2-D Isophote Profile Ellipse Fitting
The elliptical parameters for 2MASS galaxies are derived from the shape of one surface brightness isophote (per band) corresponding roughly to 3s pixel values: 20.09 mag at J, 19.34 mag at H and 18.55 mag at K. Like all the isophotes used in pipeline processing, these are uncalibrated magnitudes, which are prior to the adjustment of several tenths of a mag derived from the later calibration processing step. Consequently, the isophote at which the 2-D elliptical parameters are derived can vary from 2.6s – 3.7s, depending on the calibration correction.
The ellipse-fitting method was designed to minimize confusion from nearby sources (i.e., stars) and correlated noise features that form ‘extended’ limbs and other ‘disconnected’ false features. Since most 2MASS galaxies are small in size (<15") and the angular resolution is only ~2", we have adopted the strategy of using one isophote to represent the general elliptical shape of galaxies, which is used to compute various forms of aperture photometry (e.g., Kron, isophotal, etc) and symmetry parameters using for star-galaxy separation. The basic steps of the algorithm are:
The pixels corresponding to the isophote are determined by analyzing radial profiles covering all possible different angular directions.
The isophote is isolated by constraining the pixels determined above to give a width of one pixel to the isophote.
The best-fit ellipse to the distribution of those pixels is determined by the minimum chi-square using a grid of axial ratio, b/a, and position angle,j . The minimization method is as follows: for any given set of (b/a, j ), compute the resultant semi-major axis, rsemi, corresponding to each point on the isophote. Calculate the mean and standard deviation of the rsemi population, Drsemi. If the ellipse (b/a, j ) is perfectly matched to the isophote, the mean variance in rsemi is identically zero. Therefore, by minimizing the ratio, mean(rsemi)/ Drsemi , the best-fit ellipse is established.
The first two steps comprise what is referred to as the isovector operation. The last step is the ellipse-fitting operation. The fitting procedures are constrained by the source flux and size, described below.
Ellipse fitting is attempted on each band, J,H & K, as well as the combined "super" image (J+H+K). The "super" coadd represents the optimum signal to noise representation of the galaxy, assuming normal galaxy colors and minimal reddening. Accordingly, the derived 2-D elliptical parameter for the "super" coadd serve as the "default" fit for cases in which the individual band flux is fainter than some given limit: 14.4 at J, 13.9 at H and13.5 at K, or the SNR of the galaxy is less than 5, based on the R=10" fixed circular aperture photometry. For the case in which the semi-major axis estimated from the isovector operation is less than 5" or greater than 70", the source is assumed to be round and the parameters are set accordingly. For the case in which the axial ratio as determined in the ellipse-fitting operation is less than 0.10, the ellipse fit parameters are set to the corresponding fit from the "super" coadd. Finally, the "super" coadd values are also used when the individual band fit for one reason or another is not possible (e.g., when masked pixels are present within 1" of the peak pixel).
The final detail of note is the issue of star subtraction or masking prior to the isovector and fitting procedures. For bright galaxies (K < 12.5) in which the inclination is large (>40 deg), their is apt to be multiple point source detections strung across the disk of the galaxy (falsely induced by the sharp intensity gradient of the disk). Consequently, we do not perform any stellar masking or subtraction before the ellipse fitting step, except when the stellar number density is high (>2000 stars deg-2 for K < 14) in which case it is more beneficial to mask out nearby stars since the probability of contamination is high.
The following pix gives some examples of the isovector and ellipse fitting steps. Two high SNR galaxies and three faint galaxies are shown (in J and K only). The data comes from scan 010, 971116n.
approx 3-sigma thresholds:
threshJ = 2.1677dn
threshK = 3.9084dn
New Analysis !!
Runtime is a operational constraint to the grid size that GALWORKS uses to perform the resampling and linear interpolation. Thus, for small galaxies (radii < 7"), the method is susceptible to undersampling pixelization (i.e., blocky isophotes). Bias may be introduced (e.g., along the diagnonals) at small radii. We examine this issue more carefully by comparing the "nominal" GALWORKS method with two slightly modified methods: linear interpolation and elliptical interpolation using fine grid scales. The latter interpolation uses the "nominal" solution as input toward an ellipsoidal interpolation (i.e., model the galaxy as an ellipsoidal and interpolate accordingly to find the appropriate isophote). The modified method should work better in principle because we are not forcing a grid scale on the isophote -- the down side is that the runtime increases. The following plots and tables demonstrate the performance and comparison between methods.
Pair of bright galaxies (see bright galaxies). For the fainter of the pair, the following plots show the various components to the isophote and ellipse fit thereto (the tables below summarize the results).
1st bright galaxy: J-band 3-sigma linear & ellip-interp isophote & best-fit ellipse
white points == linear interpolated (between red/green) isophote
magenta crosses == points excluded from fitting procedure (deemed outliers compared to the "initial" or nominal solution)
Comments: Notice that there is little difference between the methods as measured from the final solutions (see tables). The finer grid scales methods seem to fit the isophote better than the nominal method (see "chi"), which is not surprising. As we will see with the next examples, the smaller the galaxy gets, the better the new methods are in comparison to the nominal. The limiting radius seems to be around 4" where the undersampling is severe (see last two examples below).
1st Faint galaxy (see three J,K pairs of faint galaxies):
2nd Faint Galaxy (see three J,K pairs of faint galaxies):
3rd Faint Galaxy (see three J,K pairs of faint galaxies):
Details of the Isovector & Ellipse Fitting Algorithm
GALWORKS determines the 2-D elliptical shape of galaxies using the 3-sigma isophote. The isophote is isolated by constructing vectors, vertex anchored to the center of the galaxy, with linear interpolation between pixels to estimate the radius (length of vector) corresponding to the 3-sigma isophote. A "mask" image is generated from this operation -- it represents the isophote, with fixed values corresponding to the 3-sigma level, a different set of values corresponding to 'inside' the isophote and finally zeros for outside the isophote. The mask image is then fed to routine "twod_ellip" that determines the best-fit ellipse to the mask (i.e., to the 3-sigma isophote).
The best fit is determined by running through the ellipse parameter space given by independent parameters, b/a (axial ratio) and p.a. (position angle). The semi-major axis corresponds to the isophote -- that is, the best fit is determined from the minimization of the semi-major axis with input being the isophote x-y locations.
1. Construct radial vectors, vertex anchored to object center, covering the entire area of the object.
2. For each vector, determine radius at which the 3-sigma level occurs. Use linear interpolation for inter-pixel values.
3. Construct isophote mask image. Image has three sets of values:
a. 3-sigma isophote itself
b. inside of 3-sigma isophote
c. outside of 3-sigma isophote.
The additional information (b and c, above) is necessary to 'clean up' the isophote itself – in some cases the isophote has a width larger than 1 pixel due to the method (using angular vectors). With three pieces of information in the mask, the isophote can be stripped to a width of one-pixel.
This is crucial toward accurate determination of the axial ratio (preliminary tests suggest our accuracy is about 0.025 or less with this method).
4. Determine best-fit ellipse to the 3-sigma isophote represented by the mask image.
axial ratio: b/a
position angle: phi
Input: delta_x and delta_y of the 3-sigma isophote pixels, relative to the center of the object.
Run through all possible values of b/a and phi.
For each combination of b/a and phi, input location of 3-sigma isophote -- get the
population of semi-major axis values for each 3-sigma location.
Minimize the output semi-major axis population:
the best fit corresponds to the lowest dispersion in the semi-major axis
determined for each isophotal value
A more robust minimization of the chi-square is to use the mean and st.
deviation of the population of semi-major axis point:
mean rsemi-major axis st. deviation in rsemi population
Instead of just minimizing the standard deviation, minimize the ratio of the st.
deviation to the mean semi-major axis:
frac = st. dev / mean rsemi
The smaller the fractional ratio, the better the ellipse fit. A poor fit has both a large st. dev and smaller rsemi, thus resulting in a larger fractional ratio. A good
fit is just the opposite since each isophote location should have the same rsemi
value (maximized), thus lower dispersion in the population as a whole.