Jagiellonian University
Doctoral Thesis
Towards advanced astronomical imaging: new techniques of data reduction and their applications
Author: Supervisor: Aleksander Kurek Dr hab. Agnieszka Pollo
A thesis submitted in fulllment of the requirements for the degree of Doctor of Philosophy in the
Faculty of Physics, Astronomy and Applied Computer Science
30 marca 2017 ii
I’ve seen things you people wouldn’t believe. Attack ships on re o the shoulder of Orion. I watched C-beams glitter in the dark near the Tannhäuser Gate
Roy Batty iii
Abstract
Goal The main goal of this thesis is to present new advanced methods of data acquisition and reduction developed in order to increase the photometric eciency and angular resolution of astronomical imaging.
Methods I present an overview of the techniques I developed during my PhD studies. They include: (1) increasing the precision of bad pix- els removal, (2) impulse noise removal, (3) precise photometry and high angular resolution imaging of extremely faint sources, (4) ef- cient exposure times planning and (5) superresolved imaging of extended distant sources.
Results It was shown that (1) is is possible to interpolate over bad pixels in the CCD ∼4× more eciently than it is done by standard meth- ods. A review of impulse noise removal techniques demonstrated (2) that the standard method (Laplacian Edge Detection) is in most cases the most ecient one, however, there are exceptions – mainly astrometric applications. Our evolutionary algorithm-based meth- ods were shown to be able to: (3) recover the surface prole of sources which are only ∼2-3 % stronger than their background; and (4) nd an optimal way to divide the available observing time to multiple exposures, so that the average photometric error of a dense eld photometry was lowered by 0.05 mag. The endeavor to employ Optical Parametric light Amplication (OPA) to high an- gular resolution astronomical imaging (5) did not succeeded so far but its limitations were demonstrated.
Keywords: high angular resolution, imaging, photometry
Streszczenie
Cele Głównym celem rozprawy doktorskiej było rozwinięcie i przetesto- wanie metod pozyskiwania i redukcji danych w celu zwiększenia dokładności pomiaru fotometrycznego oraz rozdzielczości kąto- wej obrazowania astronomicznego.
Metodyka Przedstawiam przegląd opracowanych i zaproponowanych przeze mnie technik precyzyjnych obserwacji astronomicznych. Są to tech- niki: (1) precyzyjnej redukcji wadliwych pixeli (ang. bad pixels), (2) usuwania szumu impulsowego ze zdjęć astronomicznych, (3) pre- cyzyjnej fotometrii oraz obrazowania bardzo słabych źródeł z wy- soką rozdzielczością kątową, (4) efektywnego planowania czasu ekspozycji, oraz (5) superrozdzielczego obrazowania źródeł rozcią- głych.
Wyniki Wykazano, że (1) możliwe jest ∼4× precyzyjniejsze usuwanie bad pixels, niż standardowymi metodami. Przegląd metod usuwania szumu impulsowego wykazał (2), że domyślnie używana metoda (Laplacian Edge Detection) jest zwykle najefektywniejsza, ale w nie- których sytuacjach lepiej zastosować metodę Progressive Switching Median (PSM). Zaproponowane przez nas metody bazujące na al- gorytmach ewolucyjnych są w stanie: (3) odzyskać rozkład płasz- czyznowy źródła rozciągłego, które jest jedynie 2-3% jaśniejsze niż tło; oraz (4) znaleźć optymalny podział całkowitego czasu obser- wacji na poszczególne ekspozycje w taki sposób, aby średni dla wszystkich źródeł błąd fotometryczny obniżył się o 0,05 mag. Próba zastosowania wzmacniania parametrycznego światła (OPA) do zwięk- szenia rozdzielczości kątowej obrazowania astronomicznego (5) jak dotąd nie przyniosła pozytywnych rezultatów, ale wykazaliśmy ogra- niczenia stosowania OPA w tym celu.
Słowa kluczowe: wysoka rozdzielczość kątowa, obrazowanie, fotome- tria
vii
This dissertation has been written basing on the scientic results previously reported in the following articles:
• A. Popowicz, A. R. Kurek, Z. Filus, Bad pixel modied interpo- lation for astronomical images, 2013PASP..125.1119P
• A. Popowicz, A. R. Kurek, A. Pollo, B. Smolka, Beyond the cur- rent noise limit in imaging through turbulent medium, 2015OptL...40.2181P
• A. R. Kurek, T. Pięta, Tomasz Stebel, A. Pollo, A. Popowicz, Quantum Telescopes: feasibility and constrains, 2016OptL...41.1094K
• A. Popowicz, A. R. Kurek, T. Blachowicz, V. Orlov, B. Smolka, On the eciency of techniques for the reduction of impulsive noise in astronomical images, 2016MNRAS.463.2172P
Other results presented in this dissertation are described in the fol- lowing articles which were recently submitted:
• A. R. Kurek, A. Stachowski, K. Banaszek, A. Pollo, A. Popowicz, Parametric light amplication in astronomy: a quantum optical model (MNRAS)
• A. Popowicz, A. R. Kurek, Optimization of exposure time divi- sion for multiobject photometry (PASP)
ix Acknowledgements
I would like to express my highest gratitude to my Supervisor, dr hab. Agnieszka Pollo, for enormous help and patience during writing this thesis and thorough supervision over the past 4 years. I also thank for a very friendly atmosphere and numerous useful conversations.
xi
Contents
Abstract iii
Streszczeniev
Acknowledgements ix
1 Introduction: Angular resolution in astronomical imaging1 1.1 Targets of astronomical imaging...... 4 1.1.1 Exoplanets...... 4 1.1.2 Other targets of interest...... 5 1.2 Current technological capabilities...... 10 1.3 Fundamental limits...... 11 1.4 Conclusions...... 14 1.5 The goals and outline of the thesis...... 14
2 Selected methods of increasing of the angular resolution and photometric precision in astronomical imaging 15 2.1 Bad pixel removal...... 15 2.1.1 Introduction...... 15 2.1.1.1 Present methods of bad pixel inter- polation...... 17 2.1.2 Data...... 18 2.1.3 Test of interpolation methods...... 19 2.1.3.1 Comparison test...... 19 2.1.4 Modied interpolation...... 20 2.1.4.1 Presentation of the idea...... 20 2.1.4.2 Verication...... 24 2.1.5 Conclusions...... 25 2.2 Impulse noise reduction...... 29 2.2.1 Introduction...... 29 2.2.1.1 Stationary dark current...... 29 2.2.1.2 Non stationary dark current...... 31 2.2.1.3 Clock induced charge...... 33 2.2.1.4 Cosmic rays...... 34 2.2.2 Tested methods...... 35 2.2.3 Tests...... 38 xii
2.2.4 Results...... 38 2.2.5 Conclusions...... 42 2.3 Ecient use of telescope time...... 46 2.4 Evolutionary algorithms for image restoration..... 50 2.4.1 Introduction...... 50 2.4.2 Our method...... 51 2.4.3 Summary...... 56
3 Near future 59 3.1 Extremely Large Telescopes...... 59 3.2 Space telescopes...... 61 3.3 Interferometers, including Event Horizon Telescope. 61 3.4 Hypertelscopes...... 65
4 Far future 71 4.1 Quantum Telescopes / Optical Parametric Ampli- cation...... 71 4.1.1 Introduction...... 71 4.1.2 Classical model...... 73 4.1.2.1 Simulations...... 73 4.1.2.2 QT: technological feasibility...... 76 4.1.2.3 Conclusions...... 78 4.1.3 Semiclassical model...... 78 4.1.3.1 An updated QT concept...... 79 4.1.3.2 QT – a semiclassical model...... 79 4.1.3.3 Simulations...... 82 4.1.3.4 Results...... 82 4.1.3.5 Conclusions...... 86 4.1.3.6 Discussion...... 86 4.2 Quantum and optimal: SLIVER, SPADE...... 87 4.3 Hypertelescopes in Space...... 87
5 Summary 91
Bibliography 95 xiii
List of Abbreviations
ATLAST Advanced Technology Large-Aperture Space Telescope CCD Charge Coupled Device CT Classic Telescope DL Diraction Limit EELT European Extremly Large Telescope ELT Extremly Large Telescope EMCCD Electron Multiplying CCD EPE Extrasolar Planets Encyclopedia ESI Earth Similarity Index GA Genetic Algorithm HAR High Angular Resolution HST Hubble Space Telescope IRAF Image Reduction and Analysis Facility JWST James Webb Space Telescope LI Lucky Imaging OPA Optical Parametric Amplication PCB Proxima Centauri B PSNR Peak Signal to Noise Ratio RMS Root Mean Square RMS Random Telegraph Signals SR Strehl Ratio QT Quantum Telescope SNR Signal to Noise Ratio VLBI Very Long Baseline Interferometry
1
Chapter 1
Introduction: Angular resolution in astronomical imaging
There is a constant demand to increase the angular resolution in astronomical imaging. It concerns all possible wavelength spans, where the imaging was, is and is about to be performed. The angu- lar resolution of any classical (not based on quantum mechanical tricks) imaging system is limited by its aperture diraction, i.e. so called diraction limit1. Historically, obviously the rst instrument used for astronomi- cal observations was the human eye. Below I list selected features of the human eye as an astronomical instrument. The interested reader is referred to chapter 2 of Wyszecki and Stiles, 2000 for a more detailed information on the human eye properties from as- tronomical point of view, including the light losses, stray light con- tamination, photometric specication etc.
• Spectral response of the human eye is wider at the daylight than at night. The optimal sensitivity is at 550 nm, with corre- sponds to the V band in the standard in astronomy Johnson- Cousins UBVRI photometric system (Bessell, 2005)
• The eye‘s quantum eciency is ∼10 %, which is poor in com- parison even to modern smartphone cameras2.
• The eye‘s dynamic range is 1:109, which is ∼104× higher than in the case of any existing telescope equipped with the corono- graph (Beuzit et al., 2008) and ∼3.5×104 higher than the lat- est full-frame DSLR (Digital Single-Lens Reex camera) cam- eras. This oers up to 14.8 Exposure Values (EV) of dynamic
1D. L. is well described at Hyperphysics webpage; link: http:// hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cirapp.html#c1. 2See http://www.sensorgen.info/ for a list of sensors and their QE. Chapter 1. Introduction: 2 Angular resolution in astronomical imaging
range 3,4.
• The pupil diameter varies from 3-4 mm (day) to 5-8 mm (night). The upper limit decreases with age by 1-2 mm.
• If the eye would be diraction-limited, its angular resolution would be: λ 0.55 µm 180 deg 1 mm Θ = 1.22 rad = 1.22 rad = D 8 π rad 103µm (1.1) 360000 0.0048 = 17.300 1 deg
• Several other than the diraction limit physical factors limit the resolution of the human eye. The foveal cone spacing, neural trace and physiological tests all agree that the highest resolution of the human eye is 1 arcmin (0.02◦ or 0.0003 rad; Yano and Duker, 2008). This corresponds to 0.3 m at a 1 km distance or to 1.19×1010 km at the distance of the closest known Earth-like exoplanet Proxima Centauri B (PCB). In other words, PCB is ∼9.3×106 too small for the human eye to resolve ve largest details of its surface.
Despite the invincible dynamic range, the human eye is obvi- ously not tted for astrophysical research. A major breakthrough in human perception of astrophysical bod- ies occurred in 1610: Galileo Galilei presented the rst ever tele- scope (1.5 cm refractor; see Dupré, 2003). In the forthcoming years ever larger telescopes were constructed, i.e. next telescopes by Galileo (up to 3.8 cm in 1620), as well as by Huygens, Hershel and Rosse (see King and Space, 2011 for review). This telescopes al- lowed for deeper range and faster integration of the image. But – due to the smearing eect of the atmosphere – only after 1900 the rst Californian observations brought any progress in the angular resolution. In the second half of the XX century, further progress came thanks to observatories built under a very favorable sky in Chile and Hawaii. In 1994, the Hubble Space Telescope brought another major breakthrough (Krist, Hook, and Stoehr, 2011). As a spaceborne telescope with the main mirror size of 2.4 m, it al- lowed for diraction-limited imaging of extended objects with res- olution down to 0”.05 (Sirianni et al., 2005). For an intuitive refer- ence: the largest angular diameter of a star as seen from the Earth
3According to DxOMark Sensor Scores; link: https://www.dxomark.com/. 4All such simple calculation in this Chapter are made by me unless stated otherwise or the source citation is provided. Chapter 1. Introduction: 3 Angular resolution in astronomical imaging is 0.05 arcsec (50 mas for Betelgeuse, see Uitenbroek, Dupree, and Gilliland, 1998). An interested reader can nd more examples of HST results in Martel et al., 2003. It may be interesting to note that the Strehl ratio5 of the HST is 0.8 (Gonsalves, 2014, Chap. 3 and references therein). For this reason, the actual angular resolution of the HST is slightly lower than its diraction limit. Nevertheless, in this thesis for simplicity I comply to the popular assumption that HST is a totally diraction limited imaging system. Present technology does not allow for precise resolved imag- ing of a vast majority of astrophysical targets of interest because of the large distances. Current technological limits of the diameter of the telescope primary mirror (∼40 m on the ground; ELT Science Working Group, 2006; Liske, 2011) and ∼6.5 m in the space (Gard- ner et al., 2006) are many orders of magnitude below the size which allows for resolved imaging of such celestial objects as most of stars or central parts of galaxies. As mentioned below in Sec. 1.1.2, other desired targets are even much smaller angularly. Although existing optical interferometers achieve a resolution in imaging of up to 4 mas, their imaging capability is very limited and they can operate only on very bright targets and oer small elds of view (Eisenhauer et al., 2011). Moreover, no existing telescope project is aimed at sidestepping the diraction limit of the instrument (Kulka- rni, 2016). The latest progress was brought by the construction of imaging- capable optical interferometers. Currently, the interferometer of a largest baseline is The CHARA array (ten Brummelaar et al., 2005). Its 330 m max. baseline size allowed e.g. for the rst ever imaging of two stars orbiting the common center o mas (Baron et al., 2012). Movie composed of 55 H-band frames of this system recorded from 2006 to 2010 is avalilable on Internet. CHARA‘s interferomet- ric imaging resolution is 0.0005 arcsec in the infrared, but the lim- iting magnitude is very poor: ∼8 mag, which is much too low for a vast majority of targets of interest. This drawback unavoidably aects all current optical interferometers due to their very com- plex optical path: the light from the astrophysical target has to be reected usually more than 15× before the image formation takes place.
5The Strehl ratio (SR) is a measure of the quality of optical image formation. It has a value in the range 0 to 1, where 1 corresponds to an unaberrated optical system. SR is frequently dened as the ratio of the peak aberrated image inten- sity from a point source compared to the maximum intensity attainable when using an ideal optical system limited only by diraction over the system’s aper- ture. Chapter 1. Introduction: 4 Angular resolution in astronomical imaging
Presently, radio interferometers oer higher angular resolution than optical, because it is easier to build large baseline instruments, since the image restoration can be performed o ine. In this thesis I focus mainly on UV, optical and near-infrared imaging, since that was the object of my work during PhD studies. However, I mention radio interferometers in Sec. 3.3.
1.1 Targets of astronomical imaging
1.1.1 Exoplanets
One obviously desired imaging targets are the exoplanets. Recent discovery of an Earth-like exoplanet orbiting the closest star to the Sun, Proxima Centauri (PCB), brought a new hope for the search of extraterrestial life in the Universe (Anglada-Escudé et al., 2016; Turbet et al., 2016). Below I list some quick facts, although rela- tively very litte is known about this particular object:
• PCB is ∼10-7× darker than its host star.
• The planet-star separation angle is 37 mas (Turbet et al., 2016).
• According to recent upper limits obtained with the SPHERE high-contrast imager (Beuzit et al., 2008), the mass of the
object is less than 4 MJup and the radius is less then a few RJup (Mesa et al., 2016). There is no possibility to obtain tighter constrains with presently available instruments.
Luckily, with these parameters, there still exists a high proba- bility of successful imaging as soon as in a few years (Lovis et al., 2016). It is almost certain that no closer Earth-like exoplanets are to be discovered, since there are no known stars closer than Prox- ima Centauri. Given all that, PCB seems a reasonable reference in the discussion about resolved imaging of exoplanets‘ surface. For further derivations I selected two popular and frequently up- dated online catalogs of known exoplanets:
• The Extrasolar Planets Encyclopaedia6 (EPE) hosted by Paris Astronomical Data Centre. Data version from 17 Jan. 2017.
• The Habitable Exoplanets Catalog7 (PHL) managed at Plane- tary Habitability Laboratory, University of Puerto Rico, Arecibo. Data version from 28 Mar. 2017. 6Link: http://exoplanet.eu/. 7Link: http://phl.upr.edu/projects/habitable-exoplanets-catalog. 1.1. Targets of astronomical imaging 5
A series of gures based on these data and presented in this sec- tion illustrates the main challenges and diculties in the perennial endeavour of resolved imaging of exoplanets. Fig. 1.1 is based on the EPE data and depicts the angular size of exoplanets as a function of their distance from the Sun. The ex- oplanets are divided into two sets: a) exoplanets of Terran radius and b) the remaining objects, where a vast majority has a larger, Neptune-like radius. Fig. 1.2 presents a histogram of the same data. Figures 1.3 and 1.4 are based on the PHL data, where more so- phisticated denition of exoplanets‘ similarity to Earth is used - the Earth Similarity Index (ESI). It is an open multiparameter measure of Earth-likeness and it is a function of an exoplanet’s stellar ux and its radius. It is a number between 0 (no similarity) and 1 (identical to Earth)8. The value of 0.60 used on charts was chosen discre- tionarily. Let us note here, that the vertical scales of charts are in µarcsec. For an intuitive reference, the width of a little nger at arm’s length is 1◦. This width divided by 3600 is 1 arcsec. The estimated diam- eter of PCB is ∼71 µarcsec (Seager et al., 2007), consequently the latest width needs to be divided once again – this time by 71/106 = 14 000 – to nally match the angular size of PCB. As emerging from the above divagations and presented Fig- ures, telescopes with far better angular resolution than presently available are necessary to produce resolved images of any exo- planet.
1.1.2 Other targets of interest
There are multiple other than exoplanets targets of interest in as- trophysics, both much smaller and much larger angularily. Here there is a list of the largest angularily objects of selected classes:
• The largest angularily object as seen from Earth is the Galac- tic Bulge (25◦×10◦). However, because of its low surface bright- ness, its imaging already requires an imaging system (tele- scope, camera).
• The largest angularily galaxy is not, as frequently popularly assumed, Andromeda Galaxy (160’×60’), but The Canis Major Dwarf Galaxy (CMa Dwarf) or Canis Major Overdensity (CMa Overdensity): 12◦×12◦ (Martin et al., 2004). However, there are still doubts on the true nature of this object, since it is not
8For a mathematical description see the project‘s website; link: http:// phl.upr.edu/projects/earth-similarity-index-esi. Chapter 1. Introduction: 6 Angular resolution in astronomical imaging
103 Neptune - like or other Proxima Centauri B 102 terran radius arcsec]
101
100
10-1 angular size [ 10-2 100 101 102 103 104 105 distance [LY]
Figure 1.1: Angular size of exoplanets as a function of their distance from Earth. Exoplanets are devied in two sets based on their estimated radius: Earth-like and Neptune-like. Proxima Centauri B, the closest Earth-like exoplanet, is marked by red “×”. Data from EPE.
proven that the over-densities in Monoceros are not due to the ared thick disc of the Milky Way (Lopez-Corredoira et al., 2012).
• The largest angullary emission nebula is Bernard‘s Loop, 10◦×10◦. Interestingly, its distance is still unknown, ranging from 518 (Wil- son et al., 2005) to 1434 (O’Dell et al., 2011) light years.
• Centaurus A (NGC 5128) radio lobes, although photographed so far only in the radio band of 20 cm, have an apparent size of 10◦×4◦ (Schreier, Burns, and Feigelson, 1981).
• The largest angularily supernova remnant is The Vela Super- nova Remnant, 8◦ (Cha, Sembach, and Danks, 1999).
• The angular diameter of the Moon is 29‘20“ – 34‘6“ 9. Given its distance from Earth, the Apollo Mission landing artifacts (∼4 m in size) are ∼2 mas in angular size. The HST resolution at the Moon is ∼200 meters / pix. For this reason, HST was never able to disprove the “Moon hoax” conspiracy theories. A 120 m or larger optical telescope would be necessary for that. 9Moon Fact Sheet at NASAs National Space Science Data Center (NSSDC). 1.1. Targets of astronomical imaging 7
100 all; count: 1302 terran radius; count: 382 90
80 10 largest angularily: (w/o Prox. Cent. B) 70 WASP-136 b GJ 674 b 60 51 Peg b WISE 0458+6434 b WISE 1217+16A b 50 Ross 458 (AB) c
Count beta Pic b 40 2M 0746+20 b tau Boo b HD 7924 b 30
20
10
0 0.05 0.13 0.36 1 2.72 7.39 20 55 150 400 angular size [ arcsec]
Figure 1.2: Histogram of angular sizes of known exoplanets. Ten largest angularily exoplanets are listed. Exoplanets are divided in a similar way as in Fig. 1.1. Data from EPE.
Present optical interferometers, although still incapable of provid- ing detailed images, can do very well in estimation of stellar radii.
• The angularily largest star is Betelgeuse: 49∼60 mas (mil- iarcsec; Uitenbroek, Dupree, and Gilliland, 1998) or R Doradus: 57 mas (Richichi, Percheron, and Khristoforova, 2005).
• The apparent size of Proxima Centauri is ∼495 µas (Demory et al., 2009). The size of PCB exoplanet is ∼71 µas (Seager et al., 2007). Looking from PCB, the angular size of the Sun would be ∼7.7 mas, and of Earth ∼64 µas.
• When looking at a black hole, general relativity predicts that a distant observer would see a bright photon ring enclosing a darker shadow region due to the illumination by the hot ma- terial that surrounds the event horizon. The predicted size of such a shadow for Sagittarius A* radio source (probably a Chapter 1. Introduction: 8 Angular resolution in astronomical imaging
103 Neptune - like or other Proxima Centauri B 2 10 ESI > 0.60 arcsec]
101
100
10-1 angular size [ 10-2 100 101 102 103 104 105 distance [LY]
Figure 1.3: Angular size of exoplanets as a function of their distance from Earth. Exoplanets are divided in two sets basing on Earth Simi- larity Index: Earth-like (assumed ESI > 0.60) and Neptune-like. PCB is marked using red “o”. Data from PHL.
black hole) is ∼30 µas, with approaches the resolution of cur- rent radio interferometers (see Falcke, Melia, and Agol, 2000 for in depth description of the origin of the shadow). The lat- est review on soon-to-come radio imaging of such targets is given by Goddi et al., 2016. One more target in range for such imaging is the black hole in the center of galaxy M87 (Lu et al., 2014) – much further, but also the black hole is much larger. In this case less is known about the angular size of the source, but it is estimated to be ∼7 µas (ibid.).
• The diameter of a neutron star inside Crab Pulsar is ∼20 km (Becker and Aschenbach, 1995), which gives an apparent size of ∼6.08e-05 µas. There are already some ideas for imaging even such dicult targets; see Labeyrie, 1999 and Sec. 3.4 of this thesis).
• There exist already some techniques basing on lensing events facilitating the discovery of exoplanets outside our Galaxy (In- grosso et al., 2009). The probability of such a detection is low: a 4-meter instrument watching Andromeda Galaxy (M31) for 9 months in total might pick up one or two planets. One ex- oplanet outside our Galaxy may have been already discov- ered (Setiawan et al., 2010). Its parent star HIP 13044 belongs to a group of stars that have been accreted from a disrupted 1.1. Targets of astronomical imaging 9
120 Neptune-like or other; count: 2509 ESI > 0.60; count: 90 10 largest angularily: 100 eps Eridani b GJ 674 b GJ 876 b GJ 876 c 80 GJ 832 b GJ 570 Db HD 95872 b Fomalhaut b 60 51 Peg b
WD 0806-661 B b Count
40
20
0 0.02 0.13 1 7.39 55 403 angular size [ arcsec]
Figure 1.4: Histogram of angular sizes of known exoplanets. Ten largest angularily exoplanets are listed. Exoplanets are marked in similar way as in Fig. 1.3. Data from PHL.
satellite galaxy of the Milky Way. HIP 13044 is located ∼2k LY from the Sun10, so that it is at a closer distance than some ex- oplanets plotted in the gures in this chapter. A potential PCB-class exoplanet in the closest to the Sun part of M31 would be at a distance of 2472 - 220/2 kLY = 2362 kLY from the Sun. Such a distance does not yet require employ- ment of cosmological distance measures11 (McConnachie et al., 2005; Chapman et al., 2006). Such a planet would have an angular diameter of 0.13 nas. This is 2-3× smaller than an averaged-size bacteria on the Moon as seen from Earth.
As emerging from the discussion above, for resolved imaging of other distant targets than exoplanets, far better instrumentation is also needed. 10 SIMBAD query result 11The redshift of M31 is z = -0.001001 Chapter 1. Introduction: 10 Angular resolution in astronomical imaging
1.2 Current technological capabilities
Continuing using exoplanets as a reference, Figures 1.5 and 1.6 demonstrate, how many times too small is the HST to produce a 5×5 pixels image of any known exoplanet from, respectively, EPE and PHL catalogs. These gures make it obvious that no monolithic- mirror telescope will ever be capable for resolved imaging of exo- planets. Moreover, the same applies to the expandable mirror tele- scopes, like the upcoming James Webb Space Telescope (Gard- ner et al., 2006). As shown in Fig. 1.7, even the future Advanced TechnologyLarge-Aperture Space Telescope (Postman et al., 2009; Postman et al., 2010), the successor of both HST and NGST, will not introduce any noticeable progress in this eld. The telescope of a diameter or the interferometer of baseline of at least >1 km is nec- essary to start producing resolved images of PCB (Fig. 1.8). Despite not yet being capable to perform resolved imaging of exoplanets, thanks to a larger collecting area and progress in detectors’ tech- nology, ATLAST is supposed to be ∼2000× more sensitive than HST, which may be sucient for detecting biosignatures (Sterzik, Bagnulo, and Palle, 2012; Hegde et al., 2015; Claudi and Erculiani, 2014) on extrasolar planets.
107 108
106 107
105 106
104 105 5 pix. image [m]
103 Neptune - like or other 104 Proxima Centauri B terran radius smaller than HST limit 102 103 0 1 2 3 4 5 10 10 10 10 10 10 D for 5 distance [LY]
Figure 1.5: Left axis: how many times lower is the angular resolution of HST as compared to the angular sizes of exoplanets. Right axis: re- quired telescope‘s main mirror size for 5×5 pix. images of exoplanets at 550 nm. Both as a function of the distance from Earth. Exoplanets are divided like in Fig. 1.1. PCB is marked by red “×”. Data from EPE. 1.3. Fundamental limits 11
107 108 Neptune - like or other Proxima Centauri B ESI > 0.60 106 107
105 106
4 5 10 10 5 pix. image [m]
103 104 smaller than HST limit D for 5 102 103 100 101 102 103 104 105 distance [LY]
Figure 1.6: Left axis: how many times too small is the angular resolution of HST from the angular sizes of exoplanets. Right axis: required tele- scope‘s main mirror size for 5×5 pix. images of exoplanets at 550 nm. Both as a function of the distance from Earth. Exoplanets are divided like in Fig. 1.3. PCB is marked by red “o”. Data from PHL.
1.3 Fundamental limits
There are at least two groups of fundamental limits on high angular resolution (HAR) astronomical imaging in general.
1. It is not yet clear, whether the optical homogeneity of the space, with its interstellar medium (Draine, 2011), gravity gra- dients (Krumholz and Burkhart, 2016) and gravitational waves (Fakir, 1997; Stinebring, 2013), suces to preserve the cophas- ing; or if adaptive techniques and deconvolution methods can restore it (Fish et al., 2014; Johnson and Gwinn, 2015). This is a complicated issue and up to date it was studied mainly in the contexts of ultraprecise astrometry of the stars. Cer- tainly, the amount of distortion varies with the direction. An expected average deviation of stellar positions by the gravi- tational microlensing eect in our Galaxy is ∼7 µas (Yano, 2012). On the extragalactic scales, additionally, the same eect is in- troduced by the distortion of geometry of the space by the dark matter. In summary, we see the following factors a) disturbing the Chapter 1. Introduction: 12 Angular resolution in astronomical imaging
1012
All known exoplanets 2 Proxima b 10 SWEEPS-04b, dist.: 22k LY 1011
101 1010
9
10 100
km in 1 HST pix. HST 1 in km AU in 1 ATLAST pix. ATLAST 1 in AU 108 10-1
107 100 101 102 103 104 105 Distance [LY]
Figure 1.7: Left vertical axis denotes how many km correspond to 1 HST pixel at a distance of a given exoplanet at 550 nm. Right vertical axis shows the same value for the future Advanced Technology Large- Aperture Space Telescope (ATLAST), the successor of both HST and NGST telescopes. A 9.2 m main mirror diameter version of ATLAST is assumed.
spacetime shape or b) increasing the uncertainty of the mo- mentum of the photons, and, thereby, blurring the image:
• Diraction on interstellar and intergalactic medium, • Gravitational waves, • Gravity gradients caused by: – interstellar and intergalactic medium, – stars and any baryonic objects in general, – lensing of any kind: strong, weak and microlensing; see Bastian and Hefele, 2005 for overview or Narayan and Bartelmann, 1996 for lectures, – so called femtolensing, caused by extremely small (∼10-13 - 10-16 solar mass) dark-matter objects (Gould, 1992); on extragalactic scales only, – dark matter in general (also on extragalactic scales only). 1.3. Fundamental limits 13
102
101
100
10-1
image pixels image Proxima Centauri B Centauri Proxima Hypertelescope 10-2 HST 1 1 pix. 5 5 pix. 10-3 100 101 102 103 104 105 D [m]
Figure 1.8: Number of resolved elements in PCB image at 550 nm as a function of an interferometer / hypertelescope baseline.
2. There are fundamental quantum-mechanical limits on the localization of a thermal point source in general. Theory of this issue is very intensively developed since 2015 (Ang, Nair, and Tsang, 2016; Nair and Tsang, 2016). The same limits by the rule of thumb also apply to the imaging of extended tar- gets (Tsang, 2016). Many specialists believe that of these two groups, the optical inhomogenity of the space is the lower limit on larger scales. But the total eect from these groups of limits – either each of the groups separately or both of them together – was never studied in astronomy in the context of HAR resolved imaging. Other limitations also may exist, however their existence was not proven so far. Among them there are e.g.:
• the spacetime roughness predicted by the holographic prin- ciple (Susskind, 1995), with would blur the sight only at large cosmological scales. Current results from observing targets at cosmological distances show that probably no such eect exists (Steinbring, 2016; Perlman et al., 2015).
• cosmic strings – hypothetical 1-dimensional topological de- fects which may have formed during a symmetry breaking Chapter 1. Introduction: 14 Angular resolution in astronomical imaging
phase transition of the Universe (Hindmarsh and Kibble, 1995). A cosmic string would produce a duplicate image of uctua- tions in the cosmic microwave background, and thus should have been detectable by the Planck satellite (Fraisse et al., 2008). However, a 2013 analysis of Planck data failed to nd any evidence of cosmic strings (Ade, 2014) and the latest 2015 analysis did not bring any new conclusion in this issue.
1.4 Conclusions
After presenting all this facts, it seems obvious that the need for increasing the angular resolution in imaging is everlasting. In next chapters (2,3 and4) I will describe selected current and future methods of HAR imaging in development of which I actively par- ticipated.
1.5 The goals and outline of the thesis
In this thesis, selected methods of improving astronomical imaging in the UV, visible and IR are presented. By implementing modern methods of data acquisition and reduction, a considerable gain in the data quality and scientic output can be frequently achieved. For this purpose, during my PhD studies, in collaboration with a small but well-selected group of coauthors, I developed the sub- jects listed below. The results are described in the subsequent Sections of this thesis:
• Ecient bad pixel removal method – Sec. 2.1,
• Ecient impulse noise reduction in astronomical images – Sec. 2.2,
• Ecient use of the time of telescopes – Sec. 2.3,
• Evolutionary algorithms for image restoration – Sec. 2.4,
• A possibility of employing parametric light amplication for extremely high angular resolution imaging in astronomy – Sec. 4.1. 15
Chapter 2
Selected methods of increasing of the angular resolution and photometric precision in astronomical imaging
2.1 Bad pixel removal
The research on bad pixels removal described in this Sect. was originally published in Bad Pixel Modied Interpolation for Astro- nomical Images, Popowicz, Kurek, and Filus, 2013. After ∼4 years this paper has 6 citations1. My role was the research planing, a part of data nalysis and discussion. I was also responsible for manuscript typesetting, preparation and submission. The following text is a slightly extended version of this article.
2.1.1 Introduction
Astronomical images taken with electronic image sensors are nowa- days one of the most important research tools of the modern as- tronomy (Saha, 2009; McLean, 2008). The most popular are CCD and CMOS sensors, which consist of a matrix of pixels, where the light ux is measured thanks to the photovoltaic eect (Litwiller, 2001; Janesick, 2001; Milnes, 1980). Unfortunately, not every pixel can be used eectively, mainly due to the possible imperfections located within the pixel. The most commonly encountered prob- lems are:
• high dark current rate saturating the pixel’s potential well,
1Source: NASA ADS. Chapter 2. Selected methods of increasing of the angular 16 resolution and photometric precision in astronomical imaging
• nonlinear dark current dependencies (Widenhorn, Dunlap, and Bodegom, 2010a; Dunlap et al., 2012; Popowicz, 2011; Popow- icz, 2011),
• transient events of the dark current due to the irradiation (es- pecially important for ight missions; Hopkinson, Goion, and Mohammadzadeh, 2007; Hopkinson, Dale, and Marshall, 1996),
• pixel nonlinear light response,
• CCD fabrication defects (Janesick, 2001). In professional CCD systems there are several methods developed for investigation of the so-called bad pixels (Hilbert, 2012; Hilbert and Petro, 2012). During data reduction the bad pixel masks are created to mark the defected CCD areas. This calibration is usually repeated periodically, because new bad pixels can appear over time. One way to reduce bad pixels impact on the image quality is to take many images with slight shifts of the led of view. Such a set of dithered pictures is programatically shifted back and av- eraged, ignoring the pixels from the masks (Fruchter and Hook, 2002). This approach is not possible if for any reason there is only a single image. In such cases an interpolation over bad pixels is nec- essary. The most popular method is a linear interpolation which is a standard procedure in the most widespread astronomical software package: Image Reduction and Analysis Facility (IRAF; see Massey, 1997)2. In Popowicz, Kurek, and Filus, 2013 we proposed a new method of bad pixel interpolation for astronomical images. In this thesis, in section: • 2.1.1.1 we describe the most popular interpolation methods and also introduce biharmonic interpolation algorithm as a new alternative for bad pixel correction; • 2.1.2 we present the data which was chosen for the methods comparison; • 2.1.3.1 we summarize the methodology of our tests and dis- cuss the results; • 2.1.4.1 we present our new method which is a modication of the biharmonic interpolation. The modication is dedicated especially for astronomical pictures calibration. We suggest some further enhancements which improve its eectiveness and precision.
2For an up to date overview of software used in astronomy, see Momcheva and Tollerud, 2015. 2.1. Bad pixel removal 17
• The result of our modied method compared to the original biharmonic interpolation are presented in 2.1.4.2; • We conclude in 2.1.5.
2.1.1.1 Present methods of bad pixel interpolation
Five popular interpolation algorithms were compared with respect to pixel brightness estimation in the astronomical images:
• median of the surrounding pixels – the simplest among the presented methods. The brightness of the pixel in such a case is simply replaced by the brightness of an adjacent pixel. It is to be chosen which of four adjacent pixels to select. Al- though the selection is arbitrary, it does not inuence the overall estimation quality.
• nearest neighbor interpolation – uses the median of sur- rounding pixels. In our tests, for this method, we chose the median of all 8 nearest surrounding pixels’ brightness.
• linear interpolation – the most popular method for bad pixel interpolation in astronomical images. It is implemented in widespread astronomical software package IRAF in the x- pix procedure (Massey, 1997). The interpolation is based on the linear interpolation along columns or rows of the image. In such a case simple mean value of the pixels adjacent to the bad pixel (in a row or in a column) is computed. Again, the decision of choosing row or column interpolation does not change the mean estimation error.
• cubic spline interpolation – the algorithm uses piecewise polynomials, called splines, to interpolate over the bad pixel along columns or rows. It nds the smoothest curve that passes through data points (Ahlberg, Nilson, and Walsh, 1967). In our experiments we used cubic spline interpolation, which means that we chose the third order polynomials.
• biharmonic interpolation – this method has not been used for the bad pixel interpolation yet. It is based on the linear combination of Green functions centered at each data point. Originally, the idea was applied for GEOS-3 and SEASAT al- timeter data in 1987 (Sandwell, 1987).
• our adjusted method – it is described in (Popowicz, Kurek, and Filus, 2013) and below. Chapter 2. Selected methods of increasing of the angular 18 resolution and photometric precision in astronomical imaging
Figure 2.1: An example of the SDSS astronomical image used in comparison test. Image info: 2.5 m SDSS telescope, exposure time 53.9 sec., RA 194.45516, DEC 0.005886, date: 21/03/99, lter: g. Origi- nally published in: Popowicz, Kurek, and Filus, 2013.
2.1.2 Data
As a dataset for testing the methods of interpolation we decided to use the astronomical images obtained by the Sloan Digital Sky Survey (SDSS; York et al., 2000; York et al., 2000) Data Release 73 (Abazajian et al., 2009). SDSS data are easily available through an easy to use graphical WWW interface and suits well our needs. The SDSS uses 2.5 m Ritchey-Chretien telescope located at Apache Point Observatory in New Mexico (Gunn et al., 2006). The imaging camera consists of an array of 30 CCDs with total eld of view of 3◦ and operating in drift-scan imaging mode. We used the mentioned interface, SDSS Science Archive Server,(SAS4) to download 1000 typical for SDSS astronomical images. The images were previously calibrated by standard SDSS pipeline reduction procedures. We chose SDSS g lter5 (Smith et al., 2002) images located at random positions on the sky (randomized RA and DEC coordinates). We present an exemplary image in Fig. 2.1.
3That was the latest release in the time of performing the tests. Presently version 13 is the latest. 4Link: http://www.sdss.org/dr12/. 5Link: http://skyserver.sdss.org/dr1/en/proj/advanced/color/ sdssfilters.asp. 2.1. Bad pixel removal 19
2.1.3 Test of interpolation methods
2.1.3.1 Comparison test
For the testes presented here, 1000 calibrated astronomical im- ages from SDSS data server were used. From every single im- age, two hundreds of 7×7 pixel fragments were extracted (see Fig. 2.2). The fragments containing whole or only parts of astro- nomical objects (e.g. stars, galaxies), were selected with use of a threshold-based algorithm. It prevented from choosing fragments with the image background only. The objects were not centered in each fragment. We did not use any special algorithm for ob- ject classication, so both point-like and extended objects were in- cluded in our tests. The center pixel of every fragment was chosen as an unknown and its brightness was estimated using previously mentioned interpolation methods. The brightness estimation was compared with the true value to calculate an error. The error abso- lute and relative value histograms for given interpolation methods are presented in Figs. 2.3 and 2.4. Additionally, a mean error E was computed to enable a quick comparison of the methods’ eec- tiveness. The following averaging formula was used:
N X E = |∆i|, (2.1) i=1 where: E label mean error, ∆i — i-th estimation error (absolute or relative) and N — total number of estimations. We also present point-plots (Fig. 2.5) to visualize dependency between the esti- mation error and the brightness of interpolated pixel for all tested methods. There is a clear dierence in eectiveness between the used methods. The best results were obtained with the biharmonic in- terpolation with a mean error equal to 465 ADU6 and 0.063 rela- tive, while the typically used method — linear interpolation — has an error far greater (872 ADU and 0.125 relative). The basis and an implementation of the most accurate interpolation are quite sim- ple. According to Sandwell, 1987 the biharmonic interpolation is based on computation of αi coecients (Eq. 2.2). For given values of the intermediate points, the system of equations 2.2 can be eas- ily solved as it is a linear one. The biharmonic interpolation as the most accurate method was used as a starting point for further en- hancement. It should be also noticed that there is no evidence of applying such an interpolation for astronomical images reduction.
6Analog-Digital Units – it is a measure of signal from a CCD/CMOS pixel. Chapter 2. Selected methods of increasing of the angular 20 resolution and photometric precision in astronomical imaging
Figure 2.2: Exemplar fragments, 7x7 pixels. Interpolated pixel is pointed by an arrow. Originally published in: Popowicz, Kurek, and Filus, 2013.
N q X 2 2 2 2 w(xj, yj) = αi (xj−xi) +(yj−yi) 2 ln (xj − xi) + (yj − yi) −1 , i=1 (2.2) where: w(xi, yi) — value of the interpolated function for xi, yi coor- dinates, N — number of intermediate points.
2.1.4 Modied interpolation
2.1.4.1 Presentation of the idea
The idea of the modied interpolation is based on the similarity be- tween the objects (both point sources like stars or extended ob- jects like galaxies) encountered in the astronomical images. Ac- cording to McLean, 2008, stellar light is aected by the atmo- sphere (so-called "seeing") and is blurred so that it is registered in the image not as a point, but as a point-spread-function (PSF). The PSF can be modeled by the Gaussian surface (Pan and Zhang, 2009). Usually each star in a high quality telescope system and for more than several seconds exposure shows very similar PSF. How- ever, sometimes PSF can be dierent due to the seeing variations during the exposure or due to the optical aberrations. Our modi- ed interpolation idea can be applied whether the PSF is stable or not. We construct the database of 7×7 pixel fragments from the parts of collected images not aected by the bad pixels. Because 2.1. Bad pixel removal 21
biharmonic interpolation, E=465 [ADU] cubic spline interpolation E=613 [ADU]
4 linear interpolation E=872 [ADU] 10 surrounding pixel median E=1171 [ADU] nearest neighbor E=2353 [ADU]
3 10 Count
2 10
1 10 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Estimation error [ADU]
Figure 2.3: Histogram of estimation error for the examined interpola- tion methods. Solid line – biharmonic interpolation, dashed line – cubic spline interpolation, dotted line – linear interpolation, solid bold line – surrounding pixel median, dashed bold line – nearest neighbor. Origi- nally published in: Popowicz, Kurek, and Filus, 2013.
5 10
4 10
3 10 Count
2 nearest neighbour, E=0.334 10 surrounding pixel median, E=0.154 linear interpolation, E=0.125 cubic spline interpolation, E=0.087 1 biharmonic interpolation, E=0.063 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Estimation error / pixel true brightness
Figure 2.4: Histogram of relative estimation error for the examined in- terpolation methods. Solid line – biharmonic interpolation, dashed line – cubic spline interpolation, dotted line – linear interpolation, solid bold line – surrounding pixel median, dashed bold line – nearest neighbor. Originally published in: Popowicz, Kurek, and Filus, 2013. Chapter 2. Selected methods of increasing of the angular 22 resolution and photometric precision in astronomical imaging
Figure 2.5: Dependence of estimation error on interpolated pixel brightness. Originally published in: Popowicz, Kurek, and Filus, 2013. 2.1. Bad pixel removal 23 most of the image is usually the background, a special threshold- based selection algorithm is used again to choose as many frag- ments with the objects as possible. Additionally, every fragment is rotated three times (by 90◦, 180◦ and 270◦) to extend the database and to provide for dierent orientation of extended objects. Af- ter the fragments’ acquisition, the biharmonic interpolation in ev- ery fragment is proceeded over the pixels – except for the central pixel. The brightness estimation of the central pixel is compared with the true (already known) value and the estimation error ek is computed. The biharmonic interpolation coecients αik and the corresponding estimation error ek are stored in the database as a single reference pair. The principle of supporting the interpola- tion with the database is to compare computed interpolation co- ecients with the references from the database and to nd the ref- erence which is best suited. The mean square tting error (Eq. 2.3) between the reference and current coecients is used during the search. s PN (α − α∗)2 J = i=1 ik i −−−−→k min , (2.3) k N where: Jk — mean square tting error between k-th reference and current coecients, αik — i-th coecient of k-th reference in the ∗ database, αi — i-th coecient of the interpolation of current frag- ment and k — the number of the most suitable reference. After nding the best reference, the corresponding estimation error is basically subtracted from current estimation. In sum: the method uses the biharmonic interpolation and is based on a properly created database of known fragments of the image containing astronomical objects. During the interpolation, the database is searched to nd the most suitable reference and to apply the corresponding correction to current brightness esti- mation. To improve the eciency of the method, it is desired to nd not one but a few (in the experiments the number of 5 frag- ments was used) of the most suitable references and to subtract a weighted average of the corresponding estimation errors (Eq. 2.4). v uP5 1 u i=1 J ei e = t i , (2.4) P5 1 i=1 Ji where: e — averaged estimation error, Ji — mean square tting error between i-th best tted reference and current coecients, ei - estimation error of i-th best tted reference. Another additional improvement is to normalize every fragment Chapter 2. Selected methods of increasing of the angular 24 resolution and photometric precision in astronomical imaging before any interpolation – both during the database creation and during a bad pixel correction (Eq. 2.5). The normalization enables to compare fragments of very bright objects with dimmer ones. The bad pixel estimated brightness in such a solution has to be followed by a simple renormalization (Eq. 2.6).
p − min(p) p0 = i , (2.5) i max(p) − min(p)
0 pi = pi max(p) − min(p) + min(p) , (2.6) 0 where: pi — normalized pixel brightness; pi — real pixel brightness; min(p), max(p) — minimal and maximal real brightness among the pixels in a fragment.
2.1.4.2 Verication
Here the database was created using fragments from the rst 10 pictures from the previously mentioned (Sec. 2.1.2) SDSS image set. Remaining images (990 pictures) were used as a verication set. The results are presented for three modied interpolation types: a) without averaging and normalization, b) with averaging, c) with averaging and normalization – and for the original biharmonic in- terpolation without modication (Figs. 2.6. and 2.7). In another test, it was examined if the modied interpolation can be useful for a single image correction. The database had to be created from the parts of an image that were not aected by faulty pixels. Six ran- domly chosen images from the SDSS survey were used. 80% of the 7×7 pixel fragments were used as a database and the remain- ing 20% were the test set. The results are resented in Fig. 2.8. Ad- ditionally, we compared the eectiveness of the method for the individual frame correction using database constructed from the same frame (minimal database) and from other 20 frames (large database). We analyzed 50 images for this test. The results are presented in Figs. 2.9 and 2.10. The mean error for the test with a large database showed about 50% decrease after using the modied interpolation idea. Accord- ing to the histograms (Figs. 2.6 and 2.7), the number of small errors rose and the number of large errors decreased as the modied interpolation was used. The proposed enhancements improved also the results of the method. It was shown that both enhance- ments reduced the mean error by about the same value: 20 ADU. However, in terms of relative error, the normalization improved the 2.1. Bad pixel removal 25
4 x 10 2.5 modified interpolation (no enhancement) E=266 [ADU] modified interpolation 2 (with normalization) E=243 [ADU] modified interpolation (with normalization and weighting) E=222 [ADU] biharmonic interpolation 1.5 (no modification) E=465 [ADU]
Count 1
0.5
0 50 100 150 200 250 300 350 400 450 500 Estimation error [ADU]
Figure 2.6: Histogram of estimation error for modied biharmonic in- terpolations. Solid line – biharmonic interpolation without modication, solid bold line – modied biharmonic interpolation without enhance- ments, dashed bold line – modied biharmonic interpolation with nor- malization, dash-dot solid line – modied interpolation with normaliza- tion and weighting. Originally published in: Popowicz, Kurek, and Filus, 2013. precision only slightly (form 0.044 to 0.042) in comparison to the weighting improvement (from 0.042 to 0.032). The modied interpolation seems to work well also as a single image correction, where the database has to be created from frag- ments of the same image. However, with such a minimal database, the estimation improvement will be strongly dependent on the im- age and the positions of bad pixels (Fig. 2.8). For our set of 50 im- ages, the use of larger database improved the estimation error no- ticeably (Figs. 2.9 and 2.10). However, for the images aected by strong PSF variation (e.g. due to the seeing problems), the results would be dierent.
2.1.5 Conclusions
A comparison of interpolation methods of bad-pixels correction in astronomical images was presented. The images from the Sloan Digital Sky Survey were used as an examination set. The bihar- monic interpolation as the most accurate method was enhanced with the idea of supporting it with a database of known astronom- ical image fragments. The test with a large database and a mini- mal database proved the eectiveness of the method as a pixel’s brightness estimator and its superiority over other examined in- terpolation methods. Moreover, the biharmonic interpolation has Chapter 2. Selected methods of increasing of the angular 26 resolution and photometric precision in astronomical imaging
4 x 10 2.5 biharmonic interpolation (no modification), E=0.063 modified intrpolation (no enhancement), E=0.044 2 modified interpolation (with normalization), E=0.042 modified interpolation (with normalization and weighting), E=0.032 1.5
Count 1
0.5
0 0 0.05 0.1 0.15 Estimation error / pixel true brightness
Figure 2.7: Histogram of relative estimation error for modied bihar- monic interpolations. Solid line – biharmonic interpolation without modication, solid bold line – modied biharmonic interpolation with- out enhancements, dashed bold line – modied biharmonic interpo- lation with normalization, dash-dot solid line – modied interpolation with normalization and weighting. Originally published in: Popowicz, Kurek, and Filus, 2013. not been used for the astronomical images interpolation yet. With the supporting idea applied, its accuracy was ∼4× higher than for the linear interpolation, which is typically used for the astronomical image calibration. It should be added that the modied interpola- tion idea is exible and it could be applied to any current or future interpolation method. We suggest to consider implementing presented method into data reduction pipelines – especially of large sky surveys. For this kind of application the method should be especially ecient. 2.1. Bad pixel removal 27
Figure 2.8: Histogram of estimation error for modied and not modied interpolation in a single image correction. E1 – mean estimation error for not modied biharmonic interpolation, E2 – mean estimation error for modied biharmonic interpolation with enhancements. Solid line - biharmonic interpolation, dashed line - modied interpolation with en- hancements. Originally published in: Popowicz, Kurek, and Filus, 2013. Chapter 2. Selected methods of increasing of the angular 28 resolution and photometric precision in astronomical imaging
10000 biharmonic interpolation, E=323 [ADU] large database, E=158 [ADU] 8000 individual frame database, E=266 [ADU]
6000
Count 4000
2000
0 0 100 200 300 400 500 600 700 800 900 1000 Estimation error [ADU]
Figure 2.9: Histogram of estimation error for individual frame correc- tion. Solid line – biharmonic interpolation (no modication), dashed line – modied biharmonic interpolation with 20 frames database, dot- ted line – modied biharmonic interpolation with database constructed from the same frame. Originally published in: Popowicz, Kurek, and Filus, 2013.
6000 biharmonic interpolation, E=0.05 large database, E=0.026 5000 individual frame database, E=0.043
4000
3000 Count 2000
1000
0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Estimation error / pixel true brightness
Figure 2.10: Histogram of relative estimation error for individual frame correction. Solid line – biharmonic interpolation (no modi- cation), dashed line – modied biharmonic interpolation with 20 frames database, dotted line – modied biharmonic interpolation with database constructed from the same frame. Originally published in: Popowicz, Kurek, and Filus, 2013. 2.2. Impulse noise reduction 29
2.2 Impulse noise reduction
The research on impulse noise removal described is this Sect. was originally published in On the eciency of techniques for the re- duction of impulsive noise in astronomical images, Popowicz, A. and Kurek, A. et al., 2016. I was responsible for planning of the research, preparation, parallelization and optimization of the nu- merical code of all the methods to be tested; as well as for sta- tistical analysis, consulting, discussions, typesetting and Arxiv.org submission.
2.2.1 Introduction
The impulsive noise in astronomical images originates from vari- ous sources. It develops as a result of thermal generation in pix- els, collision of cosmic rays with image sensor or may be induced by high readout voltage in Electron Multiplying CCD (EMCCD). It is usually eciently removed by employing the dark frames or by averaging several exposures. But there are circumstances, when either the observed objects or positions of impulsive pixels evolve and therefore each obtained image has to be ltered indepen- dently. Below we present an overview of impulsive noise ltering meth- ods and compare their eciency for the purpose of astronomi- cal image enhancement. The employed set of noise templates consists of dark frames obtained from CCD and EMCCD cameras working on the ground and in the Space. The experiments con- ducted on synthetic and real images allowed us for drawing nu- merous conclusions about the usefulness of several ltering meth- ods for various: (1) widths of stellar proles, (2) signal to noise ratios, (3) noise distributions and (4) applied imaging techniques. The in- terested reader is referred to (Popowicz, A. and Kurek, A. et al., 2016) for much more detailed description of the tests. The results of presented evaluation are especially valuable for selection of the most ecient ltering schema in astronomical image processing pipelines. There are several types of the impulse noise.
2.2.1.1 Stationary dark current
This type of noise is visible as distinctively bright spots / pixels or smudges in an image. The most prominent source of such in- tensity spikes is the dark current, which is produced by faults in Chapter 2. Selected methods of increasing of the angular 30 resolution and photometric precision in astronomical imaging silicon, like point defects (vacancies and intrinsic impurity atoms) or spatial defects (dislocations and clustered vacancies; see Hua et al., 1998). While the impurity-based defects and dislocations are created mainly during the CCD fabrication, the vacancies and their clusters are induced by energetic protons hitting the CCD ma- trix (Hopkinson, 2001; Hopkinson, 1999a; Hopkinson, 1999b). The number of defective pixels increases during the sensor lifetime, what can be observed especially in case of CCDs working in se- vere space environment (Hopkinson, 2000; Penquer et al., 2008). The only way to, at least partially, remove the defects from silicon crystalline, is annealing, which is regularly employed e.g. in the HST (Bautz et al., 2005; Sirianni et al., 2007). The silicon defects acts as a very ecient charge generation centers, which adds un- wanted bias during the light registration. Such centers have the activation energy Ea (amount of energy needed for an electron to release the atom nucleus) within the band gap of silicon, so that they are able to capture the electrons from the valence band and transfer them to the conduction band, increasing the charge ac- cumulated in a pixel. The number of thermally generated electrons per time interval depends on the activation energy of the defect Ea and on tem- perature (Widenhorn et al., 2002):
−E G = G exp a , (2.7) d kT where Gd is the dark current generation rate, G is a parameter, k is the Boltzmann constant and T is the temperature in Kelvins.
A straightforward way to identify both parameters (Ea and G) in- volves a linear approximation of logarithmic dependency of the dark current versus temperature. The centers (sometimes called trapping sites) located in the middle of the silicon energetic band gap (i.e. Ea = 0.55 eV, which is half of 1.1 eV silicon band gap) are usually the most ecient generation centers, since for such de- fects total probability of thermal transfer from the valence to con- duction band using a trapping site achieves maximum. The distribution of dark current in CCD matrices varies from one sensor to another, as it depends on the type of defects. For the fabrication-induced impurities, the quantization of dark cur- rent histogram is observed (Webster et al., 2010; McColgin et al., 1992; McGrath et al., 1987). The visible distinct peaks are related to the presence of 1, 2 or more defects of the same type within a pixel structure. On the other hand, the dislocations manifest their presence by a continuous distribution of dark charge, since their 2.2. Impulse noise reduction 31 generation properties depend on size and location within pixel’s electric eld (Popowicz, 2014). The distribution of dark charge of CCDs working on the ground is therefore composed of (1) discrete peaks related to the point defects and (2) a continuous background caused by the dislocations. A purely continuous distribution may be observed in sensors working in the Space. It appears due to the dominance of clusters of point defects induced by energetic particles, mainly protons. The dark current is also generated during the short readout period. It is especially visible in interline CCDs, where the pixels are divided into the light-sensitive and charge-transfer parts. Al- though the transfer part is usually better shielded, some defects may still appear and the associated dark charge, generated during the readout, is spread along column. The overall oset is usually low, since the accumulation time is limited to the readout of a sin- gle CCD row. An example of such bias structure is presented in Fig. 2.11. Latest CCD sensors are optimized to achieve the lowest pos- sible dark current by means of fabrication purity and by utilization of various pixel structures, which may be virtually free from de- fects (Bogaart et al., 2009). This results in a lower number of hot pixels and reduced average dark current. In the most advanced cameras equipped with extremely strong cooling (down to 70K), the dark current problem is negligible. However, the small obser- vatories with less sophisticated devices still have to compensate for it. It is usually done by subtraction of a dark frame, which is ob- tained with the same exposure time and at the same temperature as the astronomical image, but with a sensor protected from any light source. The calibration frames are often stored, since dark current generation rate is considered to be stable for a pixel.
2.2.1.2 Non stationary dark current
There are some circumstances, when the dark current intensity is not predictable and the correction using dark frame is insu- cient. Some of the defects show so-called random telegraph sig- nals (RTS; Bogaerts, Dierickx, and Mertens, 2002), which means that the parameter G in Eq. 2.7 uctuates between many meta- stable discrete values including often a calm state (G = 0, i.e. no dark current generation). Such problems are observed mainly in Chapter 2. Selected methods of increasing of the angular 32 resolution and photometric precision in astronomical imaging
Figure 2.11: Dark frame from KAI 11002M CCD matrix depicting the hot pixels and the column osets due to the dark current generation. Orig- inally published in: Popowicz, A. and Kurek, A. et al., 2016. space missions, where the sensors are heavily bombarded by en- ergetic protons. They cause a specic type of induced defect (phos- phorus-vacancy, P-V pair) in the form of electric dipole, which ran- domly re-orientates within electric eld, thus changing its gener- ation properties (Elkin and Watkins, 1968; Hopkins and Hopkinson, 1995). In addition to the RTS behavior, for which the standard dark frame subtraction cannot be applied, there are some dark current nonlinearities recently extensively investigated in several works: Popowicz, 2011b; Popowicz, 2011a; Widenhorn, Dunlap, and Bode- gom, 2010b. The authors conrm that the generation rate from ei- ther a single defect or from dislocations depends on current amount of charge collected in a pixel. The electrons kept in pixel’s poten- tial well disturbs the local electric eld and gradually decreases the eciency of thermal activity in defects (it is somewhat similar to the brighter-fatter eect; see Antilogus et al., 2014). It results in a lower number of thermally generated electrons in the dark frame than during the light registration. This eect leads to systematic er- rors introduced by dark frame subtraction, which can be overcome only by extended characterization of CCD’s defects using optical methods (Popowicz, 2013). 2.2. Impulse noise reduction 33
2.2.1.3 Clock induced charge
CIC is present in data collected by electron-multiplying CCDs. The EMCCDs are the image sensors utilized in observational techniques which require very low readout noise, so that each photo-induced electron can be counted (Robbins and Hadwen, 2003). The ap- plications of EMCCDs include Lucky Imaging (Law, Mackay, and Baldwin, 2006), speckle interferometry and adaptive optics, in which the images are registered with very short exposure times (several milliseconds) to retrieve the images less degraded by atmospheric turbulence (Saha, 2007; Saha, 2015). Such sensors are also em- ployed when the number of photons received in a pixel is very low, like in spectroscopy and in fast or narrow-band photometry (Tulloch and Dhillon, 2011; Popowicz, A., Kurek, A. et al., 2015). The idea of electron multiplication is based on the eect of im- pact ionization, which takes place in horizontal readout register driven by very high voltage (70 V and above). When n electrons enter the output register, the nal number of electrons m at the output is governed by the following formula (Robbins and Had- wen, 2003):
(m − n + 1)n−1 m − n + 1 P (m) = exp − , (2.8) (n − 1)! g − 1 + 1/nn g − 1 + 1/n where P (m) is the probability of receiving m output electrons and g is the average register gain. Unfortunately, very high electric eld in readout horizontal reg- ister induces additional unwanted charge. The electrons in va- lence band are swept rapidly during high-voltage switching, thus occasionally some of them gain enough energy to be transferred to the conduction band. Since the chance to generate more than two electrons for a given pixel is negligible (so n = 1 in Eq. 2.8), the distribution of CIC noise, after the electron multiplication, shows an exponential distribution:
1 m P (m) = exp − . (2.9) g g
The CIC spikes, in contrast to the dark current, appear in ran- dom pixels, therefore the phenomenon can not be mitigated by any calibration frame. Due to the skewness of distribution, the av- eraging of images is not recommended as the averaged frame will be biased. Summarizing, for most of the applications of EMCCDs, i.e. in high resolution and extremely fast imaging, the CIC noise can be Chapter 2. Selected methods of increasing of the angular 34 resolution and photometric precision in astronomical imaging calibrated either by image ltering techniques or by some new fabrication technologies. The following conclusions can be drawn about the spatial dependency of noise:
• the two pixels (on both left and right side), closest to the im- pulse, gain additional charge,
• the charge of next pixels is slightly lowered,
• there is only small and diminishing impact on further pixels in a row,
• there is no evidence of cross-talk between rows.
The phenomenon was not observed when EM mode was o. This implies that there must be some mutual inuence between elec- tric elds of cells within a row. Therefore, similarly to the dark cur- rent in columns of interline CCD sensors, the CIC noise in EMCCDs is denitely neither spatially independent nor uncorrelated.
2.2.1.4 Cosmic rays
Mostly the electrons, which have not enough energy to inict per- manent damage, but introduce temporary eects visible in form of smudges in images (see Fig. 2.12). The artifacts are created due to the energy transfer from a particle to CCD electrons in a valence band. As the particles come from dierent directions and move variously within the CCD internal structure, the cosmic ray impacts show various shapes, often even imitating the astronom- ical sources. The problem is noticeable in cameras employed in high altitude observatories or operating in the Space. Similarly to the Gaussian noise, the cosmic ray impacts can be minimized by image averaging using e.g. a sigma clipping7 or a median operation. However, this technique is not applicable if the imaged scene changes rapidly like e.g. in the case of SOHO satel- lite8 (Domingo, Fleck, and Poland, 1995) registering Solar corona phenomena (Fig. 2.12). Also, due to the required increase of ob- servational time, it is impractical to repeat very long exposures, therefore the cosmic rays cancellation has to be performed sepa- rately in each frame.
7Description based on popular Astropy Python package for astron- omy. Link: http://docs.astropy.org/en/stable/api/astropy.stats. sigma_clip.html. 8Link: https://sohowww.nascom.nasa.gov/. 2.2. Impulse noise reduction 35
Figure 2.12: Exemplary acquisitions of cosmic ray impacts – solar corona as registered by SOHO satellite during an exemplary outburst (2012). Originally published in: Popowicz, A. and Kurek, A. et al., 2016.
2.2.2 Tested methods
The methods of impulse noise removal usually employ the inten- sity replacement of a pixel utilizing simple nonlinear operations in a local sliding window (Dougherty and Astola, 1994). We tested the eciency of the methods described below. See also Tab. 2.1 for bibliometric details indication popularity of the methods, Tab. 2.2 for the description of used parameters in each method and Tab. 2.3 for the description of used cameras. • Alpha-Trimmed Mean (ATM) – the α-trimmed mean of pixels intensities in a window W is obtained by:
1. sorting the pixels, 2. removing a xed fraction of pixels at the upper and lower ends of such sorted set, 3. computing the average η of remaining ones:
N−α 1 X η = x , (2.10) N − 2α i i=α+1 where α is the number of pixels removed from the sorted set. W has radius r, thus it includes N = (2r + 1) × (2r + 1) pixels. Chapter 2. Selected methods of increasing of the angular 36 resolution and photometric precision in astronomical imaging
• Boundary Discriminative Noise Detection (BDND) – the pix- els in window W , excluding the central one, are sorted ac- cording to their intensities into a vector, in which the dier- ences between each consecutive pair are calculated. The maximum of such dierences in each half of the vector are found, thus the corresponding intensity levels of a lower and upper limit are dened. If the intensity of central window pixel is below or above the intensity of previously dened bound- aries, then the same procedure is performed for a larger win- dow, whose size is dened by radius. If the pixel again falls out of specied intensity range, it is classied as faulty and replaced by the median of its neighboring pixels classied as uncorrupted.
• Center-Weighted Median (CWM) – the intensity of a central pixel in window W is replaced by the weighted median of all the pixels intensities in the window. The central pixel has higher weight, which is obtained by repeating c times its in- tensity before the median calculation.
• Directional-Weighted Median (DWM) – utilizes the statistics of pixels intensities aligned in four main directions (up-down, left-right and two diagonal), with helps to preserve the edges of the image. The algorithm searches for the direction, which includes the pixels of similar intensities, thus it detects possi- ble edges. If the pixel diers signicantly from the intensities in any direction (a detection threshold T is employed), then it is considered as impulsive. The lter output is calculated as a weighted median of pixels in a sliding window (like in CWM), where the weights are higher (w = 2) for the previously detected best direction (i.e. the one with the lowest gradient) than for all remaining directions (w = 1, simple non-weighted median). The algorithm is repeated iteratively decreasing the detection threshold.
• Iterative Truncated Arithmetic Mean (ITM) – a mean of in- tensities in a local window is calculated and dynamic thresh- old is obtained, utilizing the mean of the following absolute dierences: N 1 X τ = |x − µ|, (2.11) N i i=1
where xi is the intensity of i-th pixel in local window W and µ is the mean intensity of the pixels in W . In each step a trun- cation procedure is performed: if the intensity of any window 2.2. Impulse noise reduction 37
pixel is below or above the dened interval µ ± τ, then it is re- placed by the corresponding boundary value: µ + τ or µ − τ. Eventually, the ltered intensity of a central pixel is the mean of such a truncated set.
• Laplacian Edge Detection (LED) – is a standard in astronomy method of ltering-out the cosmic rays. It is also an important and well known part of the IRAF package9 (Massey, 1997). The algorithm utilizes the convolution of sub-sampled image with the Laplacian of Gaussian to highlight sharp edges as- sociated with either impulsive pixels or cosmic-rays occur- rences. Each pixel in such ltered image is compared with the expected noise to properly identify the impulses. The star-like objects (roundish) are distinguished by the analysis of their symmetry. In each iteration the agged pixels are re- placed by the median of their undisturbed neighbors.
• Lower-Upper-Middle (LUM) – (family of rank-order-based l- ters) the pixels in a local window are sorted and the algorithm checks, if the central pixel belongs either to the lower (L), up- per (U) or middle set (M). The size of lower and upper sets is tunable by k parameter k = 1, .., (N − 1)/2, where N is num- ber of pixels in a local window. If the pixel is within L or U set, then it is replaced by the median intensity of M set.
• Median (MED)
• Peak and Valley (PAV) – the algorithm compares the pixel intensity with the intensities of pixels in the local window. If the central pixel is the brightest or the dimmest one, then it is replaced respectively by the highest or lowest intensity of the pixels in local neighborhood. I.e., the most extreme values are being replaced by the second-most extreme ones.
• Recursive Peak and Valley (RPAV) – the pixel is detected as impulsive the same way it is done in the original concept, however the intensity estimation is performed using a recur- sive maximum-minimum method. The gray scale value is es- timated as an average of the minimum and maximum inten- sity obtained over pixels in specied subsets within the slid- ing window.
• Progressive Switching Median (PSM) – each pixel intensity is compared with the median of neighboring pixels in a local
9Link: http://iraf.noao.edu/. Chapter 2. Selected methods of increasing of the angular 38 resolution and photometric precision in astronomical imaging
window. If the dierence is larger than predened threshold T , then such a pixel is considered as noisy. The map of impul- sive pixels is created prior to the ltering phase. Next, each faulty pixel is replaced by the median of the pixels intensi- ties in a local window, but excluding those pixels, which were previously identied as impulsive.
• Tri-State Median (TSM) – after the image is processed by both a simple median and CWM lters, the pixel’s intensity is compared with the outcomes of those lters. If the devi- ation from the median lter output is lower than a specied threshold T , then the pixel remains unchanged. If the dier- ence from the CWM output is larger than T , the output of a median lter is used for replacement. In the other case, the output of CWM lter is used. This approach utilizes the fact, that the more robust approach (median ltering) is required for strongly outlying pixels, while the CWM should be em- ployed if the pixel is only slightly brighter than its neighbor- hood.
2.2.3 Tests
We created various synthetic stellar proles based on Gaussian PSFs and contaminated it with noise. For the tests based on real images, we acquired these images on-sky, using the cameras listed in Tab. 2.3. We evaluated the photometric and astrometric per- formance after applying various ltering algorithms. There was a large number of tests, since each method has its own parameters and we had to evaluate a vast majority of its combinations. De- tailed description of these tests is beyond the scope of this thesis, but can be found in Popowicz, A. and Kurek, A. et al., 2016.
2.2.4 Results
The results are presented in Figs.:
• 2.13 – eciency in the photometry for synthetic stellar pro- les. There is no surprise here: most commonly used LED method won most of the tests.
• 2.14 – eciency in the astrometry for synthetic stellar proles. In most cases PSM turned out to be more ecient than LED and any other method. 2.2. Impulse noise reduction 39 — 19 10 98 101 142 182 281 193 637 352 490 Citations Reference Jiang, 2012 Windyga, 2001 Xu and Lai, 1998 Ko and Lee, 1991 Ng and Ma, 2006 van Dokkum, 2001 Dong and Xu, 2007 Bednar and Watt, 1984 Wang and Zhang, 1999 Chen, Ma, and Chen, 1999 Hardie and Boncelet, 1993 Dougherty and Astola, 1994 2017. Median Full lter name Peak and Valley Tri-State Median Lower-Upper-Middle Alpha-Trimmed Mean Center-Weighted Median Laplacian Edge Detection Recursive Peak and Valley Directional-Weighted Median Progressive Switching Median Iterative Truncated Arithmetic Mean Boundary Discriminative Noise Detection DWM LUM RPAV CWM ITM MED PSM Abbreviation ATM BDND LED TSM PAV Bibliometry of the tested methods. Citation report according to Thomphon Reuters Web of Science as checked in January Table 2.1: Chapter 2. Selected methods of increasing of the angular 40 resolution and photometric precision in astronomical imaging al 2.2: Table h oa idw tlzdi h loihsaesur ihasd it of width side a with square are algorithms the in utilized windows local The 2016 . al., et A. Kurek, and A. Popowicz, in h aaeeso tlzdtr ihtercrepnigrneo ausue nteeprmnsdsrbdi details in described experiments the in used values of range corresponding their with lters utilized of parameters The TSM RPAV PSM PAV MED LUM LED ITM DWM CWM BDND ATM Method Parameter σ f r r T T T α k lim lim r r r r r r r r c c i i i 2 1 ubro eta ie eeiin nCMlter CWM in repetitions pixel central of number ubro eetdpxl nawindow a in pixels rejected of number ubro eta ie repetitions pixel central of number ubro iesi n set U and L in pixels of number ubro loih iterations algorithm of number iterations algorithm of number lbl ag idwradius window large global, oa,salwno radius window small local, aia betcontrast object maximal ubro iterations of number nta threshold initial threshold initial threshold initial idwradius window radius window radius window radius window radius window radius window radius window radius window Description os limit noise 100 10 ausrne(step) range Values ∼ ∼ 20 000[ 100000 1 1 000[ 100000 1 ∼ r 0 0 ∼ ∼ ∼ 1 . . 5 5 0 [ 200 1 + (2 (2 (2 ∼ ∼ r r r +1) +1) +1) 1 1 1 1 1 1 1 1 1 1 1 1 1 e 0(0 20 (0 20 ∼ 2 2 2 − e e ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ − − 2 2 2 (100) ] − − − 0(1) 10 (20) ] (20) ] (1) 9 (1) 5 (1) 5 (1) 5 (1) 5 (1) 5 (1) 5 (1) 5 (1) 5 (1) 5 (1) 5 (1) 5 (1) 5 1 1 1 . . 25) 25) (1) (1) (1) 2 r 1 + . 2.2. Impulse noise reduction 41 e σ − − − − − − e e e 9 9 . . 15 2 2 20 e 8.8 e 13.5 e Readout noise 5 5 0 20 -10 -90 Temperature 1 s 3000 s 180 s 1800 s 1000 s 1000 s × 9 Exposure time Type Back illuminated CCD EMCCD Back illuminated CCD Interilne CCD Full frame CCD Interline CCD Sensor E2V CCD42-40 Texas Instruments TC247SPD E2V CCD42-40 Kodak KAI-11002M Kodak KAF-3200ME Kodak KAI-2001M The overview of astronomical cameras utilized in the experiments described in details in Popowicz, A. and Kurek, A. et al., . 2016 Andor Luca S EMCCD Andor iKon L (CR) BRITE Toronto (BTr) SBIG 2000 Camera Andor iKon L SBIG ST-10XME Table 2.3: Chapter 2. Selected methods of increasing of the angular 42 resolution and photometric precision in astronomical imaging
Camera Impulsive pixels [%] Mean [e−] Median [e−] Andor iKon L 49 844 848 Andor Luca S EMCCD 17.4 356 216 Andor iKon L (CR) 1.04 63 24 BRITE Toronto 3.9 409 196 SBIG ST-10XME 4.6 503 74 SBIG 2000 8.3 585 111
Table 2.4: Basic statistics of impulse noise in cameras employed in tests described in Sec. 2.2.3.
• 2.15 – eciency on real images. Similarly as in the photomet- ric case, LED method won most of the tests.
As the detailed description of the methodics is long and be- yond the main scope of this thesis, for more details the interested reader is kindly encouraged to see Popowicz, A. and Kurek, A. et al., 2016.
2.2.5 Conclusions
The LED algorithm, developed originally to deal with cosmic rays while preserving symmetrical objects10, appeared to be very ef- cient solution for denoising the astronomical images. However, there are some ranges of sizes of stellar proles (σPSF ≥ 4) for which other methods should be applied. Importantly, it should be done for spectroscopic images, which are of denitely dierent type and therefore should be treated with care. Among others, the TSM al- gorithm gives good promise for reliable photometric outcomes. In contrast to the photometry, where various methods were able to provide the best results, for astrometry the PSM method signi- cantly outperformed any other approaches. It provided the most accurate outcomes for a wide range of σPSF and SNR values. As- suming proper optimization of its tunable threshold, one should in the rst place consider this denoising technique, while detecting and localizing stellar proles buried in noise. However, when stel- lar proles are narrow or noise density is high, the LED algorithm may be also employed.
10Link: http://www.astro.yale.edu/dokkum/lacosmic/. 2.2. Impulse noise reduction 43
20
10 TSM RPAV PSM 0 PAV Best count MED LUM 1 LED ITM 2 DWM 3 CWM BDND 4 ATM σ PSF
15
10
5 Best count 0 TSM RPAV 0.1 PSM PAV 0.2 MED 0.5 LUM LED 1 ITM 2 DWM CWM 5 BDND ATM SNR 10
20
10
Best count 0 TSM RPAV PSM PAV MED Andor iKon LUM LED Andor LucaS ITM DWM Andor iKon CR CWM BDND BRITE Toronto ATM SBIG ST10XMESBIG2000 Figure 2.13: Ranking of impulsive noise reduction methods based on photometric evaluation on synthetic stellar proles. The Best count corresponds to the number of occurrences a method achieved the highest accuracy (the lowest photometric error). Originally published in: Popowicz, A. and Kurek, A. et al., 2016. Chapter 2. Selected methods of increasing of the angular 44 resolution and photometric precision in astronomical imaging
40
20 TSM RPAV PSM 0 PAV Best count MED LUM 1 LED ITM 2 DWM 3 CWM BDND 4 ATM σ PSF
30
20
10 Best count 0 TSM RPAV 0.1 PSM PAV 0.2 MED 0.5 LUM LED 1 ITM 2 DWM CWM 5 BDND ATM SNR 10
30
20
10
Best count 0 TSM RPAV PSM PAV MED Andor iKon LUM LED Andor LucaS ITM DWM Andor iKon CR CWM BDND BRITE Toronto ATM SBIG ST10XMESBIG2000 Figure 2.14: Ranking of impulsive noise reduction methods based on astrometric evaluation on synthetic stellar proles. The Best count cor- responds to the number of occurrences a method achieved the highest accuracy (the lowest astrometric error). Originally published in: Popow- icz, A. and Kurek, A. et al., 2016. 2.2. Impulse noise reduction 45
20
TSM 10 RPAV PSM PAV 0 MED LUM Best count LED ITM DWM CWM Spackle imaging BDND Spectroscopy ATM
Traditional imaging
15
10
5
Best count 0 TSM RPAV PSM PAV MED Andor iKon LUM LED Andor LucaS ITM DWM Andor iKon CR CWM BDND BRITE Toronto ATM SBIG ST10XMESBIG2000
10
5 Best count 0 TSM RPAV 0.1 PSM PAV 0.2 MED 0.5 LUM LED 1 ITM 2 DWM CWM 5 BDND ATM SNR 10 Figure 2.15: Ranking of impulsive noise reduction methods based on the evaluation on real astronomical images. The Best count corre- sponds to the number of occurrences a method achieved the highest accuracy (the lowest RMS error). Originally published in: Popowicz, A. and Kurek, A. et al., 2016. Chapter 2. Selected methods of increasing of the angular 46 resolution and photometric precision in astronomical imaging
2.3 Ecient use of telescope time
The research on ecient use of telescope time described in this Sect. was submitted to Publications of the Astronomical Society of the Pacic11 magazine as Optimization of exposure time division for multiobject photometry. I was responsible for planning the re- search, preparation, optimization and parallelization of the code of the evolutionary algorithm; for discussions, consulting, manuscript typesetting, its correction and submission.
When imaging dense elds of stars (clusters etc.), astronomers used to execute multiple exposures. If the sources of interest dier much in intensity, sometimes observers choose to vary exposure times. It is supposed to be done in such a way as not to oversat- urate the brightest sources in all exposures – and also to achieve satisfactory SNR for the weakest ones. It may be a good idea to dene these times in an analytic way. There are some constrains for the optimal choice:
• Usually the time for observations is limited, so there exists some total available time, with cannot be exceeded.
• For some telescope / camera system a minimal and / or max- imal exposure time is xed.
• The total amount of exposures has to be decided.
• The length of each exposure should be computed (not guessed).
• Each camera has its readout time, with should be taken into account.
• Intensities of all sources of interest should be taken into ac- count.
• Sky noise should be taken into account as well.
• Camera readout noise12 contributes to the SNR and usually needs to be taken into account, especially in the case of weak sources.
We have not found any literature on estimation of the optimal exposure times. This problem is highly nonlinear and we devel- oped an evolutionary algorithm to propose a universal solution, with would: 11Link: http://iopscience.iop.org/journal/1538-3873. 12Janesick, 2007, sec. 3.5. 2.3. Ecient use of telescope time 47
Stars: 34, I = 9, I = 685 [e -/sec] min max
Source Strongest source Weakest source
Figure 2.16: Exemplary dense stellar eld used in simulations. A glob- ular cluster imaged sing 12” Ritchey-Chretien telescope mounted on EQ8 paralactic mount. Camera: ATIK 11000M monochromatic, full frame cooled camera, 11 Mpixel sensor, KAI-11002. Image courtesy of A. Popowicz.
• return the optimal number of exposures for a given star eld and
• estimate optimal times of those exposures.
We submitted the results to Publications of the Astronomical So- ciety of the Pacic (PASP) magazine. Below I summarize the main results. Ultimately, we plan to release a code facilitating planning of observations. For an exemplary test of eciency of our method, an image de- picted in Fig. 2.16 was used. We assumed the following observing parameters in this test:
• Strength of sources as listed in the title of Fig. 2.16 (34 stars, - Imin = 9, Imax = 685 [e /sec]).
• All available telescope‘s time is 1k sec.
• Maximal duration of a single exposure is 1k sec.
• Minimal duration of exposure is 0.1 sec.
• We execute between 2 and 20 exposures.
• Camera readout time is 30 sec. Chapter 2. Selected methods of increasing of the angular 48 resolution and photometric precision in astronomical imaging
0.7 no-saturation limit 0.6 EA results 0.5
0.4
RMSE [mag] RMSE 0.3
0.2 2 3 4 5 6 7 8 9 Number of sub-exposures
Figure 2.17: Results for image depicted in Fig. 2.16.
• Camera readout noise is 13 e−.
Results of search for optimal exposures are depicted in Fig. 2.17. For parameters listed above, our evolutionary algorithm returned four exposure times: 80.17, 254.30, 259.34 and 286.19 s. If one ob- serves this star led using these exposure times, the mean pho- tometric error for all sources decreases by ∼0.03 mag (Fig. 2.17) in comparison to a single exposure of time constrained only by the request not to oversaturate the brightest sources. In the case of a much denser eld depicted in Fig. 2.18, the pre- liminary version of our algorithm managed to improve the mean photometric error by ∼0.05 mag. 2.3. Ecient use of telescope time 49
Stars: 690, I = 8, I = 1349 [e -/sec] min max
Source Strongest source Weakest source
Figure 2.18: Another, larger exemplary dense star eld used in simula- tions. Acquisition details as in Fig. 2.16. Image courtesy of A. Popowicz Chapter 2. Selected methods of increasing of the angular 50 resolution and photometric precision in astronomical imaging
2.4 Evolutionary algorithms for image restora- tion
The research on evolutionary algorithms for image restoration de- scribed in this Sec. was originally published in Beyond the cur- rent noise limit in imaging through turbulent medium, Popowicz, A., Kurek, A. et al., 2015. I was responsible for planning the re- search, preparation of the entire numerical code, including its op- timization and parallelization; for manuscript typesetting, prepara- tion and submission. I also took part in discussions and consulting.
2.4.1 Introduction
Images obtained through a turbulent medium (like the atmosphere) suer from serious quality degradation. There are several methods developed to alleviate this problem. One of them is the adaptive optics (Beckers, 1993), which is commonly used in astronomy, mi- croscopy and surveillance. In this technique a high-speed camera monitors continuously the shape of the wavefront and a closed- loop system compensates for the wave distortions by deforming a special mirror in the optical path (Duner, 2009). This approach is very complex, expensive and requires a sophisticated equip- ment (e.g. the deformable mirror and the wavefront sensing de- vices). Moreover, the required wavefront diagnostics is possible only for the objects much brighter than the background. If there is not enough light to perform the wavefront analysis, only the tip-tilt correction can be applied (Baba et al., 1985). It is accomplished by oscillating an additional mirror in the optical path, which results in real-time image shifting (Baba et al., 1985; Sivaramakrishnan, Wey- mann, and Beletic, 1995). Another, much simpler yet ecient approach is the Lucky Imag- ing (Fried, 1978), where consecutive short exposures are analyzed, shifted to the brightest speckle and stacked together, as depicted in Fig. 2.19. Only the images with the highest Strehl ratio (see Chap.1 for denition) are included in the nal stack, hence the nal im- age is the average of the frames captured under best atmospheric conditions. The probability of obtaining such good quality and un- blurred frames is a function of the aperture size of D and the Fried parameter r0 (Tikhomirov, 1991):