Bi-Symmetric Autocorrelation & such

In the near-infrared, most galaxies appear smoothly symmetric about the major and minor axis (as opposed to the optical, where the younger population, preferentially distributed in spiral arms, dominate the light, giving a "grainy" appearance to their profiles). Exceptions: irregular galaxies (rare), and galaxies with local stellar contamination (exclusive to high stellar density regions). For the most part, galaxies appear two-fold symmtric, while multiple star groups appear asymmetric.

We can exploit this "feature" to separate galaxies from "false" galaxy detections, including double stars and triple stars. Double stars (and most triples) appear asymetric across the minor axis -- that is to say, if you center the elliptical axis on the primary component of the double star, the resultant profile (one-D or two-D) is asymetric from one side of the major axis to the other. This is also generally the case for triple stars (although you can have configurations of three stars in which the alignment is symmetric across both the minor and major axis -- but these configs are much less common).

One way to measure the "symmetry" of an object is to perform a bi-symmetric spatial autocorrelation. Divide the object in two halfs, given by the minor axis. Rotate one-half 180 degrees and multiply the resultant pieces. The autocorrelation is then normalized by the original galaxy (squared). Mathematically, this is equivalent to:

ratio (i,j) = p*(i,j) / p(i,j)
where p is the pixel value at position (i,j), and p* is the "rotated" pixel value.

In order to use low SNR points (which are significant in this exercise, since an absense of flux is meaningful), we account for the flux uncertainty in each pixel. This is done by computing a family of ratios for a given pixel:

set { ratio (i,j) } = { (p* +- sigma ) / ( p +- sigma) }

In addition to the autocorrelation, we can also compute the ratio of total (integrated) flux between symmtric half pieces. This is a sort of "reduced" autocorrelation between the half pieces.

The final piece of information to glean from bi-symetric cross-correlation is a chi-square test (as proposed by Steve Schneider). The reduced chi-square can be written as:

rchisq = (1/N) SUM [(p-p*)^2/2/sigma^2]

where p and p* are the points 180 deg apart that are being compared, N is the number of points being compared, and sigma is the pixel noise level (the factor of two is because the difference has a SQRT(2) larger error).

Case 1: 12th mag galaxy in the galactic plane

The following gif images shows the object, a real galaxy located near the galactic plane. The first panel shows the raw image (with previously processed objects blanked from the image), and the second panel shows the resultant image after stars have been subtracted. The regions in which stars have been subtracted, and regions in which there was a previously processed object, are not used in the autocorrelation or in the integrated flux ratio measurement.

• Galaxy; J = 13.7
  Galaxy; K = 12.1

What does the two-D elliptical profile look like for this object? The following plot shows the derived elliptical aperture for the J band image. The axis ratio is between 0.6 and 0.70, the position angle between 50 and 60 degrees, and the semi-major axis from 12.8 to 13.5 pixels (corresponding to the Kron radius). The red and the green points delineate the two symmetric halfs.

• J-band Elliptical Aperture
  
• K-band Elliptical Aperture
  Note: the inner 3 arc seconds is masked from further operation

This is not the final aperture used, however. We want to avoid contamination from nearby stars or objects (re: galaxies) previously processed. So we mask those areas with suspected contamination (see first gif image). The red and the white points delineate the two symmetric halfs used in the autocorrelation and integrated flux operations. (note also that we avoid pixels with values lost in the noise -- not shown in the gif below).

• J-band Final Elliptical Aperture
  
• K-band Final Elliptical Aperture
  

Results

The autocorrelation (p*/p) distribution is shown in the histogram below.

• J-band bisymmetric autocorrelation distribution
  
• K-band bisymmetric autocorrelation distribution

Histogram Statistics:

J band : mean = 0.99 +- 0.08 , median = 0.99
H band : mean = 1.00 +- 0.19 , median = 1.00
K band : mean = 0.99 +- 0.11 , median = 0.99


Integrated flux ratio:


J band : 0.99
H band : 0.76
K band : 0.76

This object -- a real galaxy, which appears to the eye quite symmetric -- is indeed spatially bisymmetric with a mean correlation ratio of 0.99 to 1.0. The integrated flux ratios are not quite so good (except for J), with ratios ranging from 0.76 to 0.99.

****** Alternate Method *************

Here we attempt a different filter method to remove extraneous low snr points. The approach is to use a integrated flux threshold criterion. Instead of checking each pixel to see if it has an SNR greater than some threshold, we integrate the pixel plus its four nearest neighbors and use the sum to compare with an N-sigma threshold. If sum(p) or sum(p*) greater than N-sigma, then the pixel combination (p,p*) is used in the ratio computation. Thus for galaxies, even low snr points will be included in the ratio computation because they "integrate up", whereas noise bumps around stars may not integrate up (depending on the severity of the N-sigma threshold).

Although this method should eliminate most low snr points (particularly around non-galaxies), some will still remain, which can skew the histogram (since we are taking the ratio of very small numbers). Consequently; as in the previous method, we allow a range in values for the ratio (with the range set by the background noise) -- see the intro of this memo for more details.

• J-band Final Elliptical Aperture
  
• K-band Final Elliptical Aperture