Biomedical Engineering Reference
In-Depth Information
φ ( 1)
t
v t φ t =
V ,
v t
where
. V is the norm of the RKHS of the velocity fields at the identity transforma-
tion. The energy of the deformation trajectory is thus E
φ t )= 1
φ ( 1)
t
V
(
0
v t
dt ,
and the optimal curves joining φ 0 =
Id to φ 1 are geodesics in the space of
diffeomorphisms. Since optimal transformations are geodesics, we know that they
are completely determined by their initial value, here the velocity field v 0 (
.As
the velocity fields are most often non-zero everywhere, it is usually more efficient
to parameterize the deformation with the initial momentum μ 0 =
x
)
K ( 1)
V
v 0 ,which
is in most cases very sparse. Notice that the momentum is an element of V ,the
dual space of V . Thus, it is a current, just as the ones we used to represent surfaces.
However, even if the mathematical structures are identical, the objects “velocity
fields” and “surfaces” are quite different and the kernels K V and K W are usually
very different. Denoting φS the action of the transformation φ on the source object
S , the registration criterion to the target object T is thus
1
φ ( 1)
t
) 2 +
V dt.
C
(
φ
)=
(
φ 1 S, T
v t
dist
0
For our surface registration problem, the objects are surfaces represented by their
associated currents
S = i δ α k
and the distance is taken in the space
of currents W . In this setting, one can show that the optimal initial momentum
of the deformation has the same point-wise support as
and
T
x k
: μ 0 ( x )= i δ β k
(i.e.
v 0 ( x )= i K V ( x, x k ) β k ) for some set of vectors β k .Thismeansthatweareleft
with a finite dimensional optimization problem [ 7 , 39 ]. Notice that the number of
unknowns is directly linked to the number of Diracs in the source surface
S
x k
,which
is usually the atlas. This emphasizes the interest of compressing the atlas before
registering it to target surfaces.
S
5.2.4
Building an Unbiased Atlas
When we have more than two surfaces, one has to choose a reference (called the
atlas) to compare all the others consistently. Choosing this atlas randomly generally
leads to biases in the forthcoming statistical analysis because the atlas shape might
be far from the mean shape. The atlas is called unbiased if it is centered with
respect to all the surfaces, i.e. if the mean deformation from the atlas to the observed
surfaced is close to zero.
For building the optimally centered atlas, we took the 'forward' strategy that
models the set of surfaces as the deformation of an unknown 'ideal' atlas plus some
residuals [ 24 - 26 ]. For the patient number i , this can be expressed as
φ i T
T i =
+
ε i ,
(5.2)
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