Biomedical Engineering Reference
In-Depth Information
5.2.2.4
Effective Approximation of Currents Using Matching
Pursuit Algorithm
Representing shapes using currents gives a nice theoretical framework that allows
us to compute simple statistics such as mean and principal modes. However, the
complexity of the distance computation is quadratic in the number of Dirac delta
currents used to represent the shape. This number can be quite high if we take highly
detailed surface meshes, even though it adds nothing for the comparison of surfaces
at a given scale
λ
W
. Therefore, in order to remain computationally as efficient as
possible, we require a low number of degrees of freedom representation of currents
that retains the information needed for the scale of the analysis. This is the essence
of the matching pursuit algorithm as described in [
26
].
In brief, the matching pursuit algorithm is a greedy approach to find the best
projection of multi-dimensional data on an over-complete family of basis functions
[
16
]. In our case, we want to find an approximation of the current
T
that solves
L
−
W
(
T
)=
γ
, for a given vector field
γ
∈
W
. This amounts to finding
N
points
(
x
k
)
)=
k
=1
δ
α
k
x
k
and vectors
(
α
k
)
such that the current
Π
(
T
is as close as possible
to
T
. If the points are known, then
Π
(
T
)
is the orthogonal projection of
T
onto
δ
x
k
;
R
3
.The
Span
(
q
=1
,
2
,
3
,k
=1
...N
)
,where
q
is the canonical basis of
T,δ
x
k
W
∗
=
,δ
x
k
W
∗
orthogonality condition:
Π
(
T
)
leads to a linear set of
3
N
equations
N
(
K
W
(
x
i
,x
p
)
α
p
)
k
=
γ
(
x
i
)
k
,
(5.1)
p
=1
which can be solved iteratively over the continuous space of Dirac delta currents to
find the point positions
as well as their associated momenta and the residual
vector field. Additionally, we can also sample the vector field on a grid
Λ
. Applying
the matching pursuit in the discrete case constrains the estimated momenta to lie on
the nodes of the grid which forces the estimated current to belong to a discrete set of
currents
W
Λ
(
x
k
)
. This simplifies the problem in both complexity as well as computation
time. Figure
5.4
shows a compressed representation of the right ventricle.
In this approximation algorithm, the interest is not to fix the number of points
N
and to look for the optimal representation (this cannot be achieved properly with
a greedy algorithm), but rather to fix an error threshold (typically 1-5 % of the
norm of the original surface) and to stop the approximation whenever the number of
points is sufficient to guaranty an error below that threshold. This typically results
in compressions of 80 % with less than a few percent of error for one surface, but
compression by order of magnitudes for the mean of several currents.
5.2.3
An Algorithm for Surface Registration Using Currents
Having defined a non-parametric representation of surfaces, we now require an
algorithm to register one surface to another. Here, the term registration is meant
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