ap-6-11.dvi Acta Polytechnica Vol. 51 No. 6/2011 Stellar Object Detection Using the Wavelet Transform E. Anisimova, P. Páta, M. Blažek Abstract Several algorithms are used nowadays for detecting stellar objects in astronomical images, for example in theDAOPHOT program package and in SExtractor (Software for source extraction). Our team has become acquainted with the wavelet transform and its good localization properties. After studying themanual forDAOPHOTandSExtractor, andbecoming familiar with the à trous algorithm used for calculating the wavelet transform, we set ourselves the task to implement an algorithm for star detection on the basis of the wavelet transform. We focused on detecting stellar objects in complex fields, such as globular clusters and galaxies. This paper describes a stellar object detection algorithm with the help of the wavelet transform, and presents our results. Keywords: stellar object detection, wavelet transform. 1 Introduction The DAOPHOT program [1,2] and SExtractor [3,4] calculate the estimated background value and per- form thresholding of each pixel: if it is more than a specified threshold and meets certain conditions, we consider it to be a light source. Otherwisewe assume that it is noise. Problems arise in the case of faint stars, whose brightness is close to the ambient back- ground. For this reason, they may not be properly detected. Multiresolution methods of image analy- sis, e.g. the wavelet transform, have therefore gained ground. The main advantage of this transform is its ability to separate light sources contained in an im- age according to their size, enabling us to analyze both large, bright objects and the small, faint stars in their neighborhood. 2 Wavelet transform of a 2D image To realize the wavelet transform of an image it is necessary to make an image convolution with a pre- selected wavelet, first by rows and then by columns. The result is an approximation of the original image and the presence of details in horizontal, vertical and diagonal direction. With increasing decomposition level we extract larger details from the image. In order not to change the scale of the low-pass or high- pass wavelet filters, the image has to be subsampled by a factor of 2. Therefore, when the wavelet trans- form is implementedbytheMallatalgorithm[5] there is for each additional degree of decomposition an im- age with dimensions twice smaller than on the pre- vious level. For detecting stellar objects, this is not a good property, because we need to have the same number of pixels on each scale in order to comply with the same coordinates. Therefore, the wavelet transform for astronomical purposes is realized by the à trous algorithm (“with holes”). 3 The à trous algorithm Wavelet transform implementation by the à trous al- gorithm involves convoluting the input image with a 2Dconvolutionkernel representinga two-dimensional scaling function, which imitates the stellar PSF [6]. To imitate the subsampling process, we have to change for each next decomposition level the filter length in such a way that 2j − 1 zeros are inserted between the coefficients,where j is thedecomposition level [6]. • During the first decomposition (we start from j = 0) we convolute the original image S0 with an unmodified kernel K0, and the result is the smoothed matrix S1. • Subtracting S1 of S0, we get wavelet coefficients for the first decomposition level corresponding to the smallest details W1 = S0 − S1. • j = j +1. • Expand the filter by 2j −1 zeros. • Calculate the smoothedmatrix S2 = S1�K1 and the wavelet coefficients for the second decompo- sition level: W2 = S1 − S2, etc. If we stop this algorithm here, the original image is the sum of S2, W1 and W2 (Figure 1). 4 Detecting stellar objects Stellar objects are detected in wavelet coefficients W1, W2, etc., representing details contained in the original image. This means that stars with the nar- 9 Acta Polytechnica Vol. 51 No. 6/2011 Fig. 1: The à trous algorithm [8] rowest radial profile are detected in W1 and with in- creasing decomposition level flatter and more exten- sive objects will be found. 1. After calculating the à trous decomposition of the imagewedetermine their significanceat each level of wavelet coefficients. This is done by noise level, or by estimating the level of the stel- lar background (we start as in the case of con- ventional algorithms). A robust estimate σ̂ is obtained with a median measurement, which is highly insensitive to isolated outliers of poten- tially high amplitudes [7]. This estimate is usu- ally used to remove noise from the image, but we will use it to determine the threshold and todistinguishbetween significant coefficients be- longing to stellar objects and insignificant coef- ficients that are part of the background. 2. The estimated noise standard deviation for each decomposition level will be used for wavelet co- efficient thresholding in a way that it will set to zero all the coefficients belonging to the interval |WJ | ≤ 3σ̂. 3. Among the non-zero coefficientswe then look for the local maxima of the wavelet coefficients. 4. The coordinates of the local maxima can be considered to be stellar detections if there are nonzero wavelet coefficients in the next decom- position level in the same places. In this way we verify whether the detected objects have a star shape, i.e. we try to eliminate the detection of hot pixels and the detection of false centers, which could be detected due to imperfect back- ground level estimation. 5 Results In this paper the proposed algorithm for stellar ob- ject detection is based on thewavelet transform. Our team also carried out stellar detection by standard algorithms (used in SExtractor, with subtraction of discovered stars [8]) as well as by the wavelet trans- form. We monitored the total number of detected stars, the time required for performing detection, the number of discovered stars corresponding to optical catalog USNO-A2.0 [9], and the percentage of the total number of stars in the catalog for a given im- age. The best results (number of real detected stars, and the time spent)were achievedusing analgorithm based on the wavelet transform. Individual images were taken by the D50 Tele- scope at the Astronomical Institute of the Academy of Sciences of theCzechRepublic atOndrejov by the HighEnergyAstrophysicsGroup. Table 1 illustrates the results for number of detected light sources for the following stellarobjects and theirneighbourhood: GRB 100902A (Gamma-Ray Burst), M5 (Globular Cluster), M67 (Open Cluster) and M51 (Galaxy). Table 2 shows the results comparedwith the USNO- A2.0 catalog. For each image there are in the first line: the number of detected stars with the coordi- nates found in the catalog, the number of detected stars not found in the catalog, the percentage of de- tected starswith coordinates found in the catalogout of the total number of objects in the catalog for the image (second row). Figure 2 compares the number of detected light sources using conventional methods based on back- ground estimation and also using the wavelet trans- form. Table 1: Number of detected light sources. Image Standard algorithm Wavelet transform GRB 100902A 402 513 M 5 1031 1918 M 67 368 499 M 51 141 1191 10 Acta Polytechnica Vol. 51 No. 6/2011 Table 2: Number of discovered stars corresponding to optical catalog USNO-A2.0 and the percentage of the total number of stars in the catalog for a given image Image Standard algorithm Wavelet transform In catalog Outside catalog [%] In catalog Outside catalog [%] GRB 100902A 367 35 62 401 112 68 Total 592 592 M 5 823 208 16 1023 895 20 Total 5213 5213 M 67 355 13 38,5 445 54 48 Total 922 922 M 51 63 78 68 88 1103 95 Total 93 93 (a) (b) (c) Fig. 2: An example of stellar object detection for an image of open cluster M67: using conventional methods based on estimation of the global (a) and local (b) background and using the wavelet transform (c) [8] 6 Conclusion The best detection results were achieved using a method based on image analysis using the wavelet transform: the total number of discovered objects and the percentage of the number of stars in the cat- alog for all images is the highest of all tested meth- ods [8]. In addition, there is an interesting situation. In simpler stellar fields (GRB 100902A, M 67) we found more objects that are also listed in the cata- log, while the situation was the opposite for a glob- ular cluster and for galaxy M 51. This is due to the fact that stars in the middle of globular clusters or galaxies are not recorded in the catalog, so all light sources found in this place will be evaluated as not belonging to the catalog. Conversely, there aremany stars in the catalog away from themiddle of a cluster that are almost indistinguishable from noise, or not at all visible, so they have not been detected. Acknowledgement This paper has been supported by grant project SGS10/285/OHK3/3T/13. References [1] DAOPHOT – Stellar Photometry Package [on- line]. Web site of the software package. [cit. 2011–28–05]. Available on the Web: http://www.star.bris.ac.uk/ mbt/daophot/ [2] Stetson, P. B.: DAOPHOT: A computer pro- gram for crowded-field stellar photometry. Pub- lications of the Astronomical Society of the Pa- cific. March 1987, 99, p. 191–222. [3] Bertin, E., Arnouts, S.: SExtractor: Soft- ware for source extraction. Astronomy and as- trophysics supplement series. June 1996, 117, p. 393–404. [4] SExtractor –Astronomical SourceExtractor [on- line]. Web site of the software package. [cit. 2011–28–05]. Available on the Web: http://sextractor.sourceforge.net/ [5] Mallat, S. G.: A Theory for Multiresolution Signal Decomposition: The Wavelet Represen- 11 Acta Polytechnica Vol. 51 No. 6/2011 tation. IEEE Transactions on Pattern Analysis and Machine Intelligence. July 1989, vol. 11, no. 7. [6] Starck, J.-L., Murtagh, F.: Astronomical Image and Data Analysis. Springer-Verlag Berlin Hei- delberg, 2002. 289 p. ISBN 3-540-42885-2. [7] Donoho, D. L., Johnstone, I. M.: Ideal Spatial Adaptation by Wavelet Shrinkage. Biometrika. September 1994, vol. 81, p. 425–455. [8] Anisimova, E.: Methods for Analysing and Pro- cessing of Astronomical Image Data. Prague, 2011. 77 p. Thesis. Czech Technical University inPrague,Faculty ofElectricalEngineering,De- partment of Radio Engineering. [9] USNO – A2.0. A Catalog of Astrometric Stan- dards [online]. [cit. 2011–05–05].Availableon the Web: http://tdc-www.harvard.edu/software/ catalogs/ua2.html Elena Anisimova Petr Páta Martin Blažek Department of Radio Engineering Faculty of Electrical Engineering Czech Technical University in Prague 12