Towards advanced astronomical imaging: new techniques of data reduction and their applications

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DOCTORAL THESIS

Towards advanced

astronomical imaging:

new techniques of data

reduction and their

applications

Author:

Aleksander KUREK

Supervisor:

Dr hab. Agnieszka POLLO

A thesis submitted in fulfillment of the requirements

for the degree of Doctor of Philosophy

in the

Faculty of Physics, Astronomy and Applied Computer

Science

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ii

I’ve seen things you people wouldn’t believe. Attack ships on fire off the shoulder of Orion. I watched C-beams glitter in the dark near the Tannhäuser Gate

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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 efficiency 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-ficient 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 efficiently 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 efficient one, however, there are exceptions – mainly astrometric applications. Our evolutionary algorithm-based meth-ods were shown to be able to: (3) recover the surface profile of sources which are only ∼2-3 % stronger than their background; and (4) find an optimal way to divide the available observing time to multiple exposures, so that the average photometric error of a dense field photometry was lowered by 0.05 mag. The endeavor to employ Optical Parametric light Amplification (OPA) to high an-gular resolution astronomical imaging (5) did not succeeded so far but its limitations were demonstrated.

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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,

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This dissertation has been written basing on the scientific results previously reported in the following articles:

• A. Popowicz, A. R. Kurek, Z. Filus, Bad pixel modified

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 efficiency 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 amplification in astronomy: a quantum optical model(MNRAS)

• A. Popowicz, A. R. Kurek, Optimization of exposure time

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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.

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4 Far future 71 4.1 Quantum Telescopes / Optical Parametric Amplifi-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

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List of Abbreviations

ATLAST Advanced Technology Large-Aperture Space

Telescope

CCD Charge Coupled Device CT Classic Telescope DL Diffraction 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 Amplification 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

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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 diffraction, i.e. so called diffraction limit1_.

Historically, obviously the first 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, 2000for a more detailed information on the human eye properties from as-tronomical point of view, including the light losses, stray light con-tamination, photometric specification 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 efficiency is ∼10 %, which is poor in com-parison even to modern smartphone cameras2_.

• The eye‘s dynamic range is 1:109_{, which is ∼10}4_×_{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 Reflex camera) cam-eras. This offers up to 14.8 Exposure Values (EV) of dynamic 1_{D. L. is well described at} _{Hyperphysics webpage}_{; link:} _http://

hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cirapp.html#c1.

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2 Chapter 1. Introduction:

Angular resolution in astronomical imaging

range3,4_.

• 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 diffraction-limited, its angular resolution would be: Θ = 1.22λ Drad = 1.22 0.55 µm 8 rad 180deg πrad 1mm 103_µm = 0.00483600 00 1deg = 17.3 00 (1.1)

• Several other than the diffraction 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;}

Yanoff 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 five}

largest details of its surface.

Despite the invincible dynamic range, the human eye is obvi-ously not fitted for astrophysical research.

A major breakthrough in human perception of astrophysical bod-ies occurred in 1610: Galileo Galilei presented the first 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 effect of the atmosphere – only after 1900 the first 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 diffraction-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 3_{According to}_{DxOMark Sensor Scores}_{; link:}_{https://www.dxomark.com/}_. 4_{All such simple calculation in this Chapter are made by me unless stated}

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is 0.05 arcsec (50 mas for Betelgeuse, see Uitenbroek, Dupree, and Gilliland, 1998). An interested reader can find 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 diffraction limit. Nevertheless, in this thesis for simplicity I comply to the popular assumption that HST is a totally diffraction 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 offer small fields of view (Eisenhauer et al.,2011). Moreover, no existing telescope project is aimed at sidestepping the diffraction 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 first ever imaging of two stars orbiting the common center off mas (Baron et al.,2012). Movie composed of 55 H-band frames of this system recorded from 2006 to 2010 is avalilableon 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 affects all current optical interferometers due to their very com-plex optical path: the light from the astrophysical target has to be reflected usually more than 15× before the image formation takes place.

5_{The 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 defined as the ratio of the peak aberrated image inten-sity from a point source compared to the maximum inteninten-sity attainable when using an ideal optical system limited only by diffraction over the system’s aper-ture.

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Presently, radio interferometers offer higher angular resolution than optical, because it is easier to build large baseline instruments, since the image restoration can be performed offline. 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.

6_Link:_{http://exoplanet.eu/}_.

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A series of figures based on these data and presented in this sec-tion illustrates the main challenges and difficulties 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) exex-oplanets of Terran radius and b) the remaining objects, where a vast majority has a larger, Neptune-like radius. Fig.1.2presents a histogram of the same data. Figures 1.3 and 1.4 are based on the PHL data, where more so-phisticated definition 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 flux 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 finger 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 finally 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 8_{For a mathematical description see the} _{project‘s website}_{; link:} _http://

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100 101 102 103 104 105

distance [LY]

10-2 10-1 100 101 102 103

angular size [

arcsec]

Neptune - like or other Proxima Centauri B terran radius

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 flared 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, althoughphotographed

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 isThe 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.

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0.05 0.13 0.36 1 2.72 7.39 20 55 150 400

angular size [ arcsec]

0 10 20 30 40 50 60 70 80 90 100

Count

all; count: 1302 terran radius; count: 382

10 largest angularily: (w/o Prox. Cent. B) WASP-136 b GJ 674 b 51 Peg b WISE 0458+6434 b WISE 1217+16A b Ross 458 (AB) c beta Pic b 2M 0746+20 b tau Boo b HD 7924 b

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

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100 101 102 103 104 105

distance [LY]

10-2 10-1 100 101 102 103

angular size [

arcsec]

Neptune - like or other Proxima Centauri B ESI > 0.60

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 difficult targets; see Labeyrie,1999and Sec.3.4of 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

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0.02 0.13 1 7.39 55 403

angular size [ arcsec]

0 20 40 60 80 100 120

Count

Neptune-like or other; count: 2509 ESI > 0.60; count: 90 10 largest angularily: eps Eridani b GJ 674 b GJ 876 b GJ 876 c GJ 832 b GJ 570 Db HD 95872 b Fomalhaut b 51 Peg b WD 0806-661 B b

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 figures 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}

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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 figures 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 Technology Large-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 field. 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 sufficient for detecting biosignatures (Sterzik, Bagnulo, and Palle,2012; Hegde et al.,2015; Claudi and Erculiani, 2014) on extrasolar planets. 100 101 102 103 104 105

distance [LY]

102 103 104 105 106 107

smaller than HST limit

103 104 105 106 107 108

D for 5

5 pix. image [m]

Neptune - like or other Proxima Centauri B terran radius

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

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100 101 102 103 104 105

distance [LY]

102 103 104 105 106 107

smaller than HST limit

103 104 105 106 107 108

D for 5

5 pix. image [m]

Neptune - like or other Proxima Centauri B ESI > 0.60

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), suffices 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 effect in our Galaxy is ∼7 µas (Yano,2012). On the extragalactic scales, additionally, the same effect is in-troduced by the distortion of geometry of the space by the dark matter.

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100 101 102 103 104 105

Distance [LY]

107 108 109 1010 1011 1012

km in 1 HST pix.

10-1 100 101 102

AU in 1 ATLAST pix.

All known exoplanets Proxima b

SWEEPS-04b, dist.: 22k 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:

• Diffraction 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,2005for overview or Narayan and Bartelmann,1996for 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

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100 101 102 103 104 105

D [m]

10-3 10-2 10-1 100 101 102

Proxima Centauri B

image pixels

Hypertelescope HST 1 1 pix. 5 5 pix.

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 effect 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 effect exists (Steinbring,2016; Perlman et al.,2015).

• cosmic strings – hypothetical 1-dimensional topological de-fects which may have formed during a symmetry breaking

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phase transition of the Universe (Hindmarsh and Kibble,1995). A cosmic string would produce a duplicate image of fluctua-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 find 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 and 4) 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 scientific 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:

• Efficient bad pixel removal method – Sec.2.1,

• Efficient impulse noise reduction in astronomical images – Sec.2.2,

• Efficient use of the time of telescopes – Sec.2.3,

• Evolutionary algorithms for image restoration – Sec.2.4, • A possibility of employing parametric light amplification for

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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 Modified 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 flux is measured thanks to the photovoltaic effect (Litwiller, 2001; Janesick,2001; Milnes,1980). Unfortunately, not every pixel can be used effectively, 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, 1_Source:_{NASA ADS}_.

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16 Chapter 2. Selected methods of increasing of the angular

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 flight missions; Hopkinson, Goiffon, 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 filed 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,2013we 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 modification of the biharmonic interpolation. The modification is dedicated especially for astronomical pictures calibration. We suggest some further enhancements which improve its effectiveness and precision.

2_{For an up to date overview of software used in astronomy, see Momcheva}

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• The result of our modified 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 influence 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 fix-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 finds 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-3and SEASAT al-timeter data in 1987 (Sandwell,1987).

• our adjusted method – it is described in (Popowicz, Kurek, and Filus,2013) and below.

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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, filter: 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 theSloan 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-Chretientelescopelocated atApache Point Observatoryin New Mexico (Gunn et al.,2006). The imaging camera consists of an array of 30 CCDs with total field 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 choseSDSS g filter5_{(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.

3_{That was the latest release in the time of performing the tests. Presently}

version 13is the latest.

4_Link:_{http://www.sdss.org/dr12/}_.

5_Link: _{http://skyserver.sdss.org/dr1/en/proj/advanced/color/}

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2.1.3 Test of interpolation methods

2.1.3.1 Comparison test

For the testes presented here, 1000 calibrated astronomical 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 classification, so both point-like and extended obob-jects 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.2.3and2.4. Additionally, a mean error E was computed to enable a quick comparison of the methods’ effec-tiveness. The following averaging formula was used:

E =

N

X

i=1

|∆i|, (2.1)

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 difference in effectiveness 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 coefficients (Eq.2.2). For given values

of the intermediate points, the system of equations2.2can 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.

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FIGURE 2.2: Exemplar fragments, 7x7 pixels. Interpolated pixel is pointed by an arrow. Originally published in: Popowicz, Kurek, and

Filus,2013. w(xj, yj) = N X i=1 αi (xj−xi)2+(yj−yi)2 2 ln q (xj− xi)2+ (yj − yi)2−1 , (2.2) where: w(xi, yi)— value of the interpolated function for xi, yi

coor-dinates, N — number of intermediate points.

2.1.4 Modified interpolation

2.1.4.1 Presentation of the idea

The idea of the modified 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 affected 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 different due to the seeing variations during the exposure or due to the optical aberrations. Our modi-fied 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 affected by the bad pixels. Because

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0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 101

102 103 104

Estimation error [ADU]

Count

biharmonic interpolation, E=465 [ADU] cubic spline interpolation E=613 [ADU] linear interpolation E=872 [ADU] surrounding pixel median E=1171 [ADU] nearest neighbor E=2353 [ADU]

FIGURE 2.3: Histogram of estimation error for the examined interpola-tion methods. Solid line – biharmonic interpolainterpola-tion, 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.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 101 102 103 104 105

Estimation error / pixel true brightness

Count

nearest neighbour, E=0.334 surrounding pixel median, E=0.154 linear interpolation, E=0.125 cubic spline interpolation, E=0.087 biharmonic interpolation, E=0.063

FIGURE 2.4: Histogram of relative estimation error for the examined in-terpolation methods. Solid line – biharmonic inin-terpolation, dashed line – cubic spline interpolation, dotted line – linear interpolation, solid bold line – surrounding pixel median, dashed bold line – nearest neighbor.

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FIGURE 2.5: Dependence of estimation error on interpolated pixel brightness. Originally published in: Popowicz, Kurek, and Filus,2013.

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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 different orientation of extended objects. Af-ter the fragments’ acquisition, the biharmonic inAf-terpolation 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 coefficients 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 interpolainterpola-tion co-efficients with the references from the database and to find the ref-erence which is best suited. The mean square fitting error (Eq.2.3) between the reference and current coefficients is used during the search. Jk = s PN i=1(αik− α∗i)2 N k −−−−→ min , (2.3) where: Jk— mean square fitting error between k-th reference and

current coefficients, αik — i-th coefficient of k-th reference in the

database, α∗

i — i-th coefficient of the interpolation of current

frag-ment and k — the number of the most suitable reference.

After finding 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 find the most suitable reference and to apply the corresponding correction to current brightness esti-mation. To improve the efficiency of the method, it is desired to find 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).

e = v u u t P5 i=1 1 Jiei P5 i=1 1 Ji , (2.4)

where: e — averaged estimation error, Ji — mean square fitting

error between i-th best fitted reference and current coefficients, ei

- estimation error of i-th best fitted reference.

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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).

p0_i = pi− min(p)

max(p) − min(p), (2.5)

pi = p0i max(p) − min(p) + min(p) , (2.6)

where: p0

i— 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 Verification

Here the database was created using fragments from the first 10 pictures from the previously mentioned (Sec. 2.1.2) SDSS image set. Remaining images (990 pictures) were used as a verification set. The results are presented for three modified interpolation types: a) without averaging and normalization, b) with averaging, c) with averaging and normalization – and for the original biharmonic in-terpolation without modification (Figs.2.6. and2.7). In another test, it was examined if the modified interpolation can be useful for a single image correction. The database had to be created from the parts of an image that were not affected 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 effectiveness 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.2.9and2.10.

The mean error for the test with a large database showed about 50% decrease after using the modified interpolation idea. Accord-ing to the histograms (Figs.2.6and2.7), the number of small errors rose and the number of large errors decreased as the modified 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

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0 50 100 150 200 250 300 350 400 450 500 0.5 1 1.5 2 2.5x 10 4

Count

modified interpolation (no enhancement) E=266 [ADU] modified interpolation

(with normalization) E=243 [ADU] modified interpolation

(with normalization and weighting) E=222 [ADU] biharmonic interpolation

(no modification) E=465 [ADU]

FIGURE 2.6: Histogram of estimation error for modified biharmonic in-terpolations. Solid line – biharmonic interpolation without modification, solid bold line – modified biharmonic interpolation without enhance-ments, dashed bold line – modified biharmonic interpolation with nor-malization, dash-dot solid line – modified 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 modified 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. 2.9and 2.10). However, for the images affected by strong PSF variation (e.g. due to the seeing problems), the results would be different.

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 effectiveness of the method as a pixel’s brightness estimator and its superiority over other examined in-terpolation methods. Moreover, the biharmonic inin-terpolation has

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0 0.05 0.1 0.15 0 0.5 1 1.5 2 2.5x 10 4

Count

biharmonic interpolation (no modification), E=0.063

modified intrpolation (no enhancement), E=0.044 modified interpolation

(with normalization), E=0.042 modified interpolation

(with normalization and weighting), E=0.032

FIGURE 2.7: Histogram of relative estimation error for modified bihar-monic interpolations. Solid line – biharbihar-monic interpolation without modification, solid bold line – modified biharmonic interpolation with-out enhancements, dashed bold line – modified biharmonic interpo-lation with normalization, dash-dot solid line – modified interpointerpo-lation 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 modified interpola-tion idea is flexible 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 efficient.

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FIGURE 2.8: Histogram of estimation error for modified and not modified interpolation in a single image correction. E1 – mean estimation error

for not modified biharmonic interpolation, E2 – mean estimation error

for modified biharmonic interpolation with enhancements. Solid line -biharmonic interpolation, dashed line - modified interpolation with en-hancements. Originally published in: Popowicz, Kurek, and Filus,2013.

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0 100 200 300 400 500 600 700 800 900 1000 0 2000 4000 6000 8000 10000

Count

biharmonic interpolation, E=323 [ADU] large database, E=158 [ADU] individual frame database, E=266 [ADU]

FIGURE 2.9: Histogram of estimation error for individual frame correc-tion. Solid line – biharmonic interpolation (no modification), dashed line – modified biharmonic interpolation with 20 frames database, dot-ted line – modified biharmonic interpolation with database construcdot-ted from the same frame. Originally published in: Popowicz, Kurek, and

Filus,2013. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 1000 2000 3000 4000 5000 6000

Count

biharmonic interpolation, E=0.05 large database, E=0.026 individual frame database, E=0.043

FIGURE 2.10: Histogram of relative estimation error for individual frame correction. Solid line – biharmonic interpolation (no modi-fication), dashed line – modified biharmonic interpolation with 20 frames database, dotted line – modified biharmonic interpolation with database constructed from the same frame. Originally published in:

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2.2 Impulse noise reduction

The research on impulse noise removal described is this Sect. was originally published in On the efficiency 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 andArxiv.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 efficiently 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 filtered indepen-dently.

Below we present an overview of impulsive noise filtering meth-ods and compare their efficiency 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 filtering meth-ods for various: (1) widths of stellar profiles, (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 efficient filtering 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

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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 efficient 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): Gd= Gexp

−Ea

kT

, (2.7)

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 efficient 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

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generation properties depend on size and location within pixel’s electric field (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 offset 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 insuffi-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 fluctuates 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

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FIGURE 2.11: Dark frame fromKAI 11002MCCD matrix depicting the hot pixels and the column offsets 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 specific type of induced defect (phos-phorus-vacancy, P-V pair) in the form of electric dipole, which ran-domly re-orientates within electric field, 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 confirm 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 field and gradually decreases the efficiency of thermal activity in defects (it is somewhat similar to the brighter-fatter effect; 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 effect 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).

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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 effect 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 final number of electrons m at the output is governed by the following formula (Robbins and Had-wen,2003): P (m) = (m − n + 1) n−1 (n − 1)! g − 1 + 1/nn exp − m − n + 1 g − 1 + 1/n , (2.8) where P (m) is the probability of receiving m output electrons and g is the average register gain.

Unfortunately, very high electric field 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: P (m) = 1 gexp − m g . (2.9)

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

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calibrated either by image filtering 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 off. This implies that there must be some mutual influence between elec-tric fields of cells within a row. Therefore, similarly to the dark cur-rent in columns of interline CCD sensors, the CIC noise in EMCCDs is definitely neither spatially independent nor uncorrelated.

2.2.1.4 Cosmic rays

Mostly the electrons, which have not enough energy to inflict per-manent damage, but introduce temporary effects 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 different 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 ofSOHO 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.

7_{Description based on popular} _Astropy _{Python package for}

astron-omy. Link:http://docs.astropy.org/en/stable/api/astropy.stats. sigma_clip.html.

Towards advanced astronomical imaging: new techniques of data reduction and their applications

DOCTORAL THESIS