Biomedical Engineering Reference
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choice of prior is intimately related to the choice of surface representation and
the specific application, but is independent of the formulation that describes the
relationship between the estimate and the data, given in Eq. (8.37).
Because the data is noisy and incomplete it is useful to introduce a simple,
low-level prior on the surface estimate. We therefore use a prior that penalizes
surface area, which introduces a second-order smoothing term in the surface
motion. That term introduces a free parameter C , which controls the relative
influence of the smoothing term. The general question of how best to smooth
surfaces remains an important, open question. However, if we restrict ourselves
to curvature-based geometric flows, there are several reasonable options in
the literature [7, 31, 97]. The following subsection, which describes the surface
representation used for our application, gives a more precise description of our
smoothing methods.
8.6.3 Surface Representation and Prior
Our goal is to build an algorithm that applies to a wide range of poten-
tially complicated shapes with arbitrary topologies—topologies that could
change as the shapes deform to fit the data. For this reason, we have imple-
mented the free-form deformation given in Eq. (8.42) with an implicit level set
representation.
Substituting the expression for d x / dt (from Eqs. (8.45) and (8.46)) into the
d s / dt term of the level set equation (Eq. (8.4a)), and recalling that n =∇ φ/ |∇ φ | ,
gives
M
∂φ
t = −|∇ φ |
e i ( x ) + C κ ( x )
(8.47)
,
i = 1
where κ represents the effect of the prior, which is assumed to be in the normal
direction.
The prior is introduced as a curvature-based smoothing on the level set
surfaces. Thus, every level set moves according to a weighted combination of
the principle curvatures, k 1 and k 2 , at each point. This point-wise motion is in the
direction of the surface normal. For instance, the mean curvature, widely used
for surface smoothing, is H = ( k 1 + k 2 ) / 2. Several authors have proposed using
Gaussian curvature K = k 1 k 2 or functions thereof [97]. Recently [98] proposed
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