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PS reprint
Yihua Yan, Bo Peng, Xizhen Zhang
Beijing Astronomical Observatory, Chinese Academy of Sciences,
Beijing 100080, China
By a wavelet decomposition, any arbitrary function is expanded into smooth localized contributions labeled by scale and position parameters (Combes et al. 1989). This is very helpful in information extraction because, in general, information is contained on both sides of the Fourier transform. Thus it is very useful and important in Radio Astronomy to discriminate noise from signals, and some efforts have been recently paid to the synthesis imaging and data processing (Yan 1993; Starck et al. 1994).
A fundamental result in aperture synthesis is the existence of a Fourier transform relationship between the sky brightness I, and the visibility V observed with an interferometer (Perley, Schwab, & Bridle 1989),
where S is the whole sky, and i is imaginary unit. Since only a finite
number of noisy samples of V are measured in practice,
itself cannot be recovered directly. Therefore an important issue is
how to recover the I image as close to the `true' distribution as
possible. A blurred image 
, or the so-called `dirty map', can be obtained
by,
where 
is the sampling function. By considering Equation (1), we further get,
where 
is an expected estimation of 
, or what we could obtain
in the noise-free case by the aperture synthesis, 
are errors, `
' denotes convolution, and B is the so-called `dirty beam'
(or, point spread function) obtained by the Fourier transform of 
.
One way to get 
from Equation (3) when the observation of 
contains
errors is by the `CLEAN' algorithm or its variants (Högbom 1974; Perley et
al. 1989), which represents a radio source by a number of point sources in an
otherwise empty field of view.
In the following we will use the wavelet decompositions to discriminate
noise from sources in the image processing.
Figure 1: A CLEANed map (left) and its reconstruction (right).
Figure 1: PS 531 Kb
A wavelet transform of a
signal 
with finite energy with respect to a wavelet 
also with
finite energy is as follows
where the bar indicates complex conjugation and the Fourier transform
of 
satisfies
From the wavelet coefficients 
we can get the inverse wavelet transform
The wavelet transform is also energy conserving like the Fourier transform, i.e.,
Thus we can use the wavelet transform to analyze signals. Unlike the Fourier transform, we can get local information at different scales (or spatial frequencies) by the wavelet transform. This makes it possible for us to analyze the signal at different scales.
Figure 2: The wavelet decompositions. From the top to bottom, left to right panels:
the wavelet modulus at scale 1, 2 and 3, and the reduced map at scale 3, respectively.
Figure 2: PS 1.1 Mb
The above analysis can be generalized to 2-D cases by tensor products or rotations of the 1-D result, etc. (Meyer 1990). In the present analysis we have implemented a 2-D discrete wavelet transform (Mallat & Hwang 1992). The original image is decomposed into a series of finite dyadic wavelet coefficients, i.e.,
and the reconstruction is obtained from those coefficients by
where 
are 2-D wavelet coefficients, 
(
is pixels), denotes the scale 
in the dyadic case,
are 2-D wavelet and scale functions, f is an image,
such as 
, and 
is the reduced component that contains
information of f at all scales that are bigger than 
, and
eventually when j=J the average of f in the case that we have used the
above mentioned wavelet (Mallat & Hwang 1992).
In the case that an error is an additive Gaussian white noise, with a `soft threshold' to process the wavelet coefficients (Donoho 1993), the de-noised reconstruction of the image by the inverse wavelet transform can be, with high probability, at least as smooth as the expected unknown `true' image. In the presence of non-white noise, luckily, wavelet methods extend to handle various problems as well (Donoho 1993), and we apply it in the present analysis.
The method has been applied to process some simulated cases as well as a number of observed areas. Figure 1 shows the image of an area in the Galactic plane and the image reconstruction after de-noising at the wavelet decompositions as demonstrated in Figure 2. It can be seen that by the wavelet transform the noises are clearly discriminated from signals as the scale varies from 1 to 3. These results indicate that the de-noising with the help of the wavelet transform is satisfactory and prospective.
This work is supported by the NNSF of China, the Radio Astronomy Open Laboratory of CAS, and the Director Foundation of BAO. We thank Profs. S. Wang, R. Nan and Y. Zheng for discussions on this work, and our other colleagues in the radio group.
Donoho, D. L. 1993, in Different Perspectives on Wavelets, ed. I. Daubechies (Providence: AMS), 173
Högbom, J. A. 1974, ApJS, 15, 417
Mallat, S., & Hwang W. L. 1992, IEEE Trans. Info. Theo., 38, 617
Meyer, Y. 1990, Ondelettes (Paris: Hermann)
Perley, R. A., Schwab F. R., & Bridle A. H. 1989, Synthesis Imaging in Radio Astronomy, (San Francisco: ASP)
Starck, J.-L., Bijaoui, A., Lopez, B., & Perrier, C. 1994, A&A, 283, 349
Wang, S. 1984, Publications of Beijing Astronomical Observatory, Sept., 1
Yan, Y. 1993, Third Symposium on Data and Image Process in Radio Astronomy, Zhoushan, Oct., 1993, and Publications of Shanghai Astronomical Observatory, Shanghai (in press and in Chinese)
Zhang, X., Zheng, Y., Cheng, H., Wang, S., Cao, A., & Peng, B. 1995, A&A, to appear