Digital Signal Processing Reference
In-Depth Information
(2
X
[
k
]
1), but this
is only valid for a sufficiently large number of counts
X
[
k
]. The necessary average
number of counts is about 20 if bias is to be avoided. Note that errors related to
small values carry the risk of removing real objects, but not of amplifying noise. For
Poisson intensities under this threshold acceptable number of counts, the Anscombe
VST loses control over the bias. In this case, an alternative approach to variance
stabilization is needed. Solutions based on the Haar transform and the MS-VST will
be discussed in Sections 6.5 and 6.6.
It was shown by Murtagh et al. (1995) that
Y
S
[
k
]
∼N
/
g
0
,
6.5 POISSON NOISE AND THE HAAR TRANSFORM
Several authors (Kolaczyk 1997; Kolaczyk and Dixon 2000; Timmermann and
Nowak 1999; Nowak and Baraniuk 1999; Bijaoui and Jammal 2001; Fryzlewicz and
Nason 2004) have suggested independently that the Haar wavelet transform is very
well suited for treating data with Poisson noise. Since a Haar wavelet coefficient is
just the difference between two random variables following a Poisson distribution,
it is easier to derive mathematical tools to remove the Poisson noise than with any
other wavelet method.
An isotropic wavelet transform seems more adapted to more regular data (such
as in astronomical or biomedical images). However, there is a trade-off to be made
between an algorithm that optimally represents the information and another that
furnishes a reliable way to treat the noise. The approach used for noise removal
differs depending on the authors. In Nowak and Baraniuk (1999), a type of Wiener
filter was implemented. Timmermann and Nowak (1999) used a Bayesian approach
with an a priori model on the original signal, and Kolaczyk and Dixon (2000) and
Bijaoui and Jammal (2001) derived different thresholds resulting from the pdf of
the wavelet coefficients. The Fisz transform (Fryzlewicz and Nason 2004) is a Haar
wavelet transform-based VST used to “Gaussianize” the noise. Then, the standard
wavelet thresholding can be applied to the transformed signal. After the threshold-
ing, the inverse Fisz transform has to be applied. Poisson denoising has also been
formulated as a penalized maximum likelihood (ML) estimation problem (Sardy
et al. 2004; Willett and Nowak 2007) within wavelet, wedgelet, and platelet dic-
tionaries. Wedgelet (platelet-) based methods are generally more effective than
wavelet-based estimators in denoising piecewise constant (smooth) images with
smooth contours.
6.5.1 Haar Coefficients of Poisson Noise
Kolaczyk (1997) proposed the Haar transform for gamma-ray burst detection in
1-D signals and extended his method to 2-D images (Kolaczyk and Dixon 2000). The
reason why the Haar transform (see Section 2.3.3) is used is essentially simplicity of
the pdf of the Haar coefficients, hence providing a resilient mathematical tool for
noise removal. Indeed, a Haar wavelet coefficient of Poisson counts can be viewed
as the difference of two independent Poisson random variables
Y
[
k
]
∼ P
(
X
[
k
]) and
Y
[
k
]
(
X
[
k
]), whose pdf is
∼ P
X
[
k
])
−
n
/
2
I
n
(2
X
[
k
]
X
[
k
])
Y
[
k
]
e
−
(
X
[
k
]
+
X
[
k
])
(
X
[
k
]
pdf(
Y
[
k
]
−
=
n
)
=
/
,
