2 The signal processing based method DETSP

Various automatic detection methods such as binarization and thinning, Sobel filters and logical feature filters give incomplete results so far (Smareglia et al. 1994). Our approach is based on signal processing methods (Bratsolis 1997).

2.1 The detection algorithm

  

Figure 1: a) Initial image $512\times512$;b) A peak picking to separate the stellar spectrum rows from spurious peaks; c) Final results of position determination; d) The row 306 as a one-dimensional signal; e) The row 306 after a median and an average filter; f) Derivative results of filtered row 306

In the case of images obtained from objective prism plates, the objects to analyze (the stellar spectra), although possessing a great variety of internal structural characteristics, show some evident common features. For the detection of a spectrum, the most important feature is its beginning that is characterized by the presence of intense gradients. So, the main problem of the detection is reduced to the determination of the beginning of the spectrum.

Let us consider an objective prism image (Fig. 1a). It contains spectra of bright and faint stars, single or overlapped, and a random noise. This is a part of the digitized spectral plate, a subimage of $N \times M$ pixels. Let I(x,y) denote the value of the pixel (x,y) where the x-axis is down the dispersion. The subimage itself can be considered as a set of M discrete one-dimensional signals I_y(x)=I(x,y), where M is the number of rows and every signal is an independent row y. The energy E_y of a particular one-dimensional signal is defined by Eq. (1).

$${E_y}={\sum_{x=1}^{N}I_{y}^{2}(x)}.$$ (1)

The row with the minimum energy min(E_y) is considered as representing a background. Now we can use the definition of the variation of energy for every row in decibels SNR_y, see Eq. (2). Analyzing this variation with an appropriate peak-picking algorithm, we can identify energy peaks corresponding to stellar spectrum rows and separate them from spurious peaks (crossed peaks in Fig. 1b).

$${SNR}_y= 20\;\log_{10} \displaystyle \left(\frac {E_y} {{\rm min}(E_y)} \right)\cdot$$ (2)

The detection of such peaks (rows) is the first step of the algorithm. The second is to fix in each such row the beginning of the spectrum image. Let us consider now the one-dimensional signal of row 306 (Fig. 1d). The grand "gap" on the right could be a defect (here) or a real absorption line. The aim of position determination is to detect the region before the great change from background on the left (Fig. 1d).

We find it by taking the derivatives and choosing the maximal. In advance, we apply to the signal an appropriate composition of median and average filters resulting in an optimally-smoothed spectrum scan (Fig. 1e) with its derivative shown in Fig. 1f. The latter is used to determine the very beginning of the spectrum. We overcome the problem of absorption lines by considering the maximal derivative in every case at a point which is leftward from the maximum value of the row. The final results of spectra detection on this subimage are illustrated in Fig. 1c.

2.2 The image sampling

It can easily be seen that the sizes of the subimage, especially down the wavelength direction, is of crucial importance for the algorithm just described. To avoid missing two spectra on the same row, the subimage should be shorter than the spectrum length. Fixing the subimage length, we have to put some restriction on its width, to take into account the local signal-to-noise ratio that gives the best detection limits for the peak-picking process. Both these imply that an image frame should be handled in pieces, subframes.

Taking a subframe from an image, there are often spectra that begin near the subframe's left or right end or are crossed by the bottom or top side. Such spectra probably could not be detected. To avoid this, a partial overlapping of the subframes should be applied in both axes. Having in mind the spectral image parameters, it is optimal to set the subframe sizes to a value about the half of the spectral length, whereas the overlapping size to about twice of the spectral width.