Self-correlation

The ESP application SELFC generates an image on which areas displaying a degree of symmetry are more easily identified. The algorithm employed examines the position of all the pixels within a given radius of the pixel currently being considered and from that creates a set of pairs consisting of pixels equidistant from, and on opposite sides of it. A sum is then made which is maximised if the pixels pairs are both above the sky value and also of similar brightness. The sum derived is normalised and inserted into the pixel on the output image corresponding to the current pixel.

The normalisation is not to the 0-1 range but instead supplies a value above or below zero. Any object with a value below zero on the output image must be a statistical fluke or arises from poor flat field, whilst objects with values above zero may be real. Given the simple normalisation employed it is difficult to determine exactly what is real statistically. However, a good guess may be made in the following way. Generate a self-correlated image (IMAGE1) from the source image using SELFC. Then, scramble the source image using the ESP application MIXUP to generate a noise equivalent image (IMAGE2). Apply SELFC to IMAGE2 and then find the modal pixel value (BACK) and its associated standard deviation (SIGMA) using HISTPEAK. The rule is then that any object brighter than BACK+3$\times$SIGMA in IMAGE1 is probably real. For the highest possible accuracy the values of BACK and SIGMA should be derived from examination of 10 scrambled versions of IMAGE1, but the calculation time involved may be substantial.

The application can be used with the following syntax:

% selfc in=ic3374 out=ic3374s diam=10 psize=0.96 back=727

The above examples perform the self-correlation on image ic3374 using a local modal pixel value for the image of 727 counts. The sampling area used is a circle of 10 arc second width and all correlation values generated will be placed in the output NDF image ic3374s.

The correlation is performed in such a way that objects of bigger or smaller than the size requested are improperly sampled. However, they will still generate a response, as the detection method does not depend critically on the size of the template. Consequently, a compromise is involved in selecting the object size. If a large object size is requested the calculations take longer and the resolution of the output image drops, but if a small object size is requested noise quickly becomes a problem and offsets the increased speed and resolution.

It might be supposed that symmetry would not be a very good basis for correlation, given the wide range of possible galaxy shapes known. Despite this, trials suggest that the method works well with a wide range of galaxy types. The only disadvantage is that two bright objects close together can give rise to spurious objects between them. This effect can be minimised by using TOPPED to remove very bright pixels from the image. Any object containing such bright pixels will already have made its presence very obvious!