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 941 942 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) ≈ Vol. 138 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 No. 3, 2009 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 943 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) 944 THANKI, RHEE, & LEPP Figure 1. Data set: NGC 2403 to NGC 4242. Vol. 138 No. 3, 2009 FRACTAL DIMENSION OF GALAXY ISOPHOTES Figure 2. Data set: NGC 4254 to NGC 5248. 945 946 THANKI, RHEE, & LEPP Vol. 138 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 No. 3, 2009 FRACTAL DIMENSION OF GALAXY ISOPHOTES 947 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). 948 THANKI, RHEE, & LEPP Vol. 138 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 No. 3, 2009 FRACTAL DIMENSION OF GALAXY ISOPHOTES 949 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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