Fractal Dimension of Galaxy Isophotes

The Astronomical Journal, 138:941–950, 2009 September
C 2009.
doi:10.1088/0004-6256/138/3/941
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
FRACTAL DIMENSION OF GALAXY ISOPHOTES
Sandip Thanki1 , George Rhee2 , and Stephen Lepp2
1
2
Nevada State College, Department of Physical Sciences, 1125 Nevada State Drive, Henderson, NV 89002, USA; [email protected]
Physics and Astronomy Department, University of Nevada, Las Vegas, Box 4002, Las Vegas, NV 89154, USA; [email protected], [email protected]
Received 2008 September 3; accepted 2009 June 27; published 2009 August 10
ABSTRACT
In this paper we investigate the use of the fractal dimension of galaxy isophotes in galaxy classification. We
have applied two different methods for determining fractal dimensions to the isophotes of elliptical and spiral
galaxies derived from CCD images. We conclude that fractal dimension alone is not a reliable tool but that
combined with other parameters in a neural net algorithm the fractal dimension could be of use. In particular,
we have used three parameters to segregate the ellipticals and lenticulars from the spiral galaxies in our
sample. These three parameters are the correlation fractal dimension Dcorr , the difference between the correlation
fractal dimension and the capacity fractal dimension Dcorr − Dcap , and, thirdly, the B − V color of the galaxy.
Key words: galaxies: fundamental parameters – methods: data analysis – techniques: image processing
measured parameters as part of a classification algorithm in an
effective manner. Further examples of automated classification
schemes applied to SDSS data include Goto et al. (2003), Park
& Choi (2005), and Park et al. (2008).
We argue in this paper that the fractal dimension or some
closely related number will be of use when implemented in a
galaxy classification algorithm using the neural net method.
The paper is organized as follows. In Section 2 we introduce
the fractal dimension and discuss methods by which it may be
measured. In Section 3 we present the results of this study and
we give our conclusions in Section 4.
1. INTRODUCTION
The classification of galaxies is an important topic for
astronomers. Various classification schemes have been proposed
based on the idea that the study of galaxy morphology reveals
important clues to the physical processes occurring in galaxies.
The most commonly used scheme is that of Hubble (1936). Since
Hubble’s time other classification schemes have been introduced
and have been described in some detail by van den Bergh (1998).
In some instances it is necessary to determine the classification of large numbers of galaxies to obtain results of astrophysical significance. The study of the morphology density relation
is a case in point (Dressler 1980). In the last century, galaxy
classification has been done mostly by trained observers using
their eyes. With the advent of large-scale astronomical surveys
it is clear that it is no longer possible to assign Hubble types of
galaxies by eye in a reasonable amount of time.
For example, the Sloan Digital Sky Survey (SDSS) provides
detailed optical images covering more than a quarter of the sky
(Gunn et al. 2006). The SDSS uses a dedicated, 2.5 m telescope
and a 120-megapixel camera that can image 1.5 deg2 of sky
at a time. Over the course of five years, SDSS-I imaged more
than 8000 deg2 of the sky in five bandpasses, detecting nearly
200 million celestial objects, and it measured spectra of more
than 675,000 galaxies, 90,000 quasars, and 185,000 stars. For
many research projects using this database it is essential to know
whether galaxies are spiral or elliptical and one will have to rely
on automated methods if this is to be done.
New survey telescopes will produce even more data. The
Large Synoptic Survey Telescope (LSST) is a funded large
aperture, ground-based, wide field survey telescope (Ivezic et al.
2008). It has an 8.4 m primary mirror with a field of view of
10 deg2 . Taking 15 s exposures with its 3.2 Giga pixel camera,
the LSST will cover the full available sky every three nights.
About 90% of the observing time will be devoted to a survey
which will observe a 20,000 deg2 region about 1000 times
(summed over all six bands) during the anticipated ten years
of operations, and yield a co-added map to r = 27.5. Thirty
terabytes of data will be produced each night. The resulting
database will include 10 billion galaxies.
It is probable that no single measured parameter will enable
one to separate galaxy types. Lahav et al. (1996) have shown
that by using artificial neural networks one can include several
2. FRACTAL DIMENSIONS
2.1. Defining Fractals and Fractal Dimensions
Fractals are sets that appear to have complex structure no
matter what scale is used to examine them. True fractals are
infinite sets and have self-similarity across scales, so that the
same quality of structure is seen as one zooms in on them.
One can construct fractal sets using simple algorithms. If
one wants to know the length of such fractal curves, it can
be derived from the construction formula. Such computations
cannot be done for fractals in nature, such as the outline of a
cloud, the outline of a leaf or coast lines. For example, there is
no construction process or a formula for the coastline of Great
Britain. The only way to get the length of the coastline is to
measure it. We can measure the coast on a geographical map by
taking rulers set at a certain length. For a scale of 1:1,000,000 m,
the ruler length of 5 cm would be 50 km. Now we can walk this
ruler along the coast. This would give a polygonal representation
of the coast of Britain. To obtain the length of the coast, we can
count the number of steps, multiply the number of steps with
5 cm and convert the result to km. Smaller settings of the rulers
would result in more detailed polygons and surprisingly bigger
values of the measurements. For a ruler size of 500 km we obtain
a coast length of 2600 km, when the ruler size decreases to
65 km, the coast length increases to 8640 km (Peitgen et al.
1992).
The motivation for applying these ideas to galaxy classification arises from the differing appearance of isophotes of images
of elliptical galaxies versus spiral galaxies. Isophotes of elliptical galaxies in CCD images appear smoother to the eye than
those spiral galaxies in CCD images. The goal of this work is
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THANKI, RHEE, & LEPP
thus to see if one can measure this effect in a quantitative way
using the fractal dimension of the isophotes of galaxies in CCD
images.
The measured fractal dimension for galaxy isophotes is
expected to be a function of the smoothing scale since different
physical properties are responsible for surface fluctuations at
different scales. Spiral galaxies have gas clumps, bubbles, and
super bubbles at different scales. Ellipticals show many globular
clusters at high resolutions.
2.2. Measuring Fractal Dimensions
Curves, surfaces, and volumes can be so complex that ordinary measurements like length, area, and volume become meaningless. However, one can measure the degree of complexity by
evaluating how fast these measurements increase if we measure
with smaller and smaller scales. The fundamental idea is to assume that the measurement and the scale do not vary arbitrarily
but are related by a power law which allows us to compute one
from the other. The power law can be stated as y ∝ x d , where x
is the scale used to measure the quantity y and d is a constant.
d is a useful quantity in describing fractal dimensions. The
concept of what a dimension is has been reviewed by MayerKress (1986), Schuster (1988), and Gershenfeld (1988) among
others.
Fractal dimensions are always smaller than the number
of degrees of freedom (Grassberger & Procaccia 1983). For
example, for two-dimensional geometrical objects the fractal
dimension will always be less than two as one might intuitively
expect. Fractal dimension can be looked at as an indication of
how close to a geometrical dimension a given set is. In practice,
there are many ways of determining the fractal dimension. In
this paper we have chosen two, the capacity dimension and the
correlation dimension as estimators of the fractal dimension. As
we shall see, the estimators do not always give the same value.
The differences depend on the nature of the data set.
2.3. Capacity Dimension
We now describe our first estimator of fractal dimension, the
capacity dimension. In order to find the capacity dimension of a
set, we assume that the number of elements covering a data set
is inversely proportional to eD , where e is the scale of covering
elements and D is a constant. For example, we try to cover a line
segment with squares of a certain size, and find that we need
three squares. If we then tried to see how many squares of half
the original size were required to cover the segment, it could
be expected to have six squares covering the segment, which is
twice the number of squares needed when the squares were at
their original size. Thus, the number of squares required to cover
the segment is inversely proportional to the size of the squares.
The covering of any smooth, continuous curve works the same
way, provided that the size of the squares is small enough so
that the curve is approximated well by straight line segments at
that scale.
Thus, for one-dimensional objects, we see that
k
,
e
where e is the side of the square, N(e) is the number of squares
of that size required to cover the set, and k is some constant.
Now suppose that we are covering a scrap of paper with little
squares. In this case, if we halve the size of the squares, it would
take four of the smaller squares to cover what one of the larger
N (e) ≈
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squares would cover, and so we would expect N (e) to increase
by a factor of four when e is halved. This is consistent with an
equation of the form
k
N (e) ≈ 2 .
e
It seems reasonable to say that, for more arbitrary sets,
k
,
eD
where D is the dimension of the set. In other words, we can hope
to measure how much of two-dimensional space some subset
of it comes near by examining how efficiently the set can be
covered by cells of different sizes.
In order to find D from N (e) ≈ k/eD , we can solve the
formula for D, by taking the limit as e → 0. This is the capacity
method of estimating D. If we further assume that the set is
scaled so that it fits into a square with side 1, then we get
N (1) = k = 1. This yields the formula
N (e) ≈
log(N (e))
.
e→0 log(1/e)
Dcap = lim
2.4. Correlation Dimension
Correlation dimension can be calculated using the distances
between each pair of points in the set of N points,
s(i, j ) = |Xi − Xj |.
A correlation function, C(r), is then calculated using
1
× (number of pairs (i, j ) with s(i, j ) < r).
N2
C(r) has been found to follow a power law similar to the one
seen in the capacity dimension: C(r) = kr D . Therefore, we can
find Dcorr with estimation techniques derived from the formula:
C(r) =
log(C(r))
.
r→0 log(r)
C(r) can be written in a more mathematical form as
Dcorr = lim
N
N
1 θ (r − |Xi − Xj |),
N→∞ N 2
j =1 i=j +1
C(r) = lim
where θ is the Heaviside step function described as
1 0 (r − |Xi − Xj |)
θ (r − |Xi − Xj |) =
.
0 0 > (r − |Xi − Xj |)
We estimate the fractal dimensions (using the two methods
described above) using a program written by John Sarraille and
Peter DiFalco. The program (called Fd3) was created using ideas
from Liebovitch & Toth (1989; Sarraille & Myers 1994). The
uncertainties in the computed fractal dimensions are estimated
to be of order 5% when tested on reasonably sized samples
of fractals whose dimensions are known exactly. We have
confirmed this by testing the program using a model galaxy
described in Section 3.3. This is an N-body model of a galaxy
with a bar designed to mimic observations of a spiral galaxy.
We constructed FITS images of this model viewed at a given
inclination angle but with the disk rotated within the plane of
the disk. We then measured the fractal dimension for various
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FRACTAL DIMENSION OF GALAXY ISOPHOTES
local surface brightness in an annulus at r to the mean surface
brightness within r
Table 1
Data set
Observatory
Palomar
Lowell
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Spiral Galaxies
Elliptical Galaxies
Bands (nm)
31
58
0
14
500, 650, and 820
450 and 650
position angles of the disk. The disk was always viewed at an
inclination angle of 50◦ . The fractal dimensions measured by
varying the azimuthal viewing angle of the disk varied by about
2.5%. This suggests to us that the formal errors provided by
the fractal program are reasonable when compared to realistic
simulations of galaxies.
3. RESULTS
3.1. The Data Set
A data set ideally suited to our purposes was presented by Frei
et al. (1996). The data set contains 113 galaxies. The galaxies
were chosen to include a wide range of morphologies. All the
galaxies in the set are nearby, well resolved, and bright with
the faintest having the total magnitude BT of 12.90. The sample
was chosen to be suitable to test automatic galaxy classification
techniques with the idea that automatic methods of classifying
galaxies are necessary to handle the huge amount of data that are
available from large survey projects, such as the SDSS. All the
images were recorded with CCDs at the Palomar Observatory
with the 1.5 m telescope and at Lowell Observatory with the
1.1 m telescope. The images are stored in FITS format. Data on
these galaxies has also been published in the Third Reference
Catalog of Bright Galaxies (de Vaucouleurs et al. 1991). Table 1
shows the number of spiral and elliptical galaxies observed
at both of the observatories along with the broadband pass
wavelengths of the filters through which they were observed.
The images were processed to a point where the flat field and
bias corrections were made and stars were removed from them.
It can be seen from Figures 1–3, which contain scaled down
versions of all the images from the catalog, that the collection
contains a wide range of galaxy morphologies.
Galaxy images show a wide range of surface brightness. In
order to find the fractal dimensions for a galaxy, we choose
a series of isophotes, extract contours around each of them
and find the fractal dimension for each contour. We calculate
the mean fractal dimension for each galaxy and plot this as
a function of galaxy type. Examination of these plots reveals
correlations with known galaxy classifications.
3.2. Selecting Isophotes
In order to produce a fractal dimension value for each galaxy,
we average the values obtained for a set of isophotes. We thus
need to choose a minimum isophote and a maximum isophote
and a number of isophotes between these two extreme values.
We have found that the method of isophote selection has a
strong influence on the effectiveness of galaxy classification
using fractals.
We selected isophotes using a modified form of the Petrosian
(1976) system. The system is based on measurements of galaxy
fluxes within a circular aperture. The apreture radius is defined
by the shape of the azimuthally averaged light profile. The
method follows that adopted by the SDSS team (Blanton et al.
2001; Yasuda et al. 2001). The “Petrosian ratio” RP at a radius
r from the center of an object is defined to be the ratio of the
1.25r
RP (r) ≡
0.8r
dr 2π r I (r )/[π (1.252 − 0.82 )r 2 ]
r
2
0 dr 2π r I (r )/(π r )
where I (r) is the azimuthally averaged surface brightness
profile. The Petrosian radius rP is defined as the radius at which
RP (rP ) equals some specified value (set to 0.2 in our case).
We computed the Petrosian radius using this method for each
galaxy image. After examining the results we selected the flux
range to be used for computing isophotes. We set the maximum
flux as the mean flux in an annulus of radius rP /4 and the
minimum flux as the mean flux in an annulus of radius 1.2rP .
The maximum flux was defined in this way so as to avoid
the peak intensities of galaxies, since the central bulges in
spiral galaxies often resemble elliptical galaxies. The minimum
contour was defined to sample the fainter outer regions of the
galaxies in our sample yet ensure that there is enough signal-tonoise. We determined this minimum level by plotting for all the
galaxies and the contours, the fractal dimension versus signal-tonoise at that contour level. The signal-to-noise is determined by
subtracting the sky and using counting statistics. We found that if
the signal-to-noise per pixel corresponding to a given flux level is
less than ∼8σ above sky, the fractal dimension starts to increase
regardless of galaxy type, in other words the fractal dimension
is influenced by noise. This determined our choice of 1.2rP as
the radius at which we measured the minimum flux. We then
selected ten flux levels ranging from the minimum to maximum
flux as defined above. The contour levels were equally spaced
in log(flux) so that the outer parts of the galaxy would get more
weight when the average fractal dimensions were calculated.
The following method was used to extract contours around
a given intensity value. All the pixel values greater than the
desired value of contour intensity were set to one, and the
values less than the desired contour intensity value were set to
zero. This divided the image into two groups, where one group
contained the pixels having intensity higher than the desired
contour intensity and the second group contained the pixels
having intensity lower than the desired contour intensity. Now,
all the pixels having the value of one that were surrounded by
at least one pixel of value zero were retained and all the other
pixels were discarded. The only pixels that remained were the
pixels at the boundaries of the groups constructing contours
around the desired intensity value. The pixel coordinates were
stored in a file.
In determining the contours we ensured that we included
the minimum number of points for reliable estimation of
the fractal dimension. The minimum number of data points
required to obtain reliable fractal dimensions is 10d , where
d is the true fractal dimension of the object (Liebovitch &
Toth 1989). Since we are dealing with geometrical objects with
unknown fractal dimensions, deciding the minimum number
of data points is difficult. Therefore, knowing that the fractal
dimension of a two-dimensional object cannot be greater than 2,
a minimum of 200 (significantly greater than 102 = 100) points
were initially used. Running the program on several galaxies
led to the conclusion that their fractal dimension, on average,
is significantly less than 1.7. This gives us the minimum of
101.7 ≈ 50. To be on the conservative side, a minimum of 80 was
chosen. At high intensity levels, closer to the center of a galaxy,
the contours start to become smaller, containing fewer data
points. The maximum intensity level chosen at fraction (0.25)
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THANKI, RHEE, & LEPP
Figure 1. Data set: NGC 2403 to NGC 4242.
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FRACTAL DIMENSION OF GALAXY ISOPHOTES
Figure 2. Data set: NGC 4254 to NGC 5248.
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THANKI, RHEE, & LEPP
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Figure 3. Data set: NGC 5322 to NGC 6503.
Figure 4. Examples of contours generated using the method described in the text are shown. Each example contains one of the 10 contours generated for different
intensity values for an elliptical galaxy (NGC 4374) and a spiral galaxy (NGC 4303).
of the Petrosian radius ensured there were always sufficient
points in the contour to reliably determine the fractal dimension.
Figure 4 shows typical contours for an elliptical and a spiral
galaxy.
3.3. Dependence of Measured Fractal Dimension on
Observational Parameters
One can expect that the fractal dimension will be affected
by purely observational factors. We have addressed this using
a disk galaxy N-body model described in detail in Rhee et al.
(2004). We used the N-body particles that constitute the disk
and include a bar. To mimic the observing process we have
inclined the disk at a specified inclination angle to the observer
and projected it on the “sky” resulting in a two-dimensional
FITS image comparable to the data files containing the galaxy
images. We added sky counts to each pixel by selecting at
random a value from a Gaussian distribution having a mean
equal to the mean sky value in a real image. The sigma of the
Gaussian was determined to be square root of the mean. We then
scaled the signal from the object according to the user specified
integration time. This process enabled us to mimic several key
observational effects in a controlled way. For the same “galaxy”
we could determine the fractal dimension as a function of pixel
size, integration time (i.e., signal-to-noise), and inclination.
We varied the inclination in our model galaxy for 30◦ to
◦
70 in increments of 10◦ . The images were smoothed with a
Gaussian having an FWHM corresponding to the average width
in pixels of the point-spread function (PSF). This was obtained
from stars in each image before foreground star removal. The
mean value is about 3 . We use four estimates of the fractal
dimension of a galaxy; the mean and median of the contour
fractal dimensions was estimated using the capacity dimensions
and the same was done for the correlation dimension. As
we varied the inclination of the galaxy from 30◦ to 70◦ , the
dimension estimates typically decreased by 0.05. The biggest
change occurred between inclination 60◦ and 70◦ , a change in
0.03. The fractal dimension estimates for the galaxy were about
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FRACTAL DIMENSION OF GALAXY ISOPHOTES
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We varied the pixel size used for imaging the galaxy. Since the
array dimensions were not changed (like a real CCD) we varied
the size that the galaxy image occupied on the array. The pixel
size was doubled in this test again to mimic the range of galaxy
sizes in our data sample. We again find a fractal dimension
increase of 0.07 as the galaxy image increases in size by a factor
of 1.7. It is not clear why this effect arises, possibly the higher
spatial resolution reveals more details in the contours that are
picked up in the fractal analysis. It is possible to correct for
this effect by scaling the fractal dimension proportionally to the
Petrosian radius with the constant of proportionality determined
by the model. However when we plot fractal dimension versus
Petrosian radius for all the galaxies in our sample we do not find
a correlation.
Finally it is intuitively clear that the fractal dimension of the
contour will decrease as the FWHM of the seeing disk increases.
Figure 5 shows a plot of fractal dimension versus PSF FWHM.
There is indeed a weak trend of decreasing fractal dimension
with increased PSF FWHM. It is clear that the effects are more
pronounced at the extremes of the PSF range, above 5 and
below 2. 5.
3.4. Capacity Dimensions and Correlation Dimensions
Figure 5. Fractal dimension vs. PSF FWHM in arcseconds for the galaxies in
our sample. The PSF was determined from stars in the image before foreground
star removal.
1.24. This change due to inclination angle is relatively small
compared to the scatter in fractal dimension between galaxies
of a given type (0.2–0.3) as we shall see. The problem with
implementing such a correction is that it can only be applied if
one knows the galaxy classification in advance.
Next we varied the integration time of our model observation.
The goal is to see how the results change as the signal-to-noise
or contrast of the galaxy against the sky background is varied.
We varied the integration time from 60 to 400 s and kept the
galaxy inclination fixed at 50◦ . The observed fractal dimension
increased slightly with integration time as one might by about
0.02. In the case of a galaxy disk with structure (our model) one
might expect the fractal dimension to increase slightly at higher
signal-to-noise.
Fractal dimensions for all 89 spiral galaxies and 14 elliptical
galaxies were computed. For all the computations the R filter
(centered around 650 nm) images were used. The R filter was
chosen because it was the one common band of all the images.
We selected ten intensity levels evenly spaced in log intensity
between a minimum radius of 0.25rp and a maximum radius of
1.2rp where rp is the Petrosian radius.
We used two estimators of fractal dimension: the correlation
and capacity dimension. We calculated the mean and median
of these two estimators using the range in intensity levels
determined from the Petrosian radii.
Figures 6 and 7 show histograms of the galaxy fractal dimensions computed using the mean of the correlation dimension.
There are clear differences between galaxy types. For example the mean fractal dimension of all the elliptical galaxies is
1.19 ± 0.04 while the mean fractal dimension of all the spiral
galaxies is 1.31 ± 0.08. The lenticular galaxies which would be
expected to resemble ellipticals from a “fractal” viewpoint have
Figure 6. Histogram of the fractal correlation dimensions for all the galaxies in our sample (left) and for just the lenticular galaxies (right).
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THANKI, RHEE, & LEPP
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Figure 7. Histograms of fractal correlation dimensions for the elliptical and spiral galaxies in the sample.
a mean fractal dimension of 1.18 ± 0.04. These histograms are
encouraging from the point of view of separating galaxy types
in that they reveal trends. It is nevertheless clear that fractal
dimension alone is not a reliable predictor of galaxy type. We
now turn to the question of whether the fractal dimension can be
combined with some other measurable quantities to the galaxies
in our sample.
It is well known that elliptical galaxies have different colors
to spiral galaxies because of the difference in their stellar
populations. The galaxies in the sample are listed in the Third
Reference Catalog of Bright Galaxies (de Vaucouleurs et al.
1991). The RC3 catalog records galaxy properties such as total
B magnitude and total Johnson B−V color. Figure 8 shows a plot
of fractal dimension versus B−V magnitude. It is immediately
obvious that the ellipticals and lenticulars are separating from
the spirals with redder colors as expected and lower fractal
dimension as expected. The color selection is doing much
of the work for us but the fractal dimension clearly helps.
If we exclude galaxies with B−V 0.8 we are left with
20 elliptical and lenticular galaxies and 21 spiral galaxies. The
color criterion alone has excluded three quarters of the spiral
galaxies while keeping all of the ellipticals and lenticulars. If
we further exclude all galaxies with fractal dimension 1.3 we
are left with the 20 elliptical + lenticular galaxies and 9 spiral
galaxies. This selection is the upper left corner of Figure 8. Can a
third parameter involving some fractal measure further separate
spirals from ellipticals? Examination of the data suggests that the
difference between two fractal dimension estimators (capacity
and correlation) is that parameter. Figure 9 shows histograms
of the difference between the correlation and capacity fractal
dimensions for spiral and elliptical galaxies (the lenticulars
again behave like ellipticals). There is again a clear separation
between elliptical and spiral galaxies using this parameter. A
cursory examination of Figure 9 suggests that Dcorr − Dcap < 0
is a good selection criterion for elliptical galaxies. This is further
emphasized by Figure 10. From the histograms it appears that the
correlation dimension of a given spiral galaxy is typically larger
than the capacity dimension. This is not the case for ellipticals.
The effect could be attributed to the fact that the estimation
of correction dimension depends on counting the number of
Figure 8. Galaxy correlation dimension vs. total asymptotic Johnson B−V color
from the RC3 catalog. The spiral galaxies are shown as crosses and the ellipticals
and lenticulars are shown as points.
pixels separated by a given distance whereas the estimation of
capacity dimension depends on counting the number of pixels
in squares with given sizes. Since the spirals are more likely
to break up in clumps in their contours, a distance dependent
method of estimating fractal dimension is expected to yield
higher dimensions. This is consistent with our results shown in
Figure 9. Sarraille & Myers (1994) point out that the correlation
dimensions is sensitive to nonuniformity in the density of the
set of points under consideration. At any rate, Dcorr − Dcap < 0
is a useful parameter for galaxy classification purposes.
We now apply this additional parameter to the galaxies that
are included in with the ellipticals in the B−V fractal dimension
plot (Figure 8). Of the nine spirals in the upper left quadrant five
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Figure 9. Histograms of the difference between correlation and capacity fractal dimensions for the elliptical and spiral galaxies in the sample.
Figure 10. Difference between correlation and capacity fractal dimensions
versus correlation dimension for the galaxies in our sample. The spiral
galaxies are shown as crosses and the ellipticals and lenticulars are shown as
points.
have Dcorr − Dcap > 0 and are thus ruled out as ellipticals using
our third parameter. We are now left with four spiral galaxies
in the “elliptical quadrant” of Figure 8. Of those four, two are
S0 galaxies which might be understandably misclassified by
our fractal method since these galaxies are defined to lack the
features (spiral arms) to which our method is sensitive. The two
remaining galaxies are indeed classified as spirals NGC 2775
and NGC 3368. We thus have 26 elliptical + lenticular galaxies
with two spirals when we use our three parameter classification.
4. CONCLUSIONS
The visual classification of the galaxies and a mathematical
quantity, the fractal dimension, are both related to the complex-
ity of shapes. In the expectation of contributing to automated
galaxy classification schemes, fractal dimensions of 89 spiral
galaxies and 14 elliptical galaxies were studied. Two of the
fractal dimensions measured, the capacity dimension and the
correlation dimension, were calculated for the contours generated around different intensity levels of the galaxy images. Average fractal dimensions for the elliptical galaxies were expected
to have lower values compared to the average fractal dimensions
for the spiral galaxies because of their less complex shapes.
It was found that the correlation dimension on its own is not
sufficient to reliably classify the galaxies in our sample but that it
does work in combination with two other parameters. These are
the B − V color and the difference between the fractal dimension
measured using two different methods (correlation and capacity
dimension).
This method will work only for galaxies with relatively large
angular sizes since FD3 needs at least 80 pixels per contour and
a PSF FWHM needs less than 3 pixels. Furthermore, one needs
to remove the foreground stars from the galaxy images for this
method to be effective.
In recent years, the use of Artificial Neural Networks has
grown significantly for classifications. They are also being used
for classifying galaxies. When constructing an artificial neural
network, one has to specify a number of input parameters using
which the network is designed to generate outputs. We have
found three such parameters that work for the sample of galaxies
presented here.
We thank the referee for a careful reading of the manuscript
and several key insights that improved the paper. We thank
John Ray for useful discussions on this project. G.R. acknowledges the support of NSF grant AST-0709055. S.T.
acknowledges support from a NASA Nevada Spacegrant
Fellowship.
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of Texas Press)
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