2 Code description

2.1 PSF estimation

The accuracy of the PSF estimate is primary in StarFinder, since the PSF array is used as a template for all the stars in the image to be analyzed. The user selects the most suitable stars, which are background subtracted, cleaned from the most contaminating sources around, centered with sub-pixel accuracy, normalized and superposed with a median operation. The centering is performed by an iterative shift of the stellar image in order to cancel the sub-pixel offset of its centroid (see Christou & Bonaccini 1996). The median operation, which is performed pixel-by-pixel, is preferred to the mean because it is less sensitive to anomalous pixels (outliers) which might appear in one or more stellar images among the selected ones. The retrieved PSF is post-processed in order to reject any residual spurious feature and to smooth the noise in the extended halo.

The PSF extraction procedure also reconstructs the core of saturated stars by replacing the corrupted pixels with the central part of the PSF estimate. Accurate positioning is achieved by means of a cross-correlation technique, while the scaling factor is determined with a least squares fit to the wings of the star to repair. For a detailed description of the procedure, see Sect. 3.5.

2.2 Standard analysis of a stellar field

At first we build a list of objects, the presumed stars, which satisfy the condition

i(x,y)>b(x,y)+t (1)

where i(x,y) is the observed intensity, b(x,y) the background emission and t a fair detection threshold. The presumed stars are analyzed one by one by decreasing intensity. To illustrate a generic step of the analysis, we consider the (n+1)-th object in the list, after the examination of the first n. A small sub-image of fixed size is extracted around the object (Fig. 1, left).

Figure 1: Left: sub-image extracted from a simulated stellar field. The crosses indicate the objects within the region of interest: a central star (to be analyzed in the current step), a brighter source (already known), a fainter one (to be examined later) and the PSF feature of a much brighter star, represented by the structure in the upper-left part of the sub-image. Right: corresponding sub-region extracted from the stellar field model, containing one replica of the PSF for each star detected so far

This sub-image may contain brighter stars formerly analyzed, fainter objects neglected in the current step and features of other stars lying outside the sub-image. The information on the brighter sources is recorded in a synthetic stellar field (Fig. 1, right), defined as the sum of two terms: a superposition of PSF replicas, one for each star detected up to this point, and an estimate of the background, assumed to be non uniform in general. The local contribution due to the brighter stars and the background, derived from the synthetic field, is subtracted from the sub-image. If a statistically significant residual remains, it is compared to the central core of the PSF by means of a correlation check. If the correlation coefficient is higher than a pre-fixed threshold then the object of interest is rated similar to the PSF and accepted. The accurate determination of its position and relative flux is attained by means of a local fit, in which the observed sub-image is approximated with the multi-component model described in Sect. 3.6. The actual size of the fitting region is comparable to the diameter of the first diffraction ring of the PSF. This choice ensures that the information represented by the shape of a high-Strehl PSF is included in the fitting process to achieve better accuracy and prevents the growth of the number of sources to be fitted together. For the central object of our example a single component fit is performed and the contribution due to the brighter stars is considered as a fixed additive term. A multi-component fit is performed when the star is in a very compact group, at separations comparable to the PSF FWHM. If the fit is acceptable, the parameters of the new detected star are saved and those of the already known sources, which have been possibly re-fitted, are upgraded. The new star and an upgrade of the re-fitted sources are added to the synthetic field.

This analysis is performed for each object in the initial list (a flow-chart illustrating the operations of StarFinder is in Fig. 2).

Figure 2: Flow-chart of the algorithm for stars detection and analysis

To achieve a better astrometric and photometric accuracy, at the end of the examination of all the objects, the detected stars are fitted again, this time considering all the known sources. This step may be iterated a pre-fixed number of times or until a convergence condition is met.

At the end of the analysis, it is possible to stop the algorithm or instead perform a new search for lost objects removing the detected stars and using an upgraded background estimate. It should be stressed that this image subtraction is just a tool to highlight significant residuals. Any further analysis is performed on the original frame, in order to take into account the effects arising from the superposition of the PSFs of neighboring sources. Generally, after 2-3 iterations of the main loop, the number of detected stars approaches a stable value.

2.3 Crowding and blending effects

A binary star with different separation values (Fig. 3) has been simulated to show how the code works with crowded sources.

Figure 3: Simulated binary stars at various separations. From left to right, top to bottom the separation is 2, 1, 0.75, 0.5 times the PSF FWHM. For all the images the flux ratio is 2:1

With a separation of 2 PSF FWHM the two components are well separated and the code analyzes them with the standard procedure described in the previous sub-section. In the other cases (separation from 1 to 0.5 PSF FWHM) the secondary component is not detected as a separate relative maximum and it is lost. However, if the separation is not as small as 0.5 FWHM, a further iteration of the main loop enables the algorithm to detect the fainter component by subtracting the brighter one. This strategy forces the two stars to pass the correlation test, the principal component as a presumed single object and the secondary in a subsequent iteration of the loop. In a way the iteration of the main loop is a de-blending strategy, because it enables the algorithm to detect stars whose intensity peak is not directly visible in the observed data. This strategy fails when:

• the binary is very noisy and the two components have similar flux. In a similar situation the principal component may not pass the correlation test, preventing also the detection of the secondary in a further iteration. These noisy blends may be recovered at the end of the overall analysis;

• The two components are almost equally bright and have a separation close to the lower limit (1/2 PSF FWHM). The residual corresponding to the secondary after subtracting the principal component may have a distorted shape and might not pass the correlation check.

The latter case may be handled by a method based on a thresholding technique. The object is cut at a prefixed level, about 20% below the central peak, and transformed to a binary array, setting to 1 all the pixel above the threshold and to 0 the pixels below. If the area of the pixels with value 1 is more extended than the PSF, the object is considered a blend and the secondary star may be detected by subtracting the brighter one; then a two-component fit allows accurate astrometry and photometry of the two sources. This strategy can be iteratively applied to multiple blends. It should be stressed that the area measurement is not reliable when the value of the cutting threshold is comparable to the noise level: for this reason the de-blending procedure is applied only to objects with a suitable signal-to-noise ratio. Moreover the area measurement is reliable when the data are adequately sampled. This de-blending procedure is applied at the end of the last iteration of the main loop, when all the resolved sources have been detected: in this way the probability that a single object may appear artificially blurred because of the contamination of still unknown sources is largely reduced.

In a normal case, like the simulated field of Sect. 4.1, two or three iterations of the main loop find almost all ($\sim$99%) the stars that StarFinder may detect. The de-blending procedure described above adds $\sim$1% more stars, without additional false detection. The number of lost objects belonging to the first category described above is negligible ($\ll 1\%$).

Normally we perform two or three iterations of the main loop and apply the de-blending strategy only in very crowded fields.