SExtractor Draft - METU Astrophysics

166.2 Filtering6.2.1 ConvolutionDetectability is generally limited at the faintest flux levels by a background noise. The powerspectrumof the noise and that of the superimposed signal can be significantly different. Somegain in the ability to detect sources may therefore be obtained simply through appropriate linearfiltering of the data, prior to segmentation. In low density fields, an optimal convolution kernelh (“matched filter”) can be found that maximizes detectability. An estimator of detectability isfor instance the signal-to-noise ratio at source position (x 0 , y 0 ) ≡ (0, 0):( SN) 2≡ ((s ∗ h)(x 0, y 0 )) 2(n ∗ h) 2 , (3)where s is the signal to be detected, n the noise, and ‘∗’ the convolution operator. Moving toFourier space, we get:( SN) 2= (∫ SH dω) 2∫ |N | 2 |H| 2 dω , (4)where S and H are the Fourier-transforms of s and h, respectively, and |N | 2 is the powerspectrumof the noise. Remarking, using Schwartz inequality, thatwe see that∫∣ ∣2SH dω∣ ≤∫ |S|2|N | 2 dω ∫Equality (maximum S/N) in (5) and (6) is achieved for|N | 2 |H| 2 dω , (5)( ) S 2 ∫ |S|2≤ dω . (6)N |N | 2S|N | ∝ |N |H∗ , that is (7)H ∝S∗|N | 2 . (8)In the case of white noise (a valid approximation for many astronomical images, especially CCDones), |N | 2 = cste ; the optimal convolution kernel for detecting stars is then the PSF flippedover the x and y directions. It may also be described as the cross-correlation with the templateof the sources to be detected (for more details see, e.g. Bijaoui & Dantel 1970, or Das 1991).There are of course a few problems with this method. First of all, many sources of unquestionableinterest, like galaxies, appear in a variety of shapes and scales on astronomical images. Aperfectly optimized detection routine should ultimately apply all relevant convolution kernelsone after the other in order to make a complete catalog. Approximations to this approach are the(isotropic) wavelet analysis mentioned earlier, or the more empirical ImCat algorithm (Kaiseret al. 1995), for both of which sources to detect are assumed to be reasonably round. The impacton memory usage and processing speed of such refinements is currently judged too severe to beapplied in SExtractor. Simple filtering does a good job in general: the topological constraintsadded by the segmentation process make the detection somewhat tolerant towards larger objects.Extended, very Low-Surface-Brightness (LSB) features found in astronomical images are oftenartifacts (flat-fielding errors, optical “ghosts” or halos). However, it is true that some of themcan be genuine objects, like LSB galaxies, or distant galaxy clusters burried in the background