Robust Automated Photometry Pipeline for Blurred Images

We propose enhancing the signal before star extraction using convolution; this solves the problem of CCA by only using the threshold for weak signals. After the enhancement, the blurred stars can be extracted even if the image is not corrected with bias, dark, and flat fields. This approach guarantees that enough stars are found to run the matching process correctly.

For a given image I, assume a circular surface $\mathbb{S}$ with a radius $R_{\mathbb{S}}$ and an annulus $\mathbb{A}$ with inner and outer radii of $R_{\mathbb{A}}$ and $R_{\mathbb{A}}+1$, respectively. Both $\mathbb{S}$ and $\mathbb{A}$ have their centroids at position (x, y). The total flux for the whole $\mathbb{S}$ is equal to the sum of $I$ minus the sky background: $\text{flux}=\sum_{\mathbb{S}}\left(I_{x,y}-B_{x,y}\right)$. The sky background level in $\mathbb{S}$ is approximately equal to the average value in $\mathbb{A}$ ($\overline{B_{\mathbb{A}}}$). Therefore, the total flux in $\mathbb{S}$ is

where $N_{\mathbb{S}}$ and $N_{\mathbb{A}}$ are the numbers of pixels in $\mathbb{S}$ and $\mathbb{A}$, respectively. When a bright star is blurred, its size becomes bigger, and the value of each pixel ($I_{x,y}$) becomes relatively low or is even drowned out by the sky’s background noise. Therefore, CCA cannot detect it. However, the total flux of a star is always large, and it can be used to detect it effectively. We designed the functional convolution kernel shown in Figure 3 to transfer the image I to flux space ($I_{\text{E}}$). Then, the threshold for determining whether a signal is at the position (x, y) is set to 3 RMS of the sky background in the flux space ($I_{\text{E}}$). The image $I_{\text{E}}$ is given by

where $h$ is the convolution kernel for signal enhancement (Figure 3).

Figure 4 shows the results of a preliminary test that we performed with this method. The observed image before processing ($I$) and enhanced image after processing ($I_{\text{E}}$) were simulated. Three blurred stars were simulated by the profile of 2D Gaussian (FWHM = 30) in image $I$, and their peak values were 1, 2, and 3 (the SNR of total flux was 17, 35, and 53) respectively. The background of image $I$ is uneven with Gaussian noise (RMS = 1). It shows that the signals that were below 1 RMS noise in $I$ were significantly over 3 RMS noise in $I_{\text{E}}$.

In theory, image $I_{\text{E}}$ yields the maximum contrast when $R_{\mathbb{S}}$ = R${}_{\text{star}}$. However, in practice, setting $R_{\mathbb{S}}$ to 9 pixels results in sufficient enhancement. The radius of the annulus is equal to 1.5 times the equivalent radius of the brightest star, which is set to 75 pixels for the 1.26-m telescope. In Figure 4, the sky background around the star appears slightly sunken, this is caused by the annulus passing through the star. However, this does not affect CCA; it may affect nearby stars in a few cases, but enough stars will remain available for matching. Some connected components extracted by CCA in the enhanced image may not be stars; these are mainly caused by pixels with large values, such as outliers. This detection problem rarely occurs as outliers are filtered before enhancement. In addition, the filtered image is only used to extract stars and not for photometry.

There may not exist many of the bright stars that are used to match images, and two images may contain different stars. For example, the SNR of a star may be reduced, making it undetectable by CCA; different directions of the FOV may result in some stars moving out of one image and other stars moving into another image or connected components in images that are wrongly identified as stars being used for matching. These issues increase the matching difficulty. Therefore, we propose a highly robust method to solve these problems.

For the stars in one image, we create a feature set ($\mathbb{D}_{i}$) for each star (see Figure 5), which is given by

where $N$ is the number of stars in the image, $i$ (or $j$) is the index of star, $\vec{c}$ is the star centroid calculated by $\sum xI_{x,y}/\sum I_{x,y}$ and $\sum yI_{x,y}/\sum I_{x,y}$ (Stone 1989), and $d_{ij}$ is the distance between star $i$ and star $j$.

We further define $s(i,i^{\prime})$ as the similarity of stars $i$ in image $I$ ($N$ stars) and star $i^{\prime}$ in image $I^{\prime}$ ($N^{\prime}$ stars), which is expressed by:

where $|\mathbb{D}_{i}\cap\mathbb{D}_{i^{\prime}}|$ is the cardinality of the intersection of $\mathbb{D}_{i}$ and $\mathbb{D}_{i^{\prime}}$, and $i$ and $i^{\prime}$ are the star indices of the two images. Two elements from $\mathbb{D}_{i}$ and $\mathbb{D}_{i^{\prime}}$ are treated as equal if their difference is less than a certain threshold (1 pixel in this study). Therefore, the value of $s(i,i^{\prime})$ ranges from 0 to $N$ (or $N^{\prime}$).

Obviously, for star i in image $I$, it is easy to find its corresponding star $i^{\prime}$ in image $I^{\prime}$ by locating the maximum value of $s(i,i^{\prime})$. In theory, at least three star pairs would be searched to calculate the shift distance and rotation angle.

Having a few different stars in the two images does not impact the matching results (see Figure 5(a) and Figure 5(c)). However, in extreme cases, if most of the stars in the two images are different, the failure rate of matching will increase. Therefore, to ensure the robustness of the system, we only use the eight brightest stars for matching. The robustness of the matching algorithm was tested by two simulated images with n stars at fixed locations and 8-n stars at random locations (n = 3, 4, 5). We repeated the simulation 10 million times; Table 3.2 shows the results. P(n) is the probability of failure in matching. In practice, the RAPP requires an image with at least three real stars; therefore, the probability of failure is P(3). However, more than three stars can usually be found in each image; therefore, the probability should be P(4) or P(5).