Next: Design-Led Software Strategy for the 2dF Survey Spectrograph
Previous: SPEFO---A Simple, Yet Powerful Program for One-Dimensional Spectra Processing
Table of Contents --- Search ---
PS reprint
Jean-Luc Starck, Eric Pantin
CEA/DSM/DAPNIA, F-91191 Gif-sur-Yvette cedex
parameter, i.e., relative weight between the goodness-of-fit
and the entropy.
Bontekoe et al. (1994) have introduced the concept of Pyramid
Maximum Entropy reconstruction which is a special application of
multi-channel maximum entropy image reconstruction techniques (Gull & Skilling 1991). They consider an image f as a weighted sum of a visible space pyramid of
resolution f=
, i=1,K, which corresponds via a set of Intrinsic
Correlation Functions (ICFs) to a
hidden-space pyramid 
, i=1,K on which the constraint of maximum
entropy is applied. A major difficulty arises when summing the contributions
corresponding to the different channels: the weights appear to be
somewhat arbitrary or at least, difficult to determine theoretically. Another difficulty they encountered lies in the choice of the default constant (model) in each channel.
Trying to advance further, we have reformulated
this idea, using the appropriate mathematical tool to decompose a signal into
channels of spectral bands, the wavelets transform. The problem of
reconstructing the restored image is automatically solved since the inverse
transform is well-known. Introducing the concept of multiscale entropy,
we show that we minimize a functional depending on the desired solution
regularized by minimizing the total amount of information contained at each
resolution. We also use the concept of multiresolution support which leads to a fixed 
for all types of images, removing the
problem of its determination. Finally, we show that this method is very simple
to use since there is no parameter to be determined by the user, and
we give significant examples of deconvolution of blurred astronomical images
showing the power of that method, especially to reconstruct weak
structures and strong ones at the same time.
The concept of entropy following Shannon's or Skilling and Gull's definition
is a global quantity calculated on the whole image I.
It is not matched to quantify the distribution of the information at different
scales of resolution. Therefore, we have proposed the concept of
multiscale entropy (Pantin & Starck 1995) of a set of
wavelet coefficients {
} by
where 
is the standard deviation of the noise in the image I,
are the wavelet coefficients, and 
is the standard deviation
of the noise at scale j.
The multiscale entropy is the addition of the entropy at each scale.
We take the absolute value of 
in that definition because the values of 
can be positive or negative and a negative signal contains also some information in
the wavelet transform. The advantage of such a definition entropy is
the fact we can use previous works concerning the wavelet transform and
image restoration (Starck et al. 1995). The noise
behavior has been studied in the wavelet transform
and we can estimate the standard deviation of the noise 
at the scale j. These estimations can be naturally introduced in our
models 
:
.
The model 
at the scale j represents
the value taken by a wavelet coefficient in the absence of any relevant
signal and in practice, it must be a value small compared to
any significant signal value.
Following Gull and Skilling procedure, we take 
as a fraction of the
noise (
).
In this application, we use the discrete à trous algorithm
(described in (Starck et al. 1995)) for its simplicity to use.
An image I(x,y) is decomposed into 
(x,y) j=1,
scales (where
is the total number of wavelet scales) and a smooth image
(x,y) and we can write 
.
If the previous definition (see Equation 1) is used for the multi-scale entropy, the regularization acts on the whole image. We want to fully reconstruct significant structures, without imposing strong regularization, while eliminating efficiently the noise. Thus the introduction of the multi-support (see Starck et al. 1995 for the support definition and estimation) in another definition of the multi-scale entropy leads to a functional that answers these requirements:
The A function of the scale j and the pixels 
is
, i.e., the reciprocal of the multi-support M.
In order to avoid some discontinuities in the A function created by
such a coarse
threshold of 3 
, one may possibly impose some
smoothness by convolving
it with a B-spline function with a FWHM varying with the scale j.
The degree of regularization will be determined at each scale j, and at
each location 
, by the value of the function A(j,x,y): If A(j,x,y) has a value
near 1 then we have strong regularization; and it is weak when A is around 0.
The entropy 
measures the amount of information only at scales
and in areas where we have a low signal-to-noise ratio.
We will show in the next section how these notions can be taken together
to yield efficient methods for filtering and image deconvolution.
We assume that the blurring process of an image is linear. In our tests, the PSF was space invariant but the method can be extended to space variant PSFs.
As in the ME method, we will minimize a functional of O, but considering an image as a pyramid of different scales of resolution in which we try to maximize its contribution to the multiscale entropy. The functional to minimize is
The solution can be found by computing the gradient
where 
, and 
is the wavelet function corresponding to the à trous algorithm. 
are
the wavelet coefficients of O at scale j,
and performing the following iterative schema
Figure 1: Raw image (left) and deconvolved one by Lucy method (right).
The contours superimposed correspond to a flux divided by 3 at each subsequent one.
Figure 1: (left) 136 Kb,
Figure 1: (right) 136 Kb
We have tested the MEM multiresolution method on astronomical
images obtained with an mid-infrared camera: TIMMI placed on the 3.6 ESO
telescope (Chile).
The object studied is the 
Pictoris dust disk (see Figure 1 at left).
The image was obtained by integrating 5h on-source. The raw image has
a peak signal to noise ratio of 200.
Since the image is strongly blurred by a combination
of seeing, diffraction (0.7 arcsec on a 3m class telescope) and additive
gaussian noise, we need to deconvolve them to get the best information
on this object, i.e., the exact profile and thickness of the disk and compare
it to models of thermal dust emission. The deconvolved image (Figure 1 at right)
shows that the disk is extended at 10 
m and asymmetrical (the right
side is more extended than the left one). The deconvolved
image using the multiresolution MEM proves to be efficient to regularize
and leads to a good reconstruction of the faint structures of the dust disk.
Compared to the classical MEM, our method has a fixed 
parameter and there is no need to determine it: it is the same
for every image.
Furthermore, this new method is flux-conservative and thus reliable
photometry can be done on the deconvolved image.
In Bontekoe et al. (1994), it was noticed that the ``models'' in the multi-channel
MEM deconvolution should be linked to a physical quantity.
We have shown here that this is the case since it is a fraction of the
standard deviation of the noise at a given scale of resolution.
Bontekoe et al. have opened a new way of thinking in terms of multiresolution
decomposition, but they didn't use the appropriate mathematical tool which
is the wavelets decomposition. Using such an approach, we have proven that many problems they
encountered are naturally solved, especially the model and the 
estimation.
Gull, S. F., & Skilling, J. 1991, MEMSYS5 User's Manual
Pantin, E., & Starck, J. L. 1996, A&A, submitted
Starck, J. L., Murtagh, F., & Bijaoui, A. 1995, in CVIP: Graphical Models and Image Processing, 57, 5, 420
