Undersampled images

At a first glance, an image can be considered to be undersampled if the source profiles are

“sharp”. The most prevalent quantity that characterizes the sharpness of the (mostly stellar) profiles is thefull width at half magnitude (FWHM). This parameter is the diameter of the contour that connects the points having the half of the source’s peak intensity. Undersampled images therefore have (stellar) profiles with small FWHM, basically comparable to the pixel size. In the following, we list the most prominent effects of such a “small” FWHM and also check what is the practical limit below which this “small” is really small. In this short section we demonstrate the yields of various effects that are prominent in the photometry for stellar profiles with small FWHMs. All of these effects worsen the quality of the photometry unless special attention is made for their reduction.

Subpixel structure

The effect of the subpixel structure is relevant when the characteristic length of the flux variations becomes comparable to the scale length of the pixel-level sensitivity variations in the CCD detector. The latter is resulted mostly by the presence of the gate electrodes on the surface of the detector, that block the photons at certain regions of a given pixel.

Therefore, this structure not only reduces the quantum efficiency of the chip but the signal depends on the centroid position of the incoming flux: the sharper the profile, the larger the dependence on the centroid positions. As regards to photometry, subpixel structure

0.00 0.01 0.02 0.03 0.04

1 2 3 4

Light curve RMS

FWHM = 1.0

1 2 3 4

FWHM = 1.5

1 2 3 4

Aperture radius (in pixels) FWHM = 2.2

1 2 3 4

FWHM = 2.7

1 2 3 4

FWHM = 3.3

1 2 3 4

FWHM = 3.9

Figure 2.2: The graphs are showing the light curve scatters for mock stars (with 1% photon noise rms) when their flux is derived using aperture photometry. The subsequent panels shows the scatter for increasing stellar profile FWHM, assuming an aperture size between 1 and 5 pixels. The thick dots show the actual measured scatter while the dashed lines represent the lower limit of the light curve rms, derived from the photon noise and the background noise.

yields a non-negligible correlation between the raw and/or instrumental magnitudes and the fractional centroid positions. Advanced detectors such asback-illuminated CCD chips reduce the side effects of subpixel structure and also have larger quantum efficiency. Fig. 2.1 shows that the effect of the subpixel structure on the quality of the photometry highly dominates for sharp stars, where FWHM.1.2 pixels.

Spatial quantization and the size of the aperture

On CCD images, aperture photometry is the simplest technique to derive fluxes of individual point sources. Moreover, advanced methods such as photometry based on PSF fitting or image subtraction also involve aperture photometry on the fit residuals and the difference images, thus the properties of this basic method should be well understood. In principle, aperture is a certain region around a source. For nearly symmetric sources, this aperture is generally a circular region with a pre-defined radius. Since the image itself is quantized (i.e. the fluxes are known only for each pixel) at the boundary of the aperture, the per pixel flux must be properly weighted by the area of the intersection between the aperture and the pixel. Aperture photometry is implemented in almost all of the astronomical data reduction software packages (see e.g. Stetson, 1987). As it is known from the literature (Howell, 1989), both small and large apertures yield small signal-to-noise ratio (SNR) or relatively high light curve scatter (or root mean square, rms). Small aperture contains small amount of flux therefore Poisson noise dominates. For large apertures, the background noise reduces the SNR ratio. Of course, the size of the optimal aperture depends on the total flux of the source as well as on the magnitude of the background noise. For fainter sources, this optimal

2.1. UNDERSAMPLED, CROWDED AND WIDE-FIELD IMAGES

0.0 0.5 1.0 1.5 2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

RMS in # of intersecting square tiles

Radius

Figure 2.3: If circles with a fixed radius are drawn randomly and uniformly to a grid of squares, the number of intersecting squares has a well-defined scatter (since the number of squares intersecting the circle depends not only the radius of the circle but on the centroid position). The plot shows this scatter as the function of the radius.

aperture is smaller, approximately its radius is in the range of the profile FWHM, while for brighter stars it is few times larger than the FWHM (see also Howell, 1989). However, for very narrow/sharp sources, the above mentioned naive noise estimation becomes misleading.

As it is seen in the subsequent panels of Fig. 2.2, the actual light curve scatter is a non-trivial oscillating function of the aperture size and this oscillation reduces and becomes negligible only for stellar profiles wider than FWHM & 4.0 pixels. Moreover, a “bad” aperture can yield a light curve rms about 3 times higher than the expected for very narrow profiles.

The oscillation has a characteristic period of roughly 0.5 pixels. It is worth to mention that this dependence of the light curve scatter on the aperture radius is a direct consequence of the topology of intersecting circles and squares. Let us consider a bunch of circles with the same radius, drawn randomly to a grid of squares. The actual number of the squares that intersect a given circle depends on the circle centroid position. Therefore, if the circles are drawn uniformly, this number of intersecting squares has a well defined scatter. In Fig. 2.3 this scatter is plotted as the function of the circle radius. As it can be seen, this scatter oscillates with a period of nearly 0.5 pixels. Albeit this problem is much more simpler than the problem of light curve scatter discussed above, the function that describes the dependence of the scatter in the number of intersecting squares on the circle radius has the same qualitative behavior (with the same period and positions of local minima). This is an indication of a non-trivial source of noise presented in the light curves if the data reduction is performed (at least partially) using the method of aperture photometry. In the case of HATNet, the typical FWHM is between ∼ 2 −3 pixels. Thus the selection of a proper aperture in the case of simple and image subtraction based photometry is essential. The methods intended to reduce the effects of this quantization noise are going to be discussed later on, see Sec. 2.10.

-0.2 -0.1 0.0 0.1 0.2

5 10 15

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

FWHM = 0.95

5 10 15

FWHM = 1.30

5 10 15

FWHM = 1.89

5 10 15

FWHM = 2.83

Figure 2.4: One-dimensional stellar profiles for various FWHMs, shifted using spline interpolation. The profiles on the upper stripe show the original profile while the plots in the middle stripe show the shifted ones. All of the profiles are Gaussian profiles (with the same total flux) and centered at x0 = 10.7. The shift is done rightwards with an amplitude of

∆x= 0.4. The plots in the lower stripe show the difference between the shifted profiles and a fiducial sampled profile centered atx=x0+ ∆x= 10.7 + 0.4 = 11.1.

Spline interpolation

As it is discussed later on, one of the relevant steps in the photometry based on image sub-traction is the registration process, when the images to be analyzed are spatially shifted to the same reference system. As it is known, the most efficient way to perform such a registra-tion is based on quadratic or cubic spline interpolaregistra-tions. Let us suppose a sharp structure (such as a narrow, undersampled stellar profile) that is shifted using a transformation aided by cubic spline interpolation. In Fig. 2.4 a series of one-dimensional sharp profiles are shown for various FWHMs between ∼ 1 and∼3 pixels, before and after the transformation. As it can be seen well, for very narrow stars, the resulted structure has values smaller than the baseline of the original profile. For extremely sharp (FWHM≈1) profiles, the magnitude of these undershoots can be as high as 10−15% of the peak intensity. Moreover, the difference between the shifted structure and a fiducial profile centered on the shifted position also has a specific oscillating structure. The magnitude of such oscillations decreases dramatically if the FWHM is increased. For profiles with FWHM ≈3, the amplitude of such oscillation is about a few thousandths of the peak intensity (of the original profile). If the photometry is performed by the technique of image subtraction, such effects yield systematics in the photometry. Attempts to reduce these effects are discussed later on (see Sec. 2.9).

2.1. UNDERSAMPLED, CROWDED AND WIDE-FIELD IMAGES

0.5 1 1.5 2 2.5 3 3.5 4

0.5 1 1.5 2 2.5 3 3.5 4

FWHM (fitted)

FWHM (real)

Figure 2.5: This plot shows how the profile FWHM is overestimated by the simplification of the fit. The continuous line shows the fitted FWHM if the model function is sampled at the pixel centers (instead of integrated properly on the pixels).

The dashed line shows the identity function, for comparison purposes.

Profile modelling

Regarding to undersampled images, one should mention some relevant details of profile modelling. In most of the data reduction processes, stellar profiles detected on CCD images are characterized by simple analytic model functions. These functions (such as Gaussian or Moffat function) have a few parameters that are related to the centroid position, peak intensity and the profile shape parameters. During the extraction of stellar sources the parameters of such model functions are adjusted to have a best fit solution for the profile.

In order to perform a self-consistent modelling, one should derive the integrated value of the model function to adjacent pixels and fit these integrals to the pixel values instead of sampling the model function on a square grid and fit these samples to the pixel values. Although the calculations of such integrals and its parametric derivatives³ are computationally expensive, neglecting this effect yields systematic offsets in the centroid positions and a systematic overestimation of the profile size (FWHM). Since the plate solution is based on the individual profile centroid coordinates, such simplification in the profile modelling yields additional systematics in the final light curves⁴ Moreover, precise profile modelling is essential in the reduction of the previously discussed spline interpolation side effect. As an example, in Fig. 2.5 we show how the fitted FWHM is overestimated by the ignorance of the proper profile modelling, if the profile model function is Gaussian.

3Parametric derivatives of the model functions are required by most of the fitting methods.

4For photometry, the final centroid positions are derived from the plate solution and a catalogue. There-fore, systematic variations in the plate solution indirectly yield systematic variations in the photometry and in the light curves.