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Point Spread Functions in Identification of Astronomical Objects RADIOENGINEERING, VOL. 25, NO. 1, APRIL 2016 169 Point Spread Functions in Identification of Astronomical Objects from Poisson Noised Image František MOJŽÍŠ1, Jaromír KUKAL1, Jan ŠVIHLÍK1;2 1Dept. of Computing and Control Eng., University of Chemistry and Technology Prague, Technická 5, 166 28 Prague, Czech Republic 2Center for Machine Perception, Department of Cybernetics, Czech Technical University in Prague, Technická 2, 166 28 Prague, Czech Republic [email protected], jaromir.kukal@fjfi.cvut.cz, [email protected] Manuscript received August 5, 2015 Abstract. This article deals with modeling of astronomi- of this system, also called Point Spread Function (PSF). PSF cal objects, which is one of the most fundamental topics in of applied imaging system is influenced by many factors astronomical science. Introduction part is focused on prob- and is composed as a convolution of particular PSFs of sys- lem description and used methods. Point Spread Function tem’s components. The knowledge or a good estimate of the Modeling part deals with description of basic models used resulting system’s PSF plays a key role in astronomical pho- in astronomical photometry and further on introduction of tometry [3, 4], when we want to know accurate information more sophisticated models such as combinations of interfer- about the observed objects. ence, turbulence, focusing, etc. This paper also contains a way of objective function definition based on the knowledge of Poisson distributed noise, which is included in astronom- ical data. The proposed methods are further applied to real astronomical data. Keywords Astronomical image, point spread function, Poisson noise, objective function 1. Introduction Astronomy is a natural science that deals with the study of objects such as planets, moons, stars, etc. Modeling of the mentioned objects is one of the most fundamental topics Fig. 1. Analyzed astronomical image. in astronomical science and the branch dealing with object identification is called astronomical photometry. Under the Astronomical images can be expressed in mathematical term modeling can we understand finding of exact infor- way as follows mation about objects parameters such like its coordinates, x(k; l) = f(k; l) + n(k; l) (1) magnitude, etc. Previously mentioned objects are far away from the where f(k; l) are the data and n(k; l) represents noise called Earth and thus they appear as a bright point on the night sky. the dark current. This type of noise is caused by thermally When an astronomical image, Fig. 1, is acquired, the situa- generated charge, due to the long exposure times. Dark cur- tion is quite different and the bright point has changed and rent can be simply removed by a dark frame, which maps as a result the image will appear as smeared pattern. This is mentioned thermally generated charge in CCD sensor. It caused by passing of original information through the imag- can be considered that this type of noise is Poisson dis- ing system [1, 2] used for an image acquisition. The result tributed [5, 6, 7] in the following way shape of captured objects is given by the impulse response n(k; l) ∼ Poisson(λ(k; l)): (2) DOI: 10.13164/re.2016.0169 SIGNALS 170 F. MOJŽÍŠ, J. KUKAL, J. ŠVIHLÍK, POINT SPREAD FUNCTIONS IN IDENTIFICATION OF ASTRONOMICAL OBJECTS . This article is intended to present commonly used PSF do not comprise some important facts. If we consider that models such as Gauss and Moffat which are popular in PSF the light passes through the optical system before incidence modelling are easy to evaluate and provide good approxi- onto the image sensor, than it is possible to make an approx- mation of analyzed objects [7, 8, 9, 10]. Another physical imation of the result system PSF by diffraction of circular phenomena known as interference, focusing and atmospheric aperture [17] turbulence are rarely used [8, 11, 12, 13]. The main aim of 2J (ka sin θ) I(θ) = I · 1 (6) this article is to compare commonly used methods with PSF’s 0 ka sin θ based on their combinations to show their advantage in PSF I modelling. where 0 is the maximum intensity of the pattern, J1 is Bessel function of the first order, k = 2π=λ, λ is the wavenumber, Optimization of described PSF models is based on au- a is the radius of the aperture and θ is the angle of observa- thors previous work dealing with detection [5] and identi- tion. fication [14] of analyzed objects based on analysis of dark and light frames using hypotheses testing. One of the key From physics, it is known that the diffraction phe- roles plays also the way of the objective function estima- nomenon described by (6) is accompanied by the interference tion. Objective functions determination based on hypothesis phenomenon [17] and thus we can call this relation as inter- of Poisson noise is described in Section 3 as well as brief ference model, which is in the frequency domain expressed as ( introduction of Harmony Search algorithm which was used 1 if ! ≤ Ω; G(!) = (7) for purpose of objective function optimization. 0 otherwise where Ω > 0 is frequency and therefore adequate PSF is Z1 2J (Ωr) 2. Point Spread Function Modeling g(r) = H fG(!)g = 2π !G(!)J (!r)d! = 1 0 Ωr Astronomical photometry based on the two- 0 dimensional fitting uses the hypothesis that the profiles (8) where H is Hankel transform [18]. of astronomical point sources which are imaged on two- dimensional arrays are commonly referred to as PSF [2, 15], Another important factor that influences resulting image is passing of the information through the atmospheric condi- x(x; y) = object(x; y) ∗ PSF(x; y) (3) tions. Thus, important phenomenon that influences result im- age is atmospheric turbulence. The interference phenomenon where ∗ is a convolution operator and x, object, PSF are occurs in any optical system with constrained view. The PSF 2D functions, which represent the result image, the original of interference is the well known Airy disc [19]. The main object and the system response, respectively. PSFs can be question is whether the interference dominates the other pro- modeled by a number of simple or more complex mathe- cesses, only affects them or is dominated by them. According matical functions that are derived from deeper knowledge of to [19] we can express frequency spectrum of the turbulence studied problem. as 2 1 S(!) = σ2 (9) 2.1 Basic PSF Models π 1 + ( !)2 where is the scale of the turbulence and σ > 0 represents Statistical models based on different PSFs are com- the standard deviation of the velocity disturbance. monly used for objects localization in astronomical science. There are usually applied two simple models, i.e., the first The role of atmospheric turbulence increases mainly in one is two-dimensional Gaussian function [3] the case of wrong weather and observation conditions. 2 2 ! (k − x0) + (l − y0) The last phenomena described in this article and that fG(k; l; p) = A · exp − (4) 2σ2 can influence result PSF is focusing [20, 21], which can be expressed as follows where A is amplitude, x0, y0 are shifts in the x − y plane, 1 σ > 0 is its standard deviation and k, l are pixel indices as (r) = · δ(r; ρ) f 2 (10) coordinates. πρ The second model is statistical model described by Mof- where ρ > 0 is focusing radius and fat [3, 16], which is a generalization of Cauchy distribution ( 1 if r ≤ ρ; δ(r; ρ) = (11) A 0 otherwise; fM(k; l; p) = (5) 2 2 β (k−x0) +(l−y0) 1 + 2 σ Z 1 2J1(!ρ) where β is a shape parameter of PDF satisfying 0 ≤ β ≤ 50. F(!) = H ff(r)g = 2π rf(r)J0(!r)dr = : 0 !ρ As mentioned above, these two presented models are (12) commonly used in astronomical photometry using (3), but RADIOENGINEERING, VOL. 25, NO. 1, APRIL 2016 171 The focusing can be easily eliminated by setting ρ ! 0+ in the case of perfect focusing of real telescope. H(!) = FG(!) · F(!) · G(!); (19) But the importance of non-perfect focusing is in its combina- tion with interference in real telescope as will be explained and finally combination of Interference, Focusing and Tur- in the next section. bulence (IFT) 2.2 Combined Models H(!) = S(!) · F(!) · G(!): (20) In this article, there are compared previously mentioned This approach was used due to the reason that ana- models, i.e., Gauss, Moffat, Interference, Turbulence and Fo- lytical convolution of S(!) to s(r) is impossible, thus the cusing with more sophisticated models given by their con- data analyzed by combined models are processed only in the volutions, excluding Moffat. Thus, we can find 7 possible frequency domain, subsequently transformed into spatial do- combinations we can use. Possible combinations are listed main and properly modified for amplitude A, x0 and y0 shifts below and always mentioned in frequency domain with re- optimization. spect to the convolution theorem, as multiplying of Fourier Parameters of particular single models and their lower images. and upper bounds can be found it Tabs. 1 and 2, respectively. Those are Interference in combination with Turbulence (IT) PSF model parameters Gauss σ Moffat σ, β H(!) = G(!) · S(!) (13) Interference Ω Focusing ρ Turbulence , σ which is model of constrained observation in the case of Tab. 1. Parameters of basic PSFs. wrong atmospheric condition. Fortunately, the effect of tur- bulence can eliminate Airy disc around every point light model bound p1 p2 p3 p4 p5 p6 LB 1 1 1 1 source. Gauss UB 2 max(x(k; l) − d(k; l)) M N max(M; N) LB 1 1 1 1 0.01 Interference plus Focusing (IF) Moffat UB 2 max(x(k; l) − d(k; l)) M N max(M; N) 50 LB 1 1 1 0.01 0.0001 0.0001 Interference H(!) = G(!) · F(!) (14) UB 2 max(x(k; l) − d(k; l)) M N 10 10 10 LB 1 1 1 0.01 0.01 Focusing which is model of constrained observation in the case UB 2 max(x(k; l) − d(k; l)) M N 10 10 LB 1 1 1 0.01 0.01 of wrong focusing.
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