In a generic variability survey, such as the HATNet project, we are primarily focusing on the detection and the quantifications of source brightness variations. The idea behind the photometry methods involving image subtraction is to derive the part of the flux that varies from image to image. It is rather easy to see that simple per-pixel arithmetic subtraction is not sufficient to derive the difference between two images. First, the centroid positions of the stars are different for each image. The magnitude of this difference depends on the precision and the systematic variations in the mount tracking, as well as other side effects such as field rotation and the intrinsic differential refraction. However, it is rather easy to overcome this problem by registering the images to the same reference system (Sec. 2.6). Second, background level may vary from image to image. Changes in the background can
2.8. IMAGE CONVOLUTION AND SUBTRACTION
be modelled by adding a constant or some slowly varying function to the (convolved) image. Third, the stellar profiles are also vary from frame to frame, due to the variations in the seeing or in the focus. In order to have the smallest residual between two images, one should not only register these to the same reference system but on at least one of the images, the profiles should be transformed to match the profiles of the other image. This profile transformation is performed as a convolution, namely the image R is transformed to R′ as
R′ = B + R ⋆ K, (2.67)
where K is the convolution kernel and the operator (·) ⋆ (·) denotes the convolution. For (astronomical) images that are sampled on discrete pixels, the operation of convolution is defined as
R′xy = X
−BK≤i,j≤+BK
R(x−i)(y−j)Kij. (2.68)
Here, the convolution kernel Kik is sampled on a grid of (2BK+ 1) × (2BK+ 1) pixels and
Ixy refers to the intensity of the pixel at (x, y). If the difference of FWHMs of the image R
and R′ are small, the kernel can be sampled on a smaller grid. In general, a kernel function with an FWHM of FK yields a profile FWHM F′ on the convolved image of
F′ ≈ q
F2+ F2
K, (2.69)
where F is the FWHM of the profiles on the image R.
Supposing two images, I and R, the main problem of the image convolution and sub- traction method is to find the appropriate kernel K with which the image R convolved, the resulting image is nearly identical to I. The first attempt to find this optimal kernel (Tomaney & Crotts, 1996) was based on an inverse Fourier transformation between the two PSFs of the images. Theoretically, inverse Fourier transformation yields the appropriate kernel, however, the practical usage of this method is limited due to the high signal-to-noise ratio that is needed by a Fourier inversion. Kochanski, Tyson & Fischer (1996) attempted to find the kernel K by minimizing the merit function
χ2∞ =X
xy
|Ixy − (R ⋆ K)xy| . (2.70)
This minimization yields a non-linear equation for the kernel K and therefore it is not com- putationally efficient. The most cited algorithm related to image subtraction was given by Alard & Lupton (1998). In this work, an additional term was added to the convolution trans- formation, which allows to fit not only the convolution transformation but the background variations:
The basic idea of Alard & Lupton (1998) was to minimize the function χ2 =X
xy
(Ixy − [Bxy + (R ⋆ K)xy])2 (2.72)
and search the kernel solution K in the form of
K =X
i
CiK(i). (2.73)
In their work, the kernels Kiwere two dimensional Gaussian functions with variable FWHMs
multiplied by polynomials. Assuming the background variations to be constant, i.e. Bxy ≡
B, minimizing equation (2.72) yields a linear set of equations for the parameters B and Ci,
thus its solution is straightforward (and efficient). Shortly after, Alard (2000) gave a more sophisticated method that allows the kernel parameters as well as the background level to vary across the image:
Ixy = B(x, y) + [R ⋆ K(x, y)]xy. (2.74)
Both the background variations and the kernel coefficients were searched as a polynomial function of the pixel coordinates, namely
B(x, y) = X 0≤k+ℓ≤Nbg Bkℓxkyℓ (2.75) and K(x, y) =X i X 0≤k+ℓ≤NK(i) CikℓK(i)xkyℓ. (2.76)
It is easy to show that finding the optimal Bkℓ and Cikℓ coefficients still requires only linear
least squares minimization. Alard & Lupton (1998) also discuss how the individual pixels used in the fit must be weighted by the Poisson noise level in order to have a consistent result. Recently, Bramich (2008) searched the optimal kernel K by assuming an alternate set of kernel base functions K(i), involving discrete kernels instead of Gaussian functions.
These discrete kernels are defined as
K(u,v) = δ(uv), (2.77) where (δ(uv))xy = ( 1 if u = x and v = y, 0 otherwise. (2.78)
The total number of base kernels is then Nkernels = (2BK + 1)2. Yuan & Akerlof (2008)
attempted to find the solution Ki, B and Kr of the equation
2.8. IMAGE CONVOLUTION AND SUBTRACTION
This method is known as cross-convolution and works properly in the cases when there is no suitable solution for equation (2.71). For instance, on the image R the profiles have such shape parameters where K > 0 and D = 0 while on the image I these parameters are K < 0 and D = 0. The method of cross-convolution has a disadvantage, namely if one finds a solution Ki and Kr for equation (2.71), Ki ⋆ G and Kr ⋆ G is also a solution (where G is
an arbitrary convolution kernel). Therefore equation (2.71) is degenerated unless additional constraints are introduced (e.g. by minimizing the kKi− Krk difference simultaneously).
2.8.1
Reference frame
The noise characteristics of the subtracted image is determined by both the reference image R and the target image I. If both images are individual frames, the generic noise level is approximately √2 times larger than that of on the individual frames. In order to reduce the noise level on the subtracted frames, the reference image R is created from several individual frames. If the number of such frames is N, the noise level of the subtracted images isp1 + 1/N ≈ 1 + 1/(2N) (supposing that both the reference frames and the target image have the same noise level). Thus, a number of N ≈ 20 − 25 frames are sufficient to increase the noise level on the subtracted image only by a few percent20.
2.8.2
Registration
As it was seen related to the difficulties of the photometry on undersampled images (Sec. 2.1.1), the interpolation of such images with sharp profiles is likely to yield artifacts, “spline undershoots” and therefore systematic residuals (Fig. 2.4). Since the FWHM of the HATNet frames is too small to clearly remove such residuals, we have used the following sophisticated registration process. First, using the stellar profile parameters and flux esti- mations yielded by the modelling described in Sec. 2.4.2, a model for the images is created, involving the program firandom (Sec. 2.12.7). This image model is then subtracted from the original image, yielding a residual with no sharp structures. The residual image is then transformed to the reference system, simultaneously with the transformation of the centroid coordinates found in the stellar profile parameter list. Using the transformed stellar profile parameters, another model image is created that is added to the transformed residual image. Since the stellar profiles can be well modelled by an analytic function, this way of image registration yields no artifacts on the transformed images, even for highly undersampled profiles. Additionally, we do not have to involve all of the stars on the image, only the
20Strictly speaking, a noisy reference frame implies a correlated noise on the subtracted frames since the
same image (or its versions derived by convolution) is subtracted from the original frames. Therefore, it is an upper limit for the noise increment in the final light curves. However, the scatter in the convolution parameters also increase the light curve noise, but this cannot be quantified in a simple way.
brighter ones, since for fainter stars the amplitude of spline undershoots are comparable to or less than the noise level.
This kind of transformation is even more relevant during the creation of the reference image R since this image is created by averaging some of the most sharpest images.
2.8.3
Implementation
Those methods discussed above that are based on the technique of linear least squares are implemented in the program ficonv (see Sec. 2.12.12). The practical details of the photometry based on the method of image subtraction are explained in Chapter 3, related to the HAT-P-7(b) planetary system.