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Astronomical Data Analysis Software and Systems IV
ASP Conference Series, Vol. 77, 1995
R. A. Shaw, H. E. Payne, and J. J. E. Hayes, eds.
Astronomical Image Compression Using the Pyramidal
Median Transform
J.­L. Starck
CEA, DSM/DAPNIA, CEA­Saclay, F­91191 Gif­sur­Yvette Cedex,
France
F. Murtagh 1
ST­ECF, ESO, Karl­Schwarzschild­Str. 2, D­85748 Garching, Germany
M. Louys
LSIT, ENSP, 7 rue de l'Universit'e, F­67084 Strasbourg Cedex, France
Abstract. We describe image compression based on the pyramidal me­
dian transform, introduced as an alternative to the use of a wavelet trans­
form. The use of a multiresolution support allows ``protection'' of as­
tronomical objects in the image. Non­support parts of the images are
subjected to noise suppression. Experimental applications are presented.
1. Introduction
Image compression is required for preview functionality in large image databases
(e.g., HST archive), for linking image and catalog information in interactive sky
atlases (e.g., Aladin), and for image data transmission, where more global views
are communicated to the user, followed by more detail if desired. We describe
an approach to astronomical image compression through noise removal. Noise
is determined on the basis of the image's assumed stochastic properties. This
approach is quite similar to the wavelet transform­based hcompress approach.
We begin by explaining why transforms other than the wavelet transform are
important for astronomical image compression.
2. Wavelet Transform and Image Compression
Practical problems related to the use of the wavelet transform include:
Negative Values. By definition, the wavelet coefficient mean is null. Every
time we have a positive structure at a scale, we have negative values sur­
rounding it. These negative values often create artifacts during the restora­
tion process, or complicate the analysis. For instance, if we threshold small
values (noise, non­significant structures, etc.) in the wavelet transform,
1 Affiliated to Astrophys. Div., Space Sci. Dept., ESA)
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and then reconstruct the image at full resolution, the structure's flux will
be modified.
Point Objects. We often have bright point objects in astronomical images
(stars, cosmic ray hits, etc.), and the convolution of a Dirac function with
the wavelet transform is equal to the wavelet transform. So at each scale,
and at each point source, we will have the wavelet. Cosmic rays can pollute
all the scales of the wavelet transform.
This leads us to develop other multiresolution tools which we now present.
3. Pyramidal Median Transform
3.1. Multiresolution Median Transform
The median transform is nonlinear, and offers advantages for robust smooth­
ing (i.e., the effects of outlier pixel values are mitigated). The multiresolution
median transform consists of a series of smoothings of the input image, with suc­
cessively broader kernels. Each successive smoothing provides a new resolution
scale.
The multiresolution coefficient values constructed by differencing images at
successive resolution scales are not necessarily of zero mean, and so the potential
artifact­creation difficulties related to this aspect of wavelet transforms do not
arise. For integer input image values, this transform can be carried out in integer
arithmetic only, which may lead to computational savings.
3.2. Pyramidal Median Transform
Computational requirements of the multiresolution median transform are high,
and these can be reduced by decimation: one pixel out of two is retained at
each scale. In the Pyramidal Median Transform (PMT), the kernel or mask
used to obtain the succession of resolution scales remains the same at each level.
The image itself, to which this kernel is applied, becomes smaller. While this
algorithm aids computationally, the reconstruction formula for the input image
is no longer immediate. Instead, an algorithm based on B­spline interpolation
can be used for reconstruction.
An iterative scheme can be proposed for reconstructing an image, based on
pyramidal multi­median transform coefficients. Alternatively, the PMT algo­
rithm, itself, can be enhanced to allow for better estimates of coefficient values,
yielding an Iterative Pyramidal Median Transform.
4. PMT and Image Compression
The principle of the method is to select the information we want to keep, by using
the PMT, and to code this information without any loss. Thus the first phase
searches for the minimum set of quantized multiresolution coefficients which
produce an image of ``high quality.'' The quality is evidently subjective, and we
will define by this term an image for which (1) there is no visual artifact in the
decompressed image, and (2) the residual (original image minus decompressed

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image) does not contain any structure. Lost information cannot be recovered,
so if we do not accept any loss, we have to compress what we take as noise, too,
and the compression ratio will be low (only 3 or 4).
The method employed involves the following sequence of operations:
1. Determination of the multiresolution support.
2. Determination of the quantized multiresolution coefficients which gives the
filtered image.
3. Coding of each resolution level using the Huang­Bijaoui method (Huang
& Bijaoui 1991). This consists of quadtree­coding each image, followed by
Huffman­coding the quadtree representation. There is no information lost
during this phase.
4. Compression of the noise, if this is desired.
5. Decompression consists of reconstituting the noise­filtered image (plus the
compressed noise, if this was specified).
Note that we can reconstruct an image at a given resolution without having to
decode the entire compressed file.
5. Example
A simulated HST WF/PC (pre­refurbishment) stellar field image described in
Hanisch (1993) was used. The image dimensions were 256 \Theta 256. Here we used
the aberrated, noisy image. With default options for the approach described in
this article (pcomp/pdecomp; i.e., 5 iterations, 3oe thresholding, 4 multiresolution
scales, and no conservation of the noise) a compressed image with 6643 bytes
was obtained from the original image of 268,800 bytes. This is a compressed
image equal to 2.5% of the original. Even with I2 storage of the input image,
we have compression to about 5%. The total intensity dropped from 412,998 to
412,980 in compressing and decompressing, i.e., a loss rate of 0.0044%.
Using the known coordinate positions of the 470 stars in this image, we
obtained the intensities at these positions in the reconstructed image and com­
pared magnitudes in the reconstructed image with magnitudes in the input im­
age. There was reasonable fidelity over about 8 magnitudes. For fainter objects,
the noise filtering causes greater difficulty, as one would expect.
6. Conclusion
The approach described here works well in practice. Further experiments are
described in the full paper. Work comparing the approach described in this paper
with other well­known astronomical image compression procedures is continuing.

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Figure 1. Difference between original and decompressed image, using
a stellar field.
References
Hanisch, R., ed. 1993, Restoration---Newsletter of ST ScI's Image Restoration
Project (Baltimore, Space Telescope Science Institute)
Huang, L., & Bijaoui, A. 1991, Experimental Astronomy, 1, 311