3D Source Detection

Vinay Kashyap

Dept. Astron. & Astrop., 5640 S. Ellis, Chicago, IL 60637

Abstract:

1. Introduction

X-ray data are in general a list of photon positions in the detector , photon arrival time , and photon energy . The general problem of source detection is to find statistically significant enhancements in the photon density in the -space. However, source detection algorithms in current general use search for sources only in a 2-D spatial field and ignore the t and E dimensions. This has two drawbacks: First, the algorithms ignore sources that are too weak when the data are considered in totality but would be significant when short time segments or passbands are considered; and second, sources with strong variations in their light curves (often the most interesting feature of the source) will not be recognized as such. Thus, a generalization of the 2D techniques to multiple dimensions is a natural area for study. Here, we restrict ourselves to detecting sources in the domain.

Straightforward generalization of 2D techniques (such as LDETECT; Harnden et al. 1984) to 3D is however fraught with problems, the most significant of which is the runaway sizes of data arrays; the data may be binned such that the total number of elements (, where each term represents the number of elements along the respective axis) exceeds the memory capacity of computer workstations. Here we will describe an algorithm for 3D local detection of X-ray sources that will work on ordinary workstations. The work on this algorithm is still in progress, but in the final section, we show an example of applying this method to ROSAT PSPC data and discuss its achievements.

2. Algorithm

Here we describe the algorithm we have developed to detect X-ray sources in 3D (i.e., in the domain) on ordinary workstations. The computational cost of this algorithm is

compared to which would be obtained by using standard Fast Fourier Transform (FFT) techniques.

We first eliminate time-gaps in the data and assign bin numbers to each photon. These binned data are then convolved with the Mexican-Hat (MH) function,

where are scale factors. The MH function has some significant advantages over the square-cell filters which are in general use: It is centrally peaked and is surrounded by a --ve shell such that the integral of the function over all space is zero---it therefore serves as a natural background subtractor; the scale factors allow it to act as a filter, enhancing structures in the data that match these sizes; both the MH function and its Fourier Transform (FT) are limited in extent, which allows us to store a smaller fraction of the arrays (see below); and finally, it is analytically manipulable, which allows us to reduce computation times significantly. In our use of the MH function as a filter to detect sources, we follow Freeman et al. (1996) and Damiani et al. (1996).

Because direct convolution is computationally expensive, we carry it out as multiplication in Fourier space. As pointed out above, it may not be possible for the entire data array to be stored in memory, so we compute the forward transform as follows:

1. From the data , generate a 2D sub-image , and compute the FFT of this image along the x- and y-axes to obtain .
2. Store a subset of , in a 2D ``image'' .
3. Fully populate , and FFT along the t-axis to obtain .
4. Multiply by the analytically computed FT of the MH function to obtain . Because FT(MH) is limited in extent, only the small fraction of those elements of F which are non-zero need be stored. In general, this implies a major saving in memory and disk requirements.
5. Carry out steps (1)--(4) until is fully determined.

We determine the appropriate threshold for flagging sources by fitting Gaussian functions to the central core of the distribution of pixel values in (for a description and justification of this method, see Freeman et al. 1996). That threshold, written as some multiple of the standard-deviation of the best-fit Gaussian, which results in false source in simulations of source-free images with the same count density (; number of counts encompassed by the +ve part of the MH function; see Damiani et al. 1996) as with the data and adopted MH. This method fails for small values of the count density (q < 1), and we are working on an improved method of determining the threshold.

3. Application

  
Figure 1: Comparison of 3D `wavelet' and 2D `standard' source detection algorithms. EXSAS/PROS-based detections are plotted as ``+'' signs, and 3D detections are plotted as ``.''s. All 2D sources are detected in 3D except one which was a marginal detection in Micela et al. 1996, and an additional 10 sources are detected in 3D.
Figure 1: PS 50 Kb

Here we apply the algorithm described above to ROSAT PSPC observation of the Pleiades Cluster. The Pleiades are particularly useful as test cases, because of the large number of sources within the field-of-view (FOV), and the existence of previously determined source lists using both ``standard'' and wavelet based techniques (Micela et al. 1996; Damiani et al. 1996; Freeman et al. 1996). Here, we restrict our attention to one set of binning and scales ( in the spatial and 400 s in the temporal; ), but caution that a complete source detection process would require calculations at numerous values of the scales (our purpose here is to illustrate the method---a detailed description is in preparation). Based on simulations, we adopted a threshold of of the best-fit Gaussian to the frequency distribution in order to select source pixels. Further, because our algorithm cannot yet handle the strong variation in vignetting seen in the PSPC, we restrict our attention to the central part of the FOV. The comparison between ``standard'' techniques (Micela et al. 1996) and our algorithm is shown in Figure 1; note that our algorithm, despite being in 3D, has found a greater number of sources than the 2D algorithm (this advantage is lost when our wavelet based algorithm is compared with other wavelet based 2D algorithms, such as Damiani et al. and Freeman et al.).

  
Figure 2: 3D representation of sources detected in ROSAT/PSPC observation of the Pleiades. The spatial scales are the same as in Figure 1, and are represented by the projection at the left-edge. Steady 3D sources appear as cylinders, while variable sources have shorter lengths. The color represents count rate in arbitrary units.
Figure 2: PS 135 Kb

The 3-dimensional representation of the detected sources is shown in Figure 2. Note that strong sources are now represented as cylinders, and variable sources are easily recognized by their short lengths along the t-axis.

Acknowledgments:

This work was supported by the AXAF Science Center, and has benefited by useful discussions with Robert Rosner, Peter Freeman, and Francesco Damiani.

References:

Micela, G., Sciortino, S., Kashyap, V., Harnden, F. R., Jr., & Rosner, R. 1996, ApJS, 102, 75

Damiani, F., Maggio, A., Micela, G., & Sciortino, S. 1996, this volume