GNU Astronomy Utilities

6.5.2.1 Quantifying signal in a mesh

Noise is characterized with a fixed background value and a certain distribution. For example, for the Gaussian distribution these two are the mean and standard deviation. When we have absolutely no signal and only noise in a data set, the mean, median and mode of the distribution are equal within statistical errors and approximately equal to the background value. For the next paragraph, let’s assume that the background is subtracted and is zero.

Data always has a positive value and will never become negative, see Figure 1 in Akhlaghi and Ichikawa (2015). Therefore, as data is buried into the noise, the mean, median and mode shift to the positive. The mean is the fastest in this shift. The median is slower since it is defined based on an ordered distribution and so is not affected by a small (less than half) number of outliers. Finally, the mode is the slowest to shift to the positive.

Inverting the argument above provides us with the basis of Gnuastro’s algorithm to quantify the presence of signal in a mesh. Namely, when the mode and median of a distribution are approximately equal, we can argue that there is no significant signal in that mesh. So we can consider the image to be made of a grid and use this argument to ‘detect’ signal in each grid element. The median is defined to be the value of the 0.5 quantile in the image. So the only necessary parameter is the minimum acceptable quantile (smaller than 0.5) for the mode in a mesh that we deem accurate. See Mesh grid options for an explanation of the options used to customize this behavior.

Since there is sufficient signal in the mesh to bias the analysis on that mesh, any grid element whose mode quantile is smaller than the minimum acceptable quantile is usually kept with a blank value and no value is given to it. Finally, when all the grid elements have been checked, we can interpolate over all the empty elements and smooth the final result to find the sky value over the full image. See Grid interpolation and smoothing.

Convolving a data set (that contains signal and noise), creates a positive skewness in it depending on the fraction of data present in the distribution and also the convolution kernel. See Section 3.1.1 in Akhlaghi and Ichikawa (2015) and Convolution process. This skewness can be interpreted as an increase in the Signal to noise ratio of the objects buried in the noise. Therefore, to obtain an even better measure of the presence of signal in a mesh, the image can be convolved with a given PSF first. This positive skew will result in more distance between the mode an median thereby enabling a more accurate detection of fainter signal, for example the faint wings of bright stars and galaxies. When convolving over a mesh grid, the pixels in each channel will be treated independently. This can be disabled with the --fullconvolution option. See Convolution kernel for the respective options.

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GNU Astronomy Utilities 0.2.28-34fb manual, October 2016.