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(based on sparsity in a multiresolution representation). The temporal filter-
ing can either be causal or non-causal. In the former case, only past frames
are used for filtering. In the latter case, future frames are needed, which can
be achieved by introducing a temporal delay.
1
2.
Spatially local methods with recursive temporal filtering
[9, 10, 14, 15]: these
methods rely on recursive filtering that takes advantage of the temporal
correlations between subsequent frames. Because usually, first order (causal)
infinite impulse response filters are used and no temporal delay is required.
3.
Spatially and temporally non-local methods
[12, 13]: these methods take ad-
vantage of repetitive structures that occur both spatially and temporally.
Because of computation time and memory restrictions, in practice these
methods make use of a
search window
(this is a spatio-temporal window
in which similar patches are being searched for). By the practical restric-
tions, the methods actually fall under the first class, however we expect that
by more effcient parallel computing architectures and larger RAM memory
the non-locality of these methods will further be extended in the future.
One popular filter that makes use of the repetitive character of structures in
video sequences and hence belongs to the third class, is the NLMeans filter [16].
Suppose that an unknown video signal
X
(
p
) is corrupted by an additive noise
process
V
(
p
), resulting in the observed video signal:
Y
(
p
)=
X
(
p
)+
V
(
p
)
(1)
p
=[
p
x
,p
y
,p
t
] is the spatio-temporal position within the video sequence.
X
(
p
),
Here,
Z
3
onto the RGB
Y
(
p
) and
V
(
p
) are functions that map values from
R
3
. The NLMeans video filter estimates the denoised value of
color space
X
(
p
)
as the weighted average of all pixel intensities in the video sequence:
X
(
p
)=
q
∈δ
w
(
p
p
+
q
)
Y
(
p
+
q
)
q
∈δ
w
(
p
,
ˆ
,
(2)
,
p
+
q
)
where
q
=[
q
x
,q
y
,q
t
] and where the weights
w
(
p
,
p
+
q
) depend on the similarity
of patches centered at positions
.
δ
is a three dimensional search
window in which similar patches are searched for. For simplicity of the notation,
we assume that
p
and
p
+
q
Y
(
p
+
q
) is everywhere defined in (2). In practice, we make use
of boundary extension techniques (e.g. mirroring) near the image boundaries.
Because of the high computational complexity of the NLMeans algorithm (the
complexity is quadratic in the number of pixels in the video sequence and linear
in the patch size) and because of the fact that the original NLMeans method
performed somewhat inferior compared to other (local) state-of-the-art denoising
method, many improvements have been proposed by different researchers. Some
of these improvements are better similarity measures [17,18,19], adaptive patch
sizes [20], and algorithmic acceleration techniques [4, 19, 21, 22].
1
A temporal delay is not desirable for certain applications, such as video communi-
cation.
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