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Gastaud, R., Starck, J.-L., & Pierre, M. 1999, in ASP Conf. Ser., Vol. 172, Astronomical Data Analysis Software and Systems VIII, eds. D. M. Mehringer, R. L. Plante, & D. A. Roberts (San Francisco: ASP), 399
Background Fluctuation Analysis from the Multiscale Entropy
R. Gastaud, J.-L. Starck, M. Pierre
DAPNIA/SEI-SAP, CEA/SACLAY, F-91191 Gif sur Yvette Cedex
Abstract:
The entropy measurement has been proposed for image restoration
several years ago in astronomy. Different entropies have been used for
this purpose. The multiscale entropy has the advantage to introduce a
noise modeling, and the concept of information at different level of
resolution. It can be used for tracking the presence of signal in a
noisy data set. We show in this paper that it can also be used for
background fluctuations analysis. A range of examples illustrates the
results. We finally apply this method on X-ray simulations (XMM-EPIC)
of diffuse large scale structure emission in order to discriminate
different cosmological models.
The concept of entropy has been introduced in astronomy for 40 years
especially for deconvolution regularization.
Many entropy definitions have been proposed in the past, but they do
not take into account the spatial correlation of the image. Some
authors tried a pyramidal approach to take a spatial correlation into
account (Bontekoe, Koper, & Kester 1994), but this adds complexity. The correct way to
do it is to compute the entropy in the wavelet space (Starck, Murtagh,
& Gastaud 1998):
![\begin{displaymath}
H(X) = \sum_{j=1}^{l} \sum_{k=1}^{N_j} h(w_{j,k})
\end{displaymath}](img2.gif) |
(1) |
with
. Here h stands for the entropy
(or information) relative to a wavelet coefficient, and H for the
multiscale entropy (MSE) of a signal or an image. For Gaussian noise,
we get
![\begin{displaymath}
H(X) = \sum_{j=1}^{l} \sum_{k=1}^{N_j} \frac{w_{j,k}^2}{2 \sigma_j^2}
\end{displaymath}](img4.gif) |
(2) |
where
is the noise at scale j. We see that the
information is proportional to the energy of the wavelet coefficients.
This definition of entropy verify the following criteria:
- The information in a flat signal is zero.
- The amount of information in a signal is independent of the
background.
- The amount of information is dependent on the noise. A given
signal Y (
) does not furnish the same information if
the noise is high or small.
- The entropy must work in the same way for a pixel which
has a value
(B being the background), and
for a pixel which has a value
.
- The amount of information is dependent on the spatial
correlation in the signal. If a signal S presents large homogeneous
features above the noise, it contains less information.
This definition has been used with success for restoration (Starck &
Murtagh 1999).
Previous equation is used to compute an average over the pixels:
![$\displaystyle E(j) = \frac{1}{N}\sum_{k=1}^{N} h(w_j)$](img9.gif) |
|
|
(3) |
E(j) gives the mean entropy at the scale j. From the mean entropy
vector, we have statistical informations on each scale separately.
Having a noise model, we are able to calculate (generally from
simulations) the mean entropy vector
resulting from
a pure noise. Then we define the normalized mean entropy vector by
![$\displaystyle E_n(j) = \frac{ E(j) }{ E_{\rm noise}(j) }$](img11.gif) |
|
|
(4) |
Now, this function of the scale can be used to analyse the background
fluctuations.
Five simulated images were created by adding 400, 200, 100, 50, 0 sources
to a noisy image. The background is a Gaussian noise of amplitude 1.
The sources are given by:
![$\displaystyle I(x,y) = I_0\exp\left(\frac{(x-a)^2+(y-b)^2}{2\sigma^2}\right)$](img12.gif) |
|
|
(5) |
with I0 = 1 and
. Defining the signal to noise ratio
(SNR) as the ratio between the standard deviation in the smallest box
which contains at least 90% of the flux of the source, and the noise
standard deviation, we have a SNR equal to 0.25: the sources are not
detectable. Several tests of this non-detectability have been made:
test in direct space (i.e., is this pixel above the Gaussian
noise?), correlation in direct space (i.e., is this sub-image a psf?),
wavelet filter: the routine mrfilter (MR/1
software). The
sources are really hard to detect: genuine sources are lost in false
detections (tens against ten thousands). The mean entropy function of
the scale has been computed for these images. The error is estimated
by drawing ten times each image. Figure 1 clearly
shows that the sources increase the entropy of the image. But it is
obvious that the positions of these sources remain unknown.
Figure 1:
Mean entropy versus the scale of 5 simulated images
containing undetect able sources and noise. Each curve corresponds to
the multiscale of one image. From top to bottom, the image contains
respectively 400, 200, 100, 50, and 0 sources, (which is confused with
the x-axis).
![\begin{figure}
\centerline {
\vbox{
\epsfig{figure=rgastaud1.eps,bbllx=3.5cm,bblly=13cm,bburx=19.5cm,bbury=25.5cm,width=12cm,height=6cm}
}}\end{figure}](img15.gif) |
The new satellite XMM (to be launched in year 2000) will open a new
era in the study of the sky at the energy from 0.1 to 10keV. This
will enable astronomers to detect large scale structures (hundreds of
megaparsecs) and help to discriminate between different cosmological
models. The problem is to detect the hot gas in filaments which are
expected to be between galaxy clusters. The emission of these
filaments, predicted from hydrodynamical simulation models, is very
faint and diffuse. On the image of filaments will be superimposed
very numerous background and foreground quasars and clusters of
galaxies. We try the multi-scale entropy method to detect the
existence of filaments. Greg Bryan, from MIT, kindly provided
simulation of the sky at different cosmological distances
(redshift). From the density and the temperature cubes given by the
simulations, a photons flux per simulation cell for the bandwidth from
0.4keV to 4keV is computed. The image of a filament located at
z=0.5 was computed with a pixel field of view of 4 arcsecond. The
size of the image is 512 x 512 pixels. This image contains both
clusters and filaments. In addition the point-like images of quasars
are simulated using the law log(N)-log(S) (Hasinger et al. 1998). The
two images were summed, then multiplied by the integration time, here
400 kilosecond. Poisson noise, PSF blurring, vignetting are added.
Two sets of images have been simulated, with and without the
filaments. As previously, the mean entropy function of the scale has
been computed. The difference between the two entropies, for the
image with filament and without filament, is plotted in
Fig. 2. The error bars e were estimated from from
five realisations of the images, for both image with(
) and
without (
) filaments
![$\displaystyle e =
2.5\,\sqrt{\sigma_1^2 + \sigma_2^2}$](img18.gif) |
|
|
(6) |
This difference is significant at the
level (98.76%)
Figure 2:
difference of mean entropy for the images with and without filament,
error bar
.
![\begin{figure}
\centerline {
\vbox{
\epsfig{figure=rgastaud2.eps,bbllx=2.5cm,bblly=13.1cm,bburx=19.5cm,bbury=25.6cm,width=\textwidth,height=6cm}
}}\end{figure}](img19.gif) |
We have seen that information must be measured from the transformed
data, and not from the data itself. The transformation chosen was a
non-orthogonal wavelet transform which is suited to astronomical
images. The mean entropy is a good indicator of the presence of
undetected sources, or faint extended structures. It is well suited to
background fluctuations analysis.
Acknowledgments
We want to thank Greg Bryan who kindly provided us
with hydrodynamical simulated data.
References
Bontekoe, T. R., Koper, E., & Kester, D. J. M.
1994, A&A, 294, 1037
Hasinger, G., Burg, R., Giacconi, R., Schmidt, M.,
Trümper, J., & Zamorani, G. 1998, A&A, 329, 482
Starck, J.-L., Murtagh, F., & Gastaud, R. 1998,
IEEE Trans. on Circuits and Systems, 45, 1118
Starck, J.-L. & Murtagh, F. 1999, Signal Processing, in
press
© Copyright 1999 Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, California 94112, USA
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