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Doctoral Thesis VYSOKEU´ CENˇ ´I TECHNICKE´ V BRNEˇ BRNO UNIVERSITY OF TECHNOLOGY FAKULTA STROJN´IHO INZENˇ YRSTV´ ´I USTAV´ MATEMATIKY FACULTY OF MECHANICAL ENGINEERING INSTITUTE OF MATHEMATICS APPLICATION OF ADAPTIVE FILTERS IN PROCESSING OF SOLAR CORONA IMAGES APLIKACE ADAPTIVN´ICH FILTRUP˚ RIˇ ZPRACOVAN´ ´I SN´IMKU˚ SLUNECNˇ ´I KORONY´ DIZERTACNˇ ´I PRACE´ DOCTORAL THESIS AUTOR PRACE´ Ing. HANA DRUCKMULLEROV¨ A´ AUTHOR VEDOUC´I PRACE´ doc. PaedDr. DALIBOR MARTISEK,ˇ Ph.D. SUPERVISOR BRNO 2014 Abstrakt Fotografov´an´ı sluneˇcn´ı kor´ony patˇr´ı mezi nejobt´ıˇznˇejˇs´ı ´ulohy astrofotografie a z´aroveˇn je jednou z kl´ıˇcov´ych metod pro studium kor´ony. Tato pr´acepˇrin´aˇs´ı ucelen´ysouhrn metod pro pozorov´an´ı sluneˇcn´ı kor´ony pomoc´ı sn´ımk˚u.Pr´aceobsahuje nutnou mate- matickou teorii, postup pro zpracov´an´ısn´ımk˚ua souhrn adaptivn´ıch filtr˚upro vizualizaci koron´aln´ıch struktur v digit´aln´ıch obrazech. D´alepˇrin´aˇs´ın´avrhnov´ych metod urˇcen´ych pˇredevˇs´ımpro obrazy s vyˇsˇs´ımobsahem ˇsumu, neˇzje bˇeˇzn´eu obraz˚ub´ıl´ekor´ony poˇr´ıze- n´ych bˇehem´upln´ych zatmˇen´ıSlunce, napˇr.pro obrazy poˇr´ızen´epomoc´ı´uzkop´asmov´ych filtr˚u.Fourier normalizing-radial-graded filter, kter´ybyl navrˇzenv r´amcit´etopr´ace,je zaloˇzenna aproximaci hodnot pixel˚ua jejich variability pomoc´ıtrigonometrick´ych poly- nom˚us vyuˇzit´ımdalˇs´ıch vlastnost´ıobrazu. Abstract Solar corona photography counts among the most complicated tasks in astrophotography. It also plays a key role for research of the solar corona. This thesis brings an a complete overview of methods for imaging the solar corona. The thesis contains necessary methe- matical background, the sequence of steps for image processing, an overview of adaptive filters used for visualization of corona structures in digital images, and new methods are proposed, especially for images which contain more noise than it is typical for images of the white corona taken during total solar eclipses, e.g. images taken with narrow-band fil- ters. The Fourier normalizing-radial-graded filter method that I proposed during my PhD study are based on approximation of pixel values and their variability with trigonometric polynomials using other properties of the image. kl´ıˇcov´aslova sluneˇcn´ı kor´ona,adaptivn´ı filtry, zpracov´an´ı obraz˚u,Fourier normalizing-radial-graded filter key words solar corona, adaptive filters, image processing, Fourier normalizing-radial-graded filter DRUCKMULLEROV¨ A,´ H. Application of adaptive filters in processing of solar corona images. Brno: Vysok´euˇcen´ıtechnick´ev Brnˇe,Fakulta strojn´ıhoinˇzen´yrstv´ı,2014. 120 s. Vedouc´ıdizertaˇcn´ıpr´acedoc. PaedDr. Dalibor Martiˇsek,Ph.D. I declare that I have written the doctoral thesis Application of adaptive filters in processing of solar corona images on my own according to advice of my supervisor doc. PaedDr. Da- libor Martiˇsek,Ph.D., and using the sources listed in references. March 31, 2014 Hana Druckm¨ullerov´a I would like to express my gratitude to prof. Shadia Rifai Habbal, Ph.D. of Institute for Astronomy, University of Hawaii at Manoa for her hospitality during my stay at IfA UH in Honolulu, Hawaii in October 2010 to January 2011 and for the fruitful discussions we have had together. I would like to express my gratitude to my father prof. RNDr. Miloslav Druckm¨uller,CSc. for showing me the beauty of total solar eclipses and passion for image processing. Finally, I would like to thank my man Ing. Vladim´ırKut´alek,Ph.D. for his programming hints and inspirational comments to my work. Contents 1 Introduction 3 2 Imaging the solar corona 5 2.1 The solar corona . .5 2.2 Brightness gradient in the solar corona . .6 2.3 Imaging possibilities for the solar corona . .8 2.4 Methods of compensation for the brightness gradient . .9 2.4.1 Mechanical and optical methods . .9 2.4.2 Mathematical methods . 12 3 Mathematical background 14 3.1 Digital image . 14 3.1.1 Digital image as a function of two variables in Cartesian coordinates 14 3.1.2 Indexing a digital image in polar coordinates . 16 3.2 Fourier transform . 18 3.2.1 Fourier transform of functions f : R2 ! C .............. 18 3.2.2 Discrete Fourier transform . 20 3.3 Image filters . 25 3.3.1 Adaptive filters . 26 3.3.2 Linear filters . 26 4 Image acquisition and processing 30 4.1 White-light images acquired during total solar eclipses . 30 4.1.1 Image acquisition . 30 4.1.2 Image calibration . 34 4.1.3 Image registration { phase correlation . 41 4.1.4 Image composition . 47 4.2 Images in spectral lines acquired during total solar eclipses . 51 4.2.1 Importance of observations in spectral lines during total solar eclipses 51 4.2.2 Principle of observation in spectral lines in the visible part of the spectrum . 53 4.2.3 Processing images in spectral lines . 55 4.3 Images from cosmic probes . 56 4.3.1 List of cosmic probes . 56 4.3.2 Processing images from cosmic probes . 61 1 5 Earlier filters for solar corona structure enhancement 63 5.1 Tangential filter . 63 5.2 Adaptive convolution . 66 5.3 Normalizing-radial-graded filter . 68 5.4 Noise adaptive fuzzy equalization method . 71 6 Fourier normalizing-radial-graded filter 74 6.1 The principle of the filter . 74 6.1.1 The basic idea of the filter . 75 6.1.2 Attenuation coefficients . 78 6.1.3 Influence of additive noise . 83 6.2 Results . 85 6.2.1 Application to total solar eclipse observations . 85 6.2.2 Application to space-based observations . 87 6.2.3 Comparison with other methods . 88 6.2.4 Other results { graphs of local brightness and contrast . 91 6.3 Software implementation { FNRGF software . 94 6.3.1 Parameters of the input file . 95 6.3.2 Parameters of the FNRGF . 96 6.3.3 Parameters of image display . 98 6.3.4 Principles of the software implementation . 99 6.3.5 FNRGF software user guide . 101 7 Conclusion 105 Bibliography 108 Used symbols and abbreviations 118 A Enclosed CD 120 Chapter 1 Introduction The Sun is the nearest star to the Earth, therefore the research of the Sun gives the mankind a unique opportunity to study the physical parameters and properties of a star. Of course, there are both smaller and much bigger stars in the Universe, but the Sun is the most important star for us. The intensity and the spectral composition of the light emitted by the Sun as well as other particles that are spread from the Sun play a vital role for the life on the Earth. Since people are using more and more complicated electronic devices both on the Earth and in the Space and some of these devices are connected in large (global) networks, it is necessary to understand how and when the phenomena on the Sun can harm them. As an example of what happened at a time when the electronics and computer networks were not as advanced as today can be the flare of March 9, 1989, when the subsequent geomagnetic storm caused the collapse of Hydro-Qu´ebec's electricity transmission system and other blackouts including a melted transformer in New Jersey, U.S.A. A very nice popular-scientific text on the danger of a massive solar flare including information about the March 1989 flare can be found on [dGre12]. A scientific article on this topic is for instance [cSta02]. Current electronic devices are even more sensitive than those in 1989. Among other methods of the solar corona research, imaging methods play a cru- cial role. Due to the problems described in Chapter 2, sophisticated image processing techniques are necessary for taking full advantage of the current mechanical and optical systems for both Earth-based and space-born observations. Chapter 2 not only covers why it is complicated to take full advantage of the corona observations, but also gives a historical overview of mechanical and optical methods of solving these problems and an overview of numerical methods used from the early times of use of computers for image processing to nowadays. Chapter 3 describes the mathematical notions and theory that are necessary in the next chapters. Image acquisition and the steps of processing images (with mathematical reasoning) both from total solar eclipses and from cosmic probes to- gether with a list of the most important cosmic probes that study the Sun is the content of Chapter 4. The final or almost-final step of the process is the enhancement of coronal structures. An overview of the most important methods which have been used for coronal structure enhancement can be found in Chapter 5. Among them is the Normalizing-radial-graded filter (NRGF), which was the inspiration for the Fourier normalizing-radial-graded filter (FNRGF) proposed, implemented, and tested on many types of data as of part of my Ph.D. study and described in Chapter 6. The filter brings new possibilities for studying the corona, because its latest implementation 3 is relatively fast and some faint structures are enhanced better than with any of the previous methods. Chapter 6 not only describes the principle of the filter, but also its software implementation and contains many examples of images processed with this filter. Chapter 7 is the Conclusion followed by the Bibliography, and the List of used symbols and abbreviations and the description of the content of the CD which is enclosed to the thesis. 4 Chapter 2 Imaging the solar corona 2.1 The solar corona The Sun can be divided into the interior and the atmosphere.
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