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reflects all of the information about the structure and determines the weights. he
influence of the location penalty has almost vanished.
What we observe during the iteration process is the unrestricted propagation of
weights within homogeneous regions. Two regions with different values of the pa-
rameter are separated as values of the statistical penalty
s
ij
increase with decreasing
variance oftheestimates θ
i
and θ
j
and alarge enoughcontrast
.he
iteration k atwhichthisoccursdependsonthesizesofthehomogeneousregions,i.e.,
the potential variance reduction, and the contrast.
he upper right plot in Fig.
.
additionally displays the intermediate estimates
θ
(
k
)
, k
θ
(
X
i
)−
θ
(
X
j
)
=
,
,
corresponding to the weighting schemes illustrated.
Examples and Applications
8.4
Wenowprovideaseriesofexamplesforadaptiveweightssmoothinginvariousse-
tups.
Application 1: Adaptive Edge-Preserving Smoothing
in 3-D
8.4.1
he algorithm described in Sect.
.
.
is essentially dimension-free. It can easily be
applied to reconstruct
-D and
-D images. We illustrate this using a
-D MR image
of a head. heupperlet image in Fig.
.
shows the
th slice of the noisy data cube,
consisting of
voxels. he image is modeled as
Y
i
=
θ
(
X
i
)+
ε
i
,
(
.
)
where X
i
arecoordinatesonthe
-Dgridandtheerrorsε
i
are again assumed to
be i.i.d. Gaussian with unknown variance σ
. he parameter of interest θ
de-
scribes a tissue-dependent underlying gray value at voxel X
i
.hereisspecialinterest
inusing these imagestoidentify tissue borders.Denoising, preferably using an edge-
preserving or edge-enhancing filter, is a prerequisite step here.
WeapplytheAWSalgorithmfromSect.
.
.
usingamaximalbandwidth h
max
(
X
i
)
.
he error variance is estimated from the data. he default value of λ provided by
condition (
.
) for smoothing in
-D with Gaussian errors is λ
=
.
. he upper
right image provides the resulting reconstruction. Note that in the smoothed im-
age the noise is removed while the detailed structure corresponding to tissue bor-
ders is preserved. Some deterioration of the image is caused by the structural as-
sumption of a local constant model. his leads to some flattening where θ
=
(
X
i
)
is
smooth.
In the bottom row of Fig.
.
, we provide the absolute values obtained ater apply-
ing a Laplacian filter to the original noisy image and to the reconstruction obtained
by AWS, respectively. We observe an enhancement of the tissue borders.
