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Posterior shape models.
[trochlear dysplasia]
We
present
a
method
to
compute
the
conditional
distribution
of
a
statistical
shape
model
given
partial
data
.
The
result
is
a
"
posterior
shape
model
"
,
which
is
again
a
statistical
shape
model
of
the
same
form
as
the
original
model
.
This
allows
its
direct
use
in
the
variety
of
algorithms
that
include
prior
knowledge
about
the
variability
of
a
class
of
shapes
with
a
statistical
shape
model
.
Posterior
shape
models
then
provide
a
statistically
sound
yet
easy
method
to
integrate
partial
data
into
these
algorithms
.
Usually
,
shape
models
represent
a
complete
organ
,
for
instance
in
our
experiments
the
femur
bone
,
modeled
by
a
multivariate
normal
distribution
.
But
because
in
many
application
certain
parts
of
the
shape
are
known
a
priori
,
it
is
of
great
interest
to
model
the
posterior
distribution
of
the
whole
shape
given
the
known
parts
.
These
could
be
isolated
landmark
points
or
larger
portions
of
the
shape
,
like
the
healthy
part
of
a
pathological
or
damaged
organ
.
However
,
because
for
most
shape
models
the
dimensionality
of
the
data
is
much
higher
than
the
number
of
examples
,
the
normal
distribution
is
singular
,
and
the
conditional
distribution
not
readily
available
.
In
this
paper
,
we
present
two
main
contributions
:
First
,
we
show
how
the
posterior
model
can
be
efficiently
computed
as
a
statistical
shape
model
in
standard
form
and
used
in
any
shape
model
algorithm
.
We
complement
this
paper
with
a
freely
available
implementation
of
our
algorithms
.
Second
,
we
show
that
most
common
approaches
put
forth
in
the
literature
to
overcome
this
are
equivalent
to
probabilistic
principal
component
analysis
(
PPCA
)
,
and
Gaussian
Process
regression
.
To
illustrate
the
use
of
posterior
shape
models
,
we
apply
them
on
two
problems
from
medical
image
analysis
:
model
-based
image
segmentation
incorporating
prior
knowledge
from
landmarks
,
and
the
prediction
of
anatomically
correct
knee
shapes
for
trochlear
dysplasia
patients
,
which
constitutes
a
novel
medical
application
.
Our
experiments
confirm
that
the
use
of
conditional
shape
models
for
image
segmentation
improves
the
overall
segmentation
accuracy
and
robustness
.
Diseases
Validation
Diseases presenting
"posterior shape model""
symptom
trochlear dysplasia
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