Difference Image Photometry

This a quick and simple introduction to the technique that we developed and published in Tomaney and Crotts 1996, which describes in much more detail the scientific motivation, implementation and results. Although this was developed for astronomical imaging, in principle it could be applied in other areas including some medical and terrestrial imaging applications.

Astronomical photometry on digital images has conventionally been done by summing the flux around individually identified stars. The sum is typically evaluated by fitting the image Point Spread Function (PSF a.k.a. the 2D impulse response) to the star's intensity distribution. In crowded stellar fields the stars can be too blended to identify individually and must be discovered through an iterative process of subtracting off the brighter stars (using the PSF estimate) to reveal fainter stars which are then subtracted in the next iteration (refining the PSF estimate with each iteration). The most widely used codes since the 1980s using this algorithm are DAOPHOT by Peter Stetson and DoPHOT and their derivatives.

However, in extremely dense stellar fields these algorithms fail because even the brightest stars are hopelessly blended. This is the situation when imaging galaxies from the ground, for example. However, if we are only interested in detecting variability in the small fraction of stars or sources in the sky that exhibit such behaviour then a sensible approach is to remove the non-varying component of the images by subtracting a high signal-to-noise reference image, thereby isolating the variable objects in the resulting difference image.

Prior to subtraction however the images must be matched in a three step process involving: of one image to the other. In typical astronomical images the most difficult step is the PSF matching stage which depends sensitively on the focus (which can vary across the camera, click here), tracking of the telescope as well as the atmospheric conditions during the exposure which blurs the stellar images and yields very different PSFs for each image (for an illustration from MACHO images click here). (In other applications the challenge might be alignment due to geometric distortion, in say medical imaging, or intensity normalisation in terrestrial imaging.)

The key to the success of the technique is correcting all these factors precisely so that genuine variability can be distinguished from artifacts in the subtraction which would otherwise completely dominate the difference image.

Here's some examples of the technique at work:

The top left image is a deep exposure of 0.2 by 0.2 sq. degrees of the nearby galaxy M31 which extends about 1 by 4 sq. degrees on the sky. The bright nucleus of the galaxy is off to the right of the image. This image contains about 10 million stars detected at high significance. The top right image is a blow-up of the indicated subimage on the left with the underlying galaxy background light subtracted out to highlight the small scale structure. Aside from a small number of bright foreground objects and stars the underlying mottled structure belongs to the stars in the galaxy. The variation in light to dark are the so-called surface brightness fluctuations (SBFs) and reflect not individual stars but the random variation in the number of stars in each pixel of the image.

The bottom left image shows the residual difference image after a reference image made from the previous night has been subtracted using this technique; essentially all of the underlying structure seen in the previous image is removed. The bottom right image repeats the process for an image taken a month later using the same reference image. Unlike the previous image this image contains many isolated point sources, indicated by the bright and dark spots, which are stars that have respectively brightened or faded over the course of the month. At this stage it becomes relatively straightforward not only to detect all the variable stars but to measure accurately their changes in intensity in a sequence of difference images.

The following figure is one of my most overused overheads, basically because it is a spectacular demonstration of how sensitive the technique can be. The upper panel images on the left are subimages from similar exposures of M31 as seen in the previous figure and taken 50 minutes apart. The lower panels are their respective difference images after a higher signal-to-noise reference image from another night (not shown) with better spatial resolution has been subtracted after the image matching steps. The residuals in these images are essentially random shot noise (close to the photon noise limit), with the exception of the bright star in the corner, but the second image reveals a clear detection of a rapidly brightening star. What is most striking is how in the upper frames the appearance of the object is completely invisible to the eye. This is because its intensity in the second image is only a fraction of the random SBF variation in intensity in the image pixels.

The plots on the right show how this object varied with time as measured from a series of difference images taken in two passbands. Note how even at a low level the behaviour of the object is reproduced in both bands. The points corresponding to the two images on the left are indicated with two arrows in the upper panel. (The flux difference between these two points corresponds to an R magnitude of 21.1 in seeing of FWHM 1.1" which is less than 1.5% of the underlying galaxy background flux at this position. One sigma variations in flux are 0.1% modulations of the background light and equivalent to a magnitude of 24.4.)

In this case this object was possibly an erupting nova, but our coverage was insufficient to classify it. However, even at its peak brightness in our measurements the object could not be detected with conventional approaches with confidence or measured with any precision.

Thus this kind of method allows astronomical variability surveys to be extended into regimes which are not limited to resolved stars, greatly expanding their potential. This is particularly true of microlensing surveys which typically need to monitor millions of stars on a nightly basis to detect microlensing events. In addition, it is not limited to simply intensity variability of objects but is also applicable to detecting moving objects.

  • DIFIMPHOT software page.

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