an outward pointing surface normal and
v
(
x
) is the velocity of the
surface at the point
x
. The derivative of
E
data
with respect to surface motion is
therefore
dt
=
[
β
1
−
β
0
]
N
M
E
p
i
,
j
,
p
ij
δ
(
R
θ
i
x
−
s
j
)
n
(
x
)
·
v
(
x
)
d
x
.
dE
data
(8.40)
S
i
=
1
j
=
1
Note that the integral over
d
x
and the
δ
functional serve merely to associate
s
j
in the
i
th scan with the appropriate
x
point. If the samples in each projection are
sufficiently dense, we can approximate the sum over
j
as an integral over the
image domain, and thus for every
x
on the surface there is a mapping back into
the
i
th projection. We denote this point
s
i
(
x
). This gives a closed-form expression
for the derivative of the derivative of
E
data
in terms of the surface velocity,
dt
=
[
β
1
−
β
0
]
N
dE
data
e
i
(
x
)
n
(
x
)
·
v
(
x
)
d
x
,
(8.41)
S
i
=
1
where
e
i
(
x
)
=
E
(
p
(
s
i
(
x
)
,θ
i
)
,
p
(
s
i
(
x
)
,θ
i
)) is the derivative of the error associ-
ated with the point
s
i
(
x
)inthe
i
th projection. The result shown in Eq. (8.41) does
not make any specific assumptions about the surface shape or its representa-
tion. Thus, this equation could be mapped onto any set of shape parameters
by inserting the derivative of a surface point with respect to those parameters.
Of course one would have to compute the surface integral, and methods for
solving such equations on parametric models (in the context of range data) are
described in [96].
For this work we are interested in
free-form
deformations, where each point
on the surface can move independently from the rest. If we let
x
t
represent the
velocity of a point on the surface, the gradient descent surface free-form surface
