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XMM-Newton Science Analysis System


ewavelet (ewavelet-3.9) [xmmsas_20070708_1801-7.1.0]

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Description

The ewavelet tasks detects sources using wavelet transforms. For this task and its description we have made use of the Chandra Detect User Guide by Dobrzycki et al. ([1]) and the articles by Damiani et al. ([2],[3]), but the implementation will be new.

Figure: Solid line: The mexican hat wavelet function. Dashed line: a gaussian function with the same parameter $\sigma$.

Wavelet transformations are an extension of Fourier transformations, but unlike the Fourier transformation functions (sines and cosines) wavelet functions have a well defined location in space. A wavelet function should have a zero normalization and satisfy the property:

$$W_{a,\sigma}(x) = \frac{1}{\sigma}W\Bigl(\frac{x-a}{\sigma}\Bigr),$$ (1)

$\sigma$ is called the scaling or dilation parameter and $a$ is the translation parameter. A good source about wavelets is Holschneider ([4]). For this task we have used the two-dimensional Mexican Hat (MH) function $W(\sigma,x,y)$, which can be derived from the two-dimensional Gaussian function

$$\phi(x,y,\sigma) = \frac{1}{2 \pi \sigma^2} \exp\Bigl( -\frac{x^2 + y^2}{2\sigma^2} \Bigr)$$ (2)

using the following relationship:

$$W(\sigma,x,y) = [(x,y) \nabla + 2] \phi(x,y) \ = \frac{1}{2 ... ...sigma^2} \Bigr] \exp\Bigl( -\frac{x^2 + y^2}{2\sigma^2}\Bigr).$$ (3)

The central part of the MH is positive and resembles a gaussian function (see Fig. 1). Outside the circle with axis length $\sqrt{2}\sigma$ the MH is negative. Since the image is convolved with the MH function the MH acts like a sort of sliding cell: the positive part being the source cell and the negative part being background area. From this it is clear that the wavelet scale should be smaller than the image itself in order to get a well defined image convolution. As a rule of thumb, the scale size should not exceed 1/8th of the image size.

Sources are detected by convolving the image with the MH function for a given scale parameter $\sigma$. Local maxima in the convolved image correspond to sources. The image is convolved using several scale parameters. In this implementation, like in most implementations, the scale size is increased for each convolution with a factor $\sqrt 2$. Unlike the Chandra software we use circular symmetric wavelet functions. The reason is that allowing $\sigma_x \neq \sigma_y$, increases the number of convolutions, which would make the task considerably slower. Moreover, it would only increase the sensitivity for elongations in two very distinct directions, ignoring source position angles in between 0$^\circ$ and 90$^\circ$.

If we assume that the source shape has a gaussian form we can analytically derive the value of the maximum correlation at a given wavelet scale $\sigma$. So the source is described by:

$$S(r) = \frac{N_{src}}{2\pi a^2} \exp(-\frac{r^2}{2a^2}) + b,$$ (4)

where we have used the symbol $a$ instead of $\sigma$ for the gaussian parameter, in order to set it apart from the wavelet scale parameter $\sigma$. We have here included a constant background term, $b$, which is the number of background photons per pixel. $N_{src}$ is the total number of photons, or another suitable normalization (e.g. total countrate). The maximum of the convolved source occurs when the source position matches the wavelet position, this gives for the maximum of the convolution:

$$C_{max} = [W*S]_{max} = \frac{1}{2\pi\sigma^2} \frac{N_{src}}{2\p... ...igl(-\frac{r^2}{2\sigma^2}\bigr) \exp\bigl(-\frac{r^2}{2 a^2}\bigr) 2\pi r dr =$$

$$\frac{N_{src}}{2\pi a^2} \frac{2}{1 + (\sigma/a)^2} \Bigl(1 - \frac{1}{1 + (\sigma/a)^2}\Bigr)$$ (5)

$C_{max}$ peaks for $a = \sigma$ with $C_{max} = N_{src}/4\pi a^2$. Note that the background does not appear in eq. (5, as the integral of the wavelet function amounts, per definition, to zero. This suggests that we can estimate source counts and countrate by comparing $C_{i,j}$ for several wavelet scales and picking the wavelet scale, $\sigma_{max}$ for which $C_{max}$ peaks. This gives an estimate for the source size $a = \sigma_{max}$ and consequently we can estimate the source normalization with $N_{src} = C_{max} 4\pi\sigma_{max}^2$. This aspect of the wavelet analysis (namely source extent estimation) is neglected by the Chandra software. In section 3.1 it will be shown how one can improve this estimate.