Graphics Reference
In-Depth Information
Table
.
.
MAE and MSE in RGB space for the images in Fig.
.
and Fig.
.
Noisy image
AWS reconstruction
Kernel smoothing
local quadratic PS
10
−
2
10
−
2
10
−
2
10
−
2
MAE
3.62
1.91
2.25
1.70
MSE
2.06
10
−
3
8.34
10
−
4
1.12
10
−
3
6.03
10
−
4
standard deviations of σ
.
,
.
and
.
,respectively,wasadded.heresulting
noisyimage,inRGBspace,isshownintheupperletofFig.
.
.heupperrightimage
showsthe reconstruction obtained using ourprocedure,usinga maximalbandwidth
h
max
=
.
is known. he error
variance is estimated fromthe image,taking the spatial correlation into account. he
statistical penalty selected by the propagation condition (
.
) for color images with
spatially independent noise is λ
=
. It is assumed that the spatial correlation ρ
=
.
. his parameter is corrected for the effect of
spatial correlation at each iteration.
Each pixel X
i
in the lower right image contains the value N
i
;thatis,thesumof
the weights defining the local model in X
i
, for the last iteration. We can clearly see
how the algorithm adapts to the structure in the image, effectively using a large local
vicinity of X
i
if the pixel belongs to a large homogeneous region and very small local
models if the pixel X
i
belongs to a very detailed structure.
Finally, we also provide the results from the corresponding nonadaptive kernel
smoother (i.e., with λ
=
.
) for comparison at the lower
let of Fig.
.
. he bandwidth was chosen to provide the minimal mean absolute
error. Table
.
provides the mean absolute error (MAE) and the mean squared error
(MSE) for the three images in Fig.
.
.
Figure
.
provides a detail, with a resolution of
=
and a bandwidth of h
=
pixels, from the noisy
original (let),the AWS reconstruction (center) and the image obtained by nonadap-
tive kernel smoothing. he AWS reconstruction produces a much-enhanced image
at the cost of flattening some smooth areas due to its local constant approximation.
Onthe other hand,the nonadaptive kernel smoother suffersfroma bad compromise
between variance reduction and introduction of blurring, or bias, near edges.
Example: Local Polynomial
Propagation-Separation (PS) Approach
8.4.4
Models (
.
) and (
.
) assume that the gray or color value is locally constant. his
assumption is essentially used in the form of the stochastic penalty s
ij
.heeffectcan
be viewed as a regularization in the sense that in the limit for h
max
,therecon-
structed image is forced to a locally constant gray value or color structure even if the
trueimageislocally smooth. hisisclearlyapparent inthedetailed reconstruction in
thecenterofFig.
.
,wherethesculpturelooksparticularly cartoon-like.Sucheffects
can be avoided if a local polynomial structural assumption is employed. Due to the
increased flexibility of such models, this comes at the price of a decreased sensitivity
to discontinuities.
