Graphics Reference
In-Depth Information
hepropagation-separationapproachfromPolzehlandSpokoiny(
)assumes
that within a homogeneous region containing X
i
=(
i
h
, i
v
)
, i.e., for X
j
U
(
X
i
)
,the
gray value or color Y
j
h
, j
v
can be modeled as
Y
j
h
, j
v
=
θ
(
X
i
)
Ψ
(
j
h
−
i
h
, j
v
−
i
v
)+
ε
j
h
, j
v
,
(
.
)
where the components of Ψ
(
δ
h
, δ
v
)
contain values of basis functions
m
m
ψ
m
,m
(
δ
h
, δ
v
)=(
δ
h
)
(
δ
v
)
(
.
)
for integers m
, m
, m
+
m
p and some polynomial order p. For a given local
model W
(
X
i
)
,estimatesofθ
(
X
i
)
are obtained by local least squares as
θ
B
−
i
(
X
i
)=
j
w
ij
Ψ
(
j
h
−
i
h
, j
v
−
i
v
)
Y
j
h
, j
v
,
(
.
)
with
B
i
=
j
w
ij
Ψ
(
j
h
−
i
h
, j
v
−
i
v
)
Ψ
(
j
h
−
i
h
, j
v
−
i
v
)
(
.
)
heparameters θ
aredefinedwithrespecttoasystemofbasisfunctionscentered
on X
i
. Parameter estimates θ
(
X
i
)
with respect to basis
functions centered at X
i
can be obtained by a linear transformation from θ
(
X
j,i
)
in the local model W
(
X
j
)
,see
Polzehl and Spokoiny (
). At iteration k, a statistical penalty can now be defined
as
X
j
(
)
λ
σ
θ
(
k
−)
θ
(
k
−)
θ
(
k
−)
θ
(
k
−)
(
k
)
ij
s
=
(
X
i
)−
(
X
j,i
B
i
(
X
i
)−
(
X
j,i
)
(
.
)
In a similar way, a memory penalty is introduced as
τ
σ
θ
(
k
)
θ
(
k
−)
θ
(
k
)
θ
(
k
−
(
k
)
ij
B
(
k
)
i
m
=
(
X
i
)−
(
X
i
)
(
X
i
)−
(
X
i
)
(
.
)
where B
i
is constructed like B
i
, employing location weights K
l
(
k
)
ij
.hemainpa-
rameters λ and τ areagainchosenbyapropagationconditionrequiringthefreeprop-
agation of weights in the specified local polynomial model. A detailed description
and discussion of the resulting algorithm and corresponding theoretical results can
befoundinPolzehlandSpokoiny(
).
We use an artificial example to illustrate the behavior of the resulting algorithm.
he let image in Fig.
.
contains gray values
(
l
)
x
y
f
(
x, y
)=
.
+
sign
(
−
)
sin
(
ϕ
)
r
.
+
sin
(
πr
)
r
<
.
with x
=
i
.
−
, y
=
j
.
−
, r
=
x
+
y
and ϕ
=
arcsin
(
x
r
)
in lo-
cations i, j
,...,
.heimageispiecewisesmoothwithsharpdiscontinuities
along diagonals and a discontinuity of varying strength around a circle. he noisy
image on the right of Fig.
.
contains additive white noise with standard deviation
σ
=
.
.
=
