COMPARISON OF DIFFUSION MODELS IN ASTRONOMICAL OBJECT LOCALIZATION

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1 COMPARISON OF DIFFUSION MODELS IN ASTRONOMICAL OBJECT LOCALIZATION Františe Mojžíš Department of Computing and Contro Engineering, ICT Prague, Technicá, 8 Prague Abstract This paper is devoted to methods of objects ocaization based on the comparison of acquired image data and dar image of used imaging system and aso focuses on modeing of detected objects. There are processed astronomica data acquired during night with ong eposure times. Object ocaization is based on the presumption that the data acquired by CCD imaging systems are Poisson distributed. For objects ocaization in astronomica science are commony used statistica modes based on the different density or mass functions of probabiity distributions. There are usuay appied Gaussian and Moffat functions, which are compared together. For purpose of modes parameters optimization was used Matab fmincon function. Introduction Objects detection [] and their eact ocaization is one of the most fundamenta topics in astronomica images processing [, ]. Anayzed data are usuay acquired during the night when ight conditions are poor. Thus it is necessary to use ong eposure times. For data acquisition, an astronomica CCD camera [] is used. CCD sensor [] is a source of severa noises []. Suppose that it wors as an photon counter, then it is ogica that the images are contaminated by photon counting noise []. This is resut of the fact, that the ight is used as an information carrier and thus we must consider its behavior as a stream of photons. Objects ocaization, as described further in the artice, is based on estimation of used systems response to the the impuse, i.e., estimation of Point Spread Function (PSF). These are modeed by different diffusion modes. Methods appied in this artice are based on optimization of the objective function. This is derived by Maimum Lieihood Estimate [, 7] (MLE) method and it describes reation between anayzed images and used object mode. Objects detection and PSF modeing. Noise Mode Astronomica images can be epressed in mathematica way as foows (, ) = f(, ) + n(, ) () where f(, ) are the data and n(, ) represents noise caed the dar current. This type of noise is caused by thermay generated charge, due to the ong eposure times. Dar current shoud be simpy removed by a dar frame, which maps mentioned thermay generated charge in CCD sensor. It can be considered that this type of noise is Poisson distributed [] in the foowing way n(, ) Poisson(λ(, )) ()

2 where λ(,) is epected number of occurences in the CCD pie ce (, ) and λ R +. This caim can be verified on a sampe of the dar images by a statistica test for the Poisson probabiity distribution, which can be found in [, 8]. In the foowing tet we wi consider an average dar frame d(, ) = m m n i (, ) () i= where m is the number of dar images, n i (, ) is a noise in i-th frame and futher we can assume that d(, ) = ˆλ(, ).. PSF modeing As mentioned in introduction, modeing of astronomica objects is based on estimation of used system PSF. PSF presents response of appied imaging system on the Dirac deta function, fo discrete systems defined as unit impuse δ() = { if = otherwise () One of the most fundamenta diffusion mode is a two-dimensiona Gaussian function. Mode image with astronomica objects described by the diffusion mode can be derived from Eq. () by repacing the epression f(, ) in the foowing way (, ) = f(,, p) + n(, ) () where (, ) D M N, M and N are dimensions of the rectange region of interest D and f(,, p) is a Gaussian diffusion mode of astronomica object. The mode f(,, p) is possibe to epress by the foowing reation f(,, p) = p ep ( ( p ) + ( p ) ). () where its parameters respectivey are ampitude (p ), coordinates in y pane (p, p ) and standard deviation (p ). The second commony used PSF mode was described by Moffat [9], which a specia case of Cauchy distribution p f(,, p) = ( p() + ( p ) +( p ) p ) p (7) where first four parameters are same as in the case of Gauss function and p is shape parameter.. PSF parameters optimization In statistics, MLE is a method of estimating the parameters of a statistica mode, Eq. (). When appied to a data set and given statistica mode, MLE provides estimates for the mode s parameters. For a fied set of data and certain statistica mode, it produces a distribution that gives to the measured data the greatest probabiity, i.e., estimated parameters maimizes the ieihood function [, 7].

3 When it is supposed that the data are Poisson distributed ρ(, λ) = λ! e λ (8) then The opposite ieihood function can be written as n ρ = λ + n λ n!. (9) φ = n L = M N = = n ρ ( (, ), d(, ) + f(,, p) ) min p () where (, ) is the anayzed ight image, d(, ) presents appropriate average dar frame and f(,, p) is the diffusion mode, Eq. (), whereof parameters are estimated. Combination of Eq. (9) and () eads to the fina form of function φ φ = c + M N = = ( (, ) n ( d(, ) + f(,, p) ) ) + d(, ) + f(,, p) () where c is some constant. The constant c is ony data depending and can be set to satisfy φ and obtain φ = M N = = ( (, ) n ( d(, )+f(,, p) ) ) +d(, )+f(,, p)+(, ) n (, ) (, ) min p. For purpose of minimization, MATLAB buit-in function fmincon was appied. Resuts Anayzed astronomica objects were cassified into three casses based on the bit depth of the anayzed image. Processed data were acquired in the bit depth, thus the maimum intensity is. The interva of intensity vaues was uniformy divided into three casses, which can be written as foows: () sma object - maimum intensity in the anayzed area is ess than 8, medium object - maimum intensity in the anayzed area is higher than 8 and ess than 9, arge object - maimum intensity in the anayzed area eceeds 9 and the top is given by the system resoution properties, thus. Chosen objects that were used for an appication of proposed methods can be seen in Fig.. In Tabs. and are presented resuts of used modes optimization. These tabes contains information about φ min, φ ma vaues, mean and standard deviation of φ vaues for evauation of target function and estimated mode parameters for φ min vaue. Graphicay are mean and standard deviation vaues shown in Fig..

4 (a).. 8 Figure : Objects, (a) Sma, Medium, Large Tabe : Optimization resuts of Gauss mode object function vaues optimum parameters φ min φ ma φ average φ std p p p p sma medium arge Tabe : Optimization resuts of Moffat mode object function vaues optimum parameters φ min φ ma φ average φ std p p p p p sma medium arge (a) Figure : Graphica resuts of estimated modes for φ min (a) Sma astronomica object, Estimated Gauss mode, Estimated Moffat mode.

5 (a) Figure : Graphica resuts of estimated modes for φ min (a) Medium astronomica object, Estimated Gauss mode, Estimated Moffat mode. (a) Figure : Graphica resuts of estimated modes for φ min (a) Large astronomica object, Estimated Gauss mode, Estimated Moffat mode. φ min Gauss Moffat (a) mean vaue of φ vaues 8 8 Gauss Moffat standard deviation of φ vaues... Gauss Moffat sma medium arge objects sma medium arge objects sma medium arge objects Figure : (a) φ min vaues for used modes and anayzed objects, Mean vaue of φ vaues for used modes and anayzed objects and evauations, Standard deviation of φ vaues for used modes and anayzed objects and evauations. Concusion This paper was devoted to processing of astronomica data and focused especiay on detection, ocaization and modeing of astronomica objects. There were appied methods based on mathematica statistics, optimization methods and approaches used in astronomica photometry. From presented resuts is obvious that Gauss mode can be more suitabe for modeing

6 of astronomica objects, Fig. (a). When the mean vaues of φ and its standard deviaton are compared, Figs. (a) and, than the Moffat mode gives us better resuts in case of more function evauation. Thus, it is not easy to say which mode is better and which is worse. It depends on eact appication. These facts are the reason for further investigation of another more suitabe modes and different optimization agorithms. Acnowedgement This wor has been supported by the financia support from specific university research MSMT No /, research project MSM 7 of the Ministry of Education, Youth and Sports of the Czech Repubic. References [] Mojzis, F., Kua, J., Svihi, J., Astronomica Systems Anaysis and Object Detection, In Proceedings of nd Internationa Conference Radioeetronia. Brno (Czech Repubic),, p. -. [] Starc, J. L., Murtagh, F., Bijaoui, A., Image processing and data anaysis: The mutiscae approach, Cambridge University Press, 998. [] Starc, J. L., Murtagh F., Astronomica Image and Data Anaysis, Springer,. [] Bui C., CCD Astronomy: Construction and Use of an Astronomica CCD Camera, Wimann-Be, 99. [] Gonzaez, C. R., Woods, E. R., Digita Image Processing, Prentice Ha,. [] Pawitan, Y., In A Lieihood: Statistica Modeing and Inference Using Lieihood, Oford University Press,. [7] Severini, T. A., Lieihood Methods in Statistics, Oford University Press,. [8] Brown L. D., Zhao L. H., A new test for the Poisson distribution. 9 pages. [Onine] Cited -8-. Avaiabe at: zhao/papers/newtest.pdf. [9] A.F.J. Moffat, A Theoretica Investigation of Foca Stear Images in the Photographic Emusion and Appiction to Photographic Photometry, Astronomy and Astrophysics (99) -. Františe Mojžíš was born in Prague, Czech Repubic in 98. He received his M.Sc. from the Institute of Chemica Technoogy Prague (ICT), in. Now he is a Ph.D. student at ICT. His research interests incude image processing and image denoising.

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