Wavelet-based detection algorithms

Section 3.4.1present a brief and concise description of wavelets. In this section, each of the selected wavelet-based source detection algorithms is described. A short summary of the main characteristics of the chosen procedures is given in Table4.2.

wavdetect

wavdetect2is one of the Chandra wavelet-based detection techniques (Freeman et al.2002). It con- sists of two parts: wtransform, which convolve the data with a wavelet function, and wrecon, which analyses the detected sources.

wtransform uses the “Mexican” hat wavelet function, which has a positively valued quasi-Gaussian core surrounded by a negatively valued annulus, and an overall zero normalization. The most import- ant control parameters in wtransform are the set of scales and the significance threshold. The scale parameter determines the scale length (in pixel units) and the number of scaled transforms that will be computed on the data. The image is convolved with each of the successive larger scale values of the wavelet, obtaining several “correlation maps”. In addition, for each scale, a background level is estim- ated from the negative annulus of the wavelet function. With these background values, several simulated source-free background images are created. For each pixel, a significance probability is calculated. This is obtained by comparing the pixel value distribution with its corresponding background distribution. If the pixel significance value is lower than the provided significance threshold, the pixel is assumed to be associated with a source. For each wavelet scale, wtransform creates a list of source pixels or correlation maxima.

4.2 Comparison of wavelet-based source detection algorithms for eROSITA images Procedure Implementation Version Method

wavdetect ChandraCIAO 4.2 Wavelet

ewavelet XMM-SAS 6.12 Wavelet

mr_detect MR/1 4.0 Multiscale Vision Model mr_filter+SE MR/1+SExtractor 4.0/2.4.4 Multiresolution filtering

followed by SExtractor detection Table 4.2:Short description of the selected wavelet-based detection algorithms.

For each wavelet scale, wrecon creates a “flux image”. This image is obtained by smoothing the raw data at the corresponding wavelet scale, and then subtracting the background map created by wtrans- form. The mean value of the flux image is zero, except in the vicinity of a source, where the mean deviates considerably from zero. These regions are flagged as putative sources. Source properties are calculated inside the detection cells defined by minimizing the function | log2rPSF−log2σF|, where rPSF is the size of the PSF encircling a given fraction of the total flux and σF is the size of the object at the scale closest to the PSF size. To create a final source list, wrecon cross-correlates the list of correla- tion maxima (obtained with wtransform) with the list of putative sources in each wavelet scale. If the number of correlation maxima lying inside a detection cell is zero, then the putative source is discarded. Finally, wrecon determines different properties for the final list of sources, such as the location, counts, exposure time and count rate.

ewavelet

ewavelet3belongs to the XMM-SAS package of the XMM-Newton telescope. ewavelet is similar to wavdetect: it also uses the “Mexican” hat function as wavelet transformation, the positive part of this function also acts as a source detector and the negative part acts as a measure of background level. The most important control parameters are also a set of scales and a significance threshold.

Unlike wavdetect, ewavelet assumes that sources have a Gaussian shape, and it uses circular sym- metric wavelet functions. This reduces the number of convolutions and makes the task faster. The ewaveletalgorithm is also simpler than wavdetect, it identifies sources by just cross-correlating all the correlation maxima of each wavelet scale. The source properties, like counts and count rate, are determined from the wavelet scale for which the correlation maximum peaks.

Mixed method:mr_filter+SE

mr_filter belongs to the Multiresolution package MR/1 (Starck et al.1998). mr_filter uses the à trous(“with holes”) wavelet algorithm. This algorithm carries out discrete convolutions of the data with a filter of a successively broader kernel. This kernel is a B3-splinescaling function. The wavelet images at different scales are obtained by differencing images at successive filter scales. As a result, the data image is decomposed in a number of wavelet images of growing scales plus a final smoothed image, which correspond to the last filtered array. mr_filter identifies the significant coefficients in each wavelet image with an especial treatment for the Poisson noise known as autoconvolution or wavelet histogram method (Starck & Pierre1998), which estimate the PDF of the noise in the wavelet space assuming a flat distribution. The insignificant signal is then filtered directly in the wavelet space

using a thresholding algorithm, which consists in setting all wavelet coefficients which have a value lower than a threshold to zero. Based on the significant coefficients, a filtered image can be obtained by an iterative reconstruction algorithm, which recovers the flux and shape of the structures in the data. The control parameters are also a set of scales for the wavelet transformation and a significance threshold for selecting the significant coefficients.

The source detection on the filtered images is performed by the SExtractor4 _{software (Bertin &}

Arnouts 1996). Originally, SExtractor was developed to detect objects in optical data, but since the multiresolution filtering removes most of the noise in an X-ray image and produces a smooth back- ground, SExtractor can also be applied to filtered X-ray images. Roughly speaking, SExtractor works as follows: it divides the image into boxes, where the sources are removed and a background is calculated. This process is repeated until the mean value in each box converges to a constant value. By applying median filtering, possible bright boxes are removed. Full background and background- RMS maps are obtained by bicubic-spline interpolation, which smooths the values between boxes. The background-RMS is a map of the background noise. This final background image is subtracted from the original image to obtain a background-free image. The background subtracted image is convolved with a compact support detection filter to bring out faint objects and divided by the background-RMS map to derive a signal-to-noise map used for the detection. Then, SExtractor isolates the objects by thresholding. SExtractor tries deblending on each isolated source to assure that the object is not com- posed of several sources. Finally, a detailed analysis of individual sources is delivered: source positions, shapes, photometry, etc.

mr_detect

mr_detect also belongs to the Multiresolution package MR/1 (Starck et al. 1998). mr_detect also uses the same wavelet algorithm, significant coefficient selection and image filtering methods as mr_filter. However, the mr_detect includes a source detection and characterisation algorithm, which is described in the following.

mr_detectuses the Multiscale Vision Model (MVM) for object identification. In this model, a set of connected significant wavelet coefficients in a given scale is known as a structure. Structures within different wavelet scales are connected to form objects by means of the “inter-scale relation”. This inter- scale process works as follows: a structure s1, jat scale j is connected to a structure s2, j+1at scale j+1 if s_{2, j+1}contains the pixel in s1, jwith the maximum wavelet coefficient. There are two possible scenarios in this process:

1. If at given scale mr_detect detects two objects while in a lower scale it detects only one, then the algorithm reconstructs a single object.

2. If at given scale j mr_detect detects two objects which correspond to two objects at scale j − 1 and one object at scale j − 2, then the algorithm considers those initial objects as two separate sources.

The advantage of this process is that objects can be identified in wavelet space. The objects are recon- structed iteratively and the counts associated with each object are also computed.

4.2 Comparison of wavelet-based source detection algorithms for eROSITA images

Figure 4.3: Left: Cut-out of a simulated X-ray photon image with only point-like sources. The sources have 10 counts and are marked by a blue circle. They are separated by 40_{. Right: the mr_filter filtered image (upper}

left), the wavdetect (upper right), ewavelet (lower left) and mr_detect (lower right) reconstructed images. Overlaid on the images are the detected sources by each procedure (marked by red circles), the spurious sources (green circles) and the non-detected sources (white circles).