The problem of detecting and enhancing structures in galaxies, such as shells, ripples and x-structures is not new. Some simple techniques can be used: blurring the original image by convolution with a kernel (median, modal, gaussian or rectangular) and further subtraction of this new image from the original one to find features otherwise hidden in the overall signal of the galaxy (as dust lanes) or model subtraction, as in the works of Jedrzejewski (1987) or Reduzzi et al. (1994). From our point of view, the search for structures in images is more a matter of applying several techniques to extract as much information as possible, than preferring this or that method.
Some techniques of filtering in the transform domain to obtain image enhancement are described in what follows.
In practice, the observed image gi(x,y) can be described by two additive terms:
where g0(x,y) denotes the hypothetically noise-free intensity image and n(x,y) is the noise. Noise can be estimated if one knows the sources, or by a statistical analysis of a region known to contain a nearly constant gray level; then it can be minimized by classical statistical filtering or by spatial ad hoc processing techniques.
A good intuitive approach to image enhancement is the clustering specification inspired in the suggestion of Toney (1983). It follows from what we generally expect from a processed image: object-distinctive and details-discernible. In the first one, it is expected that all the desired objects are included in the processed image and, in addition, those separate objects are as distinct from one another as possible. In the second one, which is the interesting point of this work, fine details of the objects are expected to be discernible as well as possible. Before starting the analysis, a subimage containing the object of interest copied out from the clean image is created, and which will be the initial input image function gi(x,y).
To apply a spatial filter to the input image we multiply the Fourier transform of the input image, Gi(fx,fy), by another function H(fx,fy), which defines a linear spatial filter, normally called the transfer function of the filter (see Bow 1984 and references quoted therein). Their product
is the Fourier transform of the output of the adopted filter. The
final image gf(x,y), "the expected processed image", is
obtained by the inverse Fourier transform of 
:
As well pointed out by Bow (1984), the proper choice of the filter is largely problem-dependent. He has shown that for the high-pass filters Butterworth, exponential, and trapezoidal, the high-frequency-component emphasis increases in the above order, but the preservation of low-frequency information increases in the reverse order.
The results of two such filtering operations, obtained for a low-pass filter for the transfer function,
and the high-pass filter, with the transfer function,
are here presented for illustrative purposes.
We have arbitrarily raised the cosines to the 10th and 20th power in order to make the transfer function fall off quite rapidly from its maximum value of 1 to its minimum value of 0.
The original image in the filter Gunn g of the peculiar S0 galaxy
NGC 5761 is displayed in Fig. 1 (click here). The long bridge which is probably a
remnant of the interaction of the two upper objects is clearly seen.
The almost radial spikes that appear to surround NGC 5761 are real
features and extend to the central regions of the galaxy. Several
median filters have been applied to the original image in an attempt to
enhance as much details as possible with this method. Figure 2 (click here) displays
the residual image of one of such experiments, made with a kernel of
pixels; analyzing all the experiments, it can be seen that the
aforementioned spikes do extend to the galactic center and a brighter
curved structure connects the bridge to the center. Figure 3 (click here) displays
NGC 5761 after the application of Eq. (5). We have tested the high-pass
filter for introduction of texture artifacts by use of a random-noise
image, but no such effect has been encountered. It has also been
applied to a section of the original image containing the upper left
sky portion only, and again no directional artifacts have been created
on the sky (see Lorre & Gillespie 1980). This figure also
displays the radial spike-like patterns of Figs. 1 (click here) and 2 (click here), some knotty
structures around the bulge and along the bridge, including the curved
"arm" of ESO 580-G38. No other structures stand out.
Figure 1: The Gunn g original image of NGC 5761 (center), ESO 580-G38
(top), and ESO-LV 580-G391 (bottom). In the figure, North is to the
bottom and East is on the left
Figure 2: Median filtering of NGC 5761. The vertical line is a chip
flaw enhanced by the process. Note the curved structure on the top and
the radial spike-like features in NGC 5761. A probable dusty lane below
the region where the bridge encounters the inner part of the bulge is
discernible
Figure 3: High-pass filtering of NGC 5761. The chip flaw appear as a
blob on top of NGC 5761, but the other features around it are real -
see text. A probable curved counterpart of the arm of ESO 580-G38
appears below its boxy bulge. Linear segments appear on both sides of
NGC 5761; the one on the right is probably related to the connection
with the bridge
Figure 4: The Gunn g original image of NGC 5193/5193A. North is at
the bottom and East is on the left
The original image of NGC 5193/5193A is displayed in Fig. 4 (click here), and the
residual image of the subtraction of a 
-pixel median is shown in
Fig. 5 (click here). Most of the radial spikes in the elliptical are artifacts,
but the warped disk-like feature in the lenticular galaxy is real, as
well as those elliptical's spikes that are parallel to this feature. We
have applied the same filtering procedure of above and then enhanced
the results by the application of a 
-pixel median filter. Figure 6 (click here)
displays the results, where the warped dusty disk and the parallel
spikes have been well enhanced against a less noisy background.
Figure 5: The residual image of the subtraction of the -pixel
median-filtered image of NGC 5193/5193A from the original. Note the
warped disk seen edge-on and the spikes in NGC 5193 parallel to it
Figure 6: The high-pass filtering of NGC 5193/5193A, enhanced by the
subtraction of its -pixel median filtered image. Again, the disk
and the parallel spikes have been enhanced, but a little better defined
than in Fig. 5 (click here). The vertical line is a chip flaw
Figure 7: The B original image of ESO 143-G7
Figure 8: Result of the subtraction a 
-pixel median image from
the original one - see text
Figure 9: The high-pass filtering of ESO 143-G7. The spiral structure
is preserved, enhancing other structures inside the spiral pattern
Figure 10: The original image of HRG 54103
Figure 11: The HRG 54103 processed image after high-pass filtering as
transfer function
Figure 12: High-pass filtered image of HRG 54103 - see text
Figure 13: The residual of the subtraction of a -pixel median
image from the original image of HRG 54103
Another example of this processing is the case of ESO 143-G7, of which
original B image is in Fig. 7 (click here). The most revealing median filter
applied is the one with a -pixel kernel, whose residual image is
in Fig. 8 (click here); the small rim on the far right has been enhanced as well
as the extremities of the arms and some knotty features around the
bulge. The high-pass filtering (Fig. 9 (click here)) has enhanced not only the
aforementioned rim, but also an extension of if and an interesting
bubble-like feature closer to the bulge (a large supernova bubble? two
pseudo-oscillations of a thin disk?). The application of a low-pass
transfer function suggests the existence of a thin pseudo-bar
connecting the starting points of both principal arms of the galaxy,
but the resolution of the image is poor and hence is has not been shown here.
It is possible to create as many transfer functions as one wishes,
applying them to images previously processed or not. Multiple filtering
can also be made. We present here the results of first applying Eq. (5)
to the original image of HRG 54103 (Fig. 10 (click here)), and then adopting this
new image as the transfer function to be used in Eq. (2). The processed
image is shown in Fig. 11 (click here). In this figure one can see two knots,
which we associate to satellites in the internal periphery of the bulge
of the galaxy. These two objects could be the observable examples of
the experiments in numerical simulation of satellite systems such as
globular clusters or dwarf galaxies moving within a galaxy, as in the
work of Cora et al. (1996). We roughly estimate that each knot
is 0.02 of the bulge frame area. On the opposite side of the bulge a
small plume-like feature is seen (an internal bar?). A detailed
analysis of the physical properties of this object will appear in a
future work (Faúndez-Abans & de Oliveira-Abans 1998).
Finally, manipulating the LUTs of the image display tool, it is
possible to discern a bar, a thin rim which encapsulates the nucleus,
satellite knots and the plume. In Fig. 12 (click here) an attempt to enhance this
rim by a high-pass filter in the B image is displayed. The contrast has
been set to emphasize the pseudo-ring surrounding the nuclear region
(better seen on the left side of the image); features such as the
bulge, the outer ring, and a faint halo are still discernible. For the
sake of comparison, Fig. 13 (click here) displays the residual of the subtraction
of a -pixel median image from the original one. The "rim"
that connects the lower part of the external ring to the bulge on the
right has been emphasized, as well as the knots on this ring; the white
features are artifacts introduced by this particular choice of the
median kernel.
Figure 14: The image of NGC 5193/5193A after the application of a
Fourier hologram and gaussian filter. The edge-on dust disk and the
arm have been enhanced
Another alternative that has been tested for is to adopt a Fourier hologram as transfer function. The Fourier hologram is described by:
where gh(fx,fy) is the pixel intensity, a the
amplitude, |Gi(fx,fy)|2 = Gi(fx,fy)
Gi*(fx,fy), 
, 
is a
monochromatic wavelength, 
is the inclination angle of the
reference wave to the holographic plane, and y the adopted maximum
amplitude of the holographic image in this plane. This algorithm is a
traditional example of a holographic recording set-up for an optical
instrument for image formation.
We have adapted this relationship by adopting the center of the principal object (the most prominent galaxy, but the center of the frame may also be carefully used) as the origin of coordinates of the image plane, and no reference wave has been considered. The hologram so constructed is the new transfer function.
Figure 14 (click here) shows the results obtained with the high-pass filtered hologram as a transfer function. This image has additionally been processed with a gaussian filter to enhance the structure of the edge-on dust disk of NGC 5193A. The disk extends to the top and then to the left of the image, spanning a broad region above both galaxies. The "empty" region around the nuclei are artifacts introduced by the gaussian filter chosen. The reader is referred to Fig. 5 (click here) for comparison.
Another experiment was to create a low-pass filtered hologram, to apply it to the image of NGC 5761, and then subtract the result from the original (see Fig. 15 (click here)). The spikes discussed previously (Figs. 1 (click here), 2 (click here), and 3 (click here)) have been strikingly highlighted. Again, the outer arm of ESO 580-G38 can be seen.
Figure 15: Residual image of the subtraction of the
low-pass filtered Fourier hologram of NGC 5761 from the original image.
The radial spikes have been highly enhanced
In the case of HRG 54103, we have also made the difference between the original image and that obtained by the use of a low-pass hologram as transfer function (Fig. 16 (click here)). Two satellites projected onto the bulge are discernible, both with different sizes than in Fig. 11 (click here). No diagnostics about the nucleus can be done due to the resolution of the image. With this contrast, the upper plume is not apparent.
Figure 16: After the application of the low-pass hologram as transfer
function, the two satellites in the bulge of HRG 54103 are emphasized
In an early and beautiful article, Sulentic & Lorre (1984) demonstrated the potential of image processing. They stated that "new techniques are being created all the time by astronomers with special problems to solve". This is only partially true: a lot of research is being conducted in the fields of artificial vision, signal and optical image processing, microscopy, and pattern recognition. Astronomers frequently make use of these ideas.
We have based the next experiment on the idea of background-suppressed filtering (Scholl 1995), where the object scene is enhanced by suppressing selected background features. The following mathematical relation
is the background-suppressed scene, where 
and
are the amplitudes of the Fourier transform of
the scene and background, respectively; 
is
the phase of the scene's Fourier transform.
In order to demonstrate the method's efficacy, an IRAF computer-generated image containing the letters A, O, and L as scene image (as in Fig. 5 (click here)a of the work of Chen & Chen 1995) and an equally-sized frame with the letter A only - the background image - have been used. The experiment succeeded in eliminating the letter A from the scene, as displayed in Fig. 17 (click here).
The next step then was to test this technique for its ability of differentiating galactic structures through the suppressing of selected patterns, either background or not. We have worked on our galaxies with satisfactory results (Faúndez-Abans & de Oliveira-Abans 1998).
The reader is referred to the works of Baba et al. (1994); Frieden & Bajkove (1996); Paranjape et al. (1994); Carnicer et al. (1994); Shi & Ward (1995); Brasher & Woodson (1996); Tom et al. (1996); Pluzhnik (1996); Thurnhofer & Mitra (1996) and Yung & Lai (1996) for other ideas on image identification, restoration, correlations, simulations, and noise subtraction.
Figure 17: Top: the scene image, middle: the background image to be
suppressed, and bottom: the resultant frame