Microform Edition © ProQuest LLC. ... of digital storage, the production of photographic atlases of the night sky ... processing of astronomical images, in particular operations that preface ...... Proceedings of the Fourth IEEE International Conference on Automatic Face and ..... [107] Rafael C. Gonzales and Richard E. Woods.
ENHANCEMENT CLASSIFICATION OF GALAXY IMAGES
APPROVED BY SUPERVISING COMMITTEE: Arytom Grigoryan, Ph.D., Chair Walter Richardson, Ph.D. David Akopian, Ph.D.
Accepted: Dean, Graduate School
Copyright 2014 John Jenkinson All rights reserved.
DEDICATION
To my family.
ENHANCEMENT CLASSIFICATION OF GALAXY IMAGES
by
JOHN JENKINSON, M.S.
DISSERTATION Presented to the Graduate Faculty of The University of Texas at San Antonio In Partial Fulfillment Of the Requirements For the Degree of MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
THE UNIVERSITY OF TEXAS AT SAN ANTONIO College of Engineering Department of Electrical and Computer Engineering December 2014
UMI Number: 1572687
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ACKNOWLEDGEMENTS
My most sincere regard is given to Dr. Artyom Grigoryan for giving me the opportunity to learn to research and for being here for the students, to Dr. Walter Richardson, Jr. for teaching complex topics from the ground up and leading this horse of a student to mathematical waters applicable to my research, to Dr. Mihail Tanase for being the study group that I have never had, and to Dr. Azima Mottaghi for constant motivation, support and the remark, "You can finish it all in one day." Additionally, this work was progressed through discussions with Mehdi Hajinoroozi, Skei, hftf, and pavonia. I also acknowledge the UTSA Mexico Center for their support of this research.
December 2014 iv
ENHANCEMENT CLASSIFICATION OF GALAXY IMAGES
John Jenkinson, B.S. The University of Texas at San Antonio, 2014 Supervising Professor: Arytom Grigoryan, Ph.D., Chair With the advent of astronomical imaging technology developments, and the increased capacity of digital storage, the production of photographic atlases of the night sky have begun to generate volumes of data which need to be processed autonomously. As part of the Tonantzintla Digital Sky Survey construction, the present work involves software development for the digital image processing of astronomical images, in particular operations that preface feature extraction and classification. Recognition of galaxies in these images is the primary objective of the present work. Many galaxy images have poor resolution or contain faint galaxy features, resulting in the misclassification of galaxies. An enhancement of these images by the method of the Heap transform is proposed, and experimental results are provided which demonstrate the image enhancement to improve the presence of faint galaxy features thereby improving classification accuracy. The feature extraction was performed using morphological features that have been widely used in previous automated galaxy investigations. Principal component analysis was applied to the original and enhanced data sets for a performance comparison between the original and reduced features spaces. Classification was performed by the Support Vector Machine learning algorithm.
v
TABLE OF CONTENTS
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
1.2
Galaxy Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Hubble Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.2
de Vaucouleurs Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Digital Data Volumes in Modern Astronomy . . . . . . . . . . . . . . . . . . . . . 12 1.2.1
Digitized Sky Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.2
Problem Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3
Problem Description and Proposed Solution . . . . . . . . . . . . . . . . . . . . . 14
1.4
Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1
Survey of Automated Galaxy Classification . . . . . . . . . . . . . . . . . 15
1.4.2
Survey of Support Vector Machines . . . . . . . . . . . . . . . . . . . . . 17
1.4.3
Survey of Enhancement Methods . . . . . . . . . . . . . . . . . . . . . . 18
Chapter 2: Morphological Classification and Image Analysis . . . . . . . . . . . . . . . 20 2.1
Astronomical Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2
Image enhancement measure (EME) . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3
Spatial domain image enhancement . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1
Negative Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2
Logarithmic Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 28 vi
2.4
2.5
2.6
2.3.3
Power Law Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.4
Histogram Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.5
Median Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Transform-based image enhancement . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.1
Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2
Enhancement methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Image Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5.1
Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5.2
Rotation, Shifting and Resizing . . . . . . . . . . . . . . . . . . . . . . . 53
2.5.3
Canny Edge Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Data Mining and Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.6.1
Feature Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6.2
Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . 64
2.6.3
Support Vector Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.7
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.8
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Appendix A: Project Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 A.1 Preprocessing and Feature Extraction codes . . . . . . . . . . . . . . . . . . . . . 85 A.2 SVM Classification codes with data . . . . . . . . . . . . . . . . . . . . . . . . . 92 A.2.1
Original data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
A.2.2
Enhanced data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Vita
vii
LIST OF TABLES
Table 1.1
Hubble’s Original Classification of Nebulae Table . . . . . . . . . . . . . .
3
Table 2.1
Morphological Feature Descriptions . . . . . . . . . . . . . . . . . . . . . 64
Table 2.2
Feature Values Per Class . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Table 2.3
Galaxy list and relation between NED classification and current project classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Table 2.4
Summary of classification results for original and enhanced data. Accuracy improved by 12.924% due to enhancement. . . . . . . . . . . . . . . . . . 81
viii
LIST OF FIGURES
Figure 1.1
Hubble Tuning Fork Diagram. Image from http://www.physast.uga.edu/ rls/astro1020/ch20/ch26_fig26_9.jpg. . . . . . . . . . . . . . . . . . . . . .
Figure 1.2
Plate scan of Elliptical and Irregular Nebulae from Mount Wilson Observatory originally included in Hubble’s paper, Extra-galactic Nebulae. . . . .
Figure 1.3
2
4
Plate scan of Spiral and Barred Spiral Nebulae from Mount Wilson Observatory originally included in Hubble’s paper, Extra-galactic Nebulae. . . . .
6
Figure 1.4
A plane projection of the revised classification scheme. . . . . . . . . . . . 10
Figure 1.5
A 3-Dimensional representation of the revised classification volume and notation system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Figure 1.6
Sloan Digital Sky Survey coverage map. http://www.sdss.org/sdss-surveys/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Figure 2.1
Schmidt Camera of Tonantzintla. Permission to use image from the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE). . . . . . . . . 20
Figure 2.2
Plate Sky Coverage. Permission to use image from the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE). . . . . . . . . . . . . . . . . 21
Figure 2.3
Digitized plate AC8431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 2.4
Marked plate scan AC8431 . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Figure 2.5
Plate scan AC8409 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Figure 2.6
Marked plate scan AC8409 . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Figure 2.7
Cropped galaxies from plate scans AC8431 and AC8409 read left to right and top to bottom: NGC 4251, 4274, 4278, 4283, 4308, 4310, 4314, 4393, 4414, 4448, 4559, 3985, 4085, 4088, 4096, 4100, 4144, 4157, 4217, 4232, 4218, 4220, 4346, 4258. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Figure 2.8
Negative, log and power transformations. . . . . . . . . . . . . . . . . . . 28
ix
Figure 2.9
Top to bottom: Galaxy NGC4258 and its Negative Image. . . . . . . . . . . 29
Figure 2.10 Logarithmic and nth root transformations. . . . . . . . . . . . . . . . . . . 30 Figure 2.11 γ-power transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Figure 2.12 Galaxy NGC 4217 power law transformations. . . . . . . . . . . . . . . . . 32 Figure 2.13 Histogram processing to enhance Galaxy NGC 6070. . . . . . . . . . . . . 34 Figure 2.14 Top to Bottom: Histogram of original and enhanced image. . . . . . . . . . 35 Figure 2.15 Illustration of the median of a set of points in different dimensions. . . . . . 36 Figure 2.16 Signal-flow graph of determination of the five-point transformation by a vector x = (x0 , x1, x2, x3, x4 ). . . . . . . . . . . . . . . . . . . . . . . . . 43 Figure 2.17 Network of the x-induced DsiHT of the signal z. . . . . . . . . . . . . . . . 44 Figure 2.18 Intensity values and spectral coefficients of Galaxy NGC 4242. . . . . . . . 46 Figure 2.19 Butterworth lowpass filtering performed in the Fourier (frequency) domain.
47
Figure 2.20 α-rooting enhancement of Galaxy NGC 4242. . . . . . . . . . . . . . . . . 47 Figure 2.21 Top: Galaxy PIA 14402, Bottom: NGC 5194, both processed by Heap transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Figure 2.22 Computational scheme for galaxy classification. . . . . . . . . . . . . . . . 49 Figure 2.23 Background subtraction of Galaxy NGC 4274 by manual and Otsu’s thresholding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Figure 2.24 Morphological opening for star removal from Galaxy NGC 5813. . . . . . 54 Figure 2.25 Rotation of Galaxy image NGC 4096 by galaxy second moment defined angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Figure 2.26 Resizing of Galaxy NGC 4220. . . . . . . . . . . . . . . . . . . . . . . . . 59 Figure 2.27 Canny edge detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Figure 2.28 PCA rotation of axes for a bivariate Gaussian distribution. . . . . . . . . . 65 Figure 2.29 Pictorial representation of the development of the geometric margin. . . . . 69 Figure 2.30 Maximum geometric margin. . . . . . . . . . . . . . . . . . . . . . . . . . 70 Figure 2.31 SVM applied to galaxy data. . . . . . . . . . . . . . . . . . . . . . . . . . 73 x
Figure 2.32 Classification iteration class pairs. . . . . . . . . . . . . . . . . . . . . . . 77 Figure 2.33 PCA feature space iteration 1 classification. . . . . . . . . . . . . . . . . . 78 Figure 2.34 PCA feature space iteration 2 classification. . . . . . . . . . . . . . . . . . 79 Figure 2.35 PCA feature space iteration 3 classification. . . . . . . . . . . . . . . . . . 79 Figure 2.36 PCA feature space iteration 4 classification. . . . . . . . . . . . . . . . . . 80 Figure 2.37 PCA feature space iteration 1 classification of enhanced data. . . . . . . . . 81 Figure 2.38 PCA feature space iteration 2 classification of enhanced data. . . . . . . . . 82 Figure 2.39 PCA feature space iteration 3 classification of enhanced data. . . . . . . . . 82 Figure 2.40 PCA feature space iteration 4 classification of enhanced data. . . . . . . . . 83
xi
Chapter 1: INTRODUCTION 1.1 Galaxy Classification Why classify galaxies? It is an inherent characteristic of man to classify objects. Our country’s government classifies families according to annual income to establish tax laws. Medical doctor’s classify our blood’s type making successful transfusion possible. Organic genes are classified by genetic engineers so that freeze resistant DNA from a fish can be used to "infect" a tomato cell making the tomato less susceptible to cold. Words in the English language are assigned to the categories noun, verb, adjective, adverb, pronoun, preposition, conjunction, determiner, and exclamation, allowing for the structured composition of sentences. Differential equations are classified as ordinary (ODEs) and partial (PDEs) with ODEs having sub-categories: linear homogeneous, exact differential equations, n-th order equations, etc..., which allowing easy of study and for solution methods to be developed for certain classes such as the method of undetermined coefficients for ordinary linear differential equations with variable coefficients. If we say that a system is linear, there is no need to mention that the system’s input-output relationship is observed to be additive and homogeneous. Classification pervades every industry, and enables improved communication, organization and operation within society. For galaxies classification in particular, astrophysicists think that to understand the formation and subsequent evolution of galaxies one must first distinguish between the two main morphological classes of massive systems: spirals and early-type systems which are also called ellipticals. Galaxies with spiral arms, for example, are normally rotating disk of stars, dust and gas with plenty of fuel for future star formation. Ellipticals, however, are normally more mature system which long ago finished forming stars. The galaxies’ histories are also revealed; dust lane early-type galaxies are starbust systems formed in gas-rich mergers of smaller spiral galaxies. A galaxy’s classification can reveal information about its environment. A morphology-density relationship has been observed in many studies; spiral galaxies tend to be located in low-density environments and ellipticals in more dense environments [1, 2, 3].
1
There are many physical parameters of galaxies that are useful for their classification, but this paper considers the classification of galaxies by their morphology, a word derived from the Greek word morph, meaning shape or form. 1.1.1
Hubble Scheme
Hubble’s scheme was visually popularized by the "tuning fork" diagram which displays examples of each nebulae class, described in this section, in the transition sequence from early-type elliptical to late-type spiral. The tuning fork diagram is shown in Figure 1.1. While the basic classification
Figure 1.1: Hubble Tuning Fork Diagram. Image from http://www.physast.uga.edu/ rls/astro1020/ch20/ch26_fig26_9.jpg. of galaxy morphology assigns members to the categories of elliptical and spiral, the most prominent classification scheme was introduced by Sir Edwin Hubble in his 1926 paper, "Extra-galactic Nebulae." This classification scheme is based on galaxy structure. The individual members of a class differ only in apparent size and luminosity. Originally, Hubble stated that the forms divide themselves naturally into two groups: those found in or near the Milky Way and those in moderate 2
or high altitude galactic latitudes. This paper, along with Hubble’s classification scheme will only consider the extra-galactic division: Table 1.1 shows that this scheme contains two main divisions, Table 1.1: Hubble’s Original Classification of Nebulae Table Type: Symbol Example A. Regular: N.G.C 3379 E0 1. Elliptical....................................................En 221 E2 E5 (n=1,2,...,7 indicates the ellipticity of the image) 4621 2117 E7 2. Spirals: N.G.C. a) Normal spirals............................................S (1) Early..........................................................Sa 4594 (2) Intermediate..............................................Sb 2841 5457 (3) Late...........................................................Sc N.G.C. b) Barred spirals.............................................SB (1) Early..........................................................SBa 2859 (2) Intermediate..............................................SBb 3351 (3) Late...........................................................SBc 7479 B. Irregular: ........................................................................Irr 4449 regular and irregular galaxies. Within the regular division, three main classes exist: elliptical, spirals, and barred spirals. The term nebulae and galaxies are used interchangeably with a brief discussion of the rational for this at the end of this subsection. N.G.C. and U.G.C are acronyms for New General Catalogue and Uppsala General Catalogue, respectively, and are designations for deep sky objects. Elliptical galaxies range in shape from circular through flattening ellipses to a limiting lenticular figure in which the ratio of axes is about 1 to 3 or 4. They contain no apparent structure except for their luminosity distribution which is maximum at the center of the galaxy and decreases to unresolved edges. The degree to which an elliptical nebulae is flattened is determined by the criterion, elongation, defined as (a − b)/a, where a and b are the semi major and semi minor axes, respectively, or an ellipse fitted to the nebulae. The elongation mentioned here is different than, and not to be confused with, the morphic feature elongation that is introduced later in this paper. Elliptical nebulae are designated by the symbol,"E," followed by the numerical value of ellipticity. 3
The complete series is E0, E1,. . ., E7, the last representing a definite limiting figure which marks the junction with spirals. Examples of nebulae with differing ellipticities are shown in Figure 1.2.
Figure 1.2: Plate scan of Elliptical and Irregular Nebulae from Mount Wilson Observatory originally included in Hubble’s paper, Extra-galactic Nebulae. All regular nebulae with ellipticities greater than about E7 are spirals, and no spirals are known 4
with ellipticity less than this limit. Spirals are designated by the symbol "S". Classification criteria for spiral nebulae is: (1) relative size of the unresolved nuclear region; (2) extent to which the arms are unwound; (3) degree of resolution in the arms. Relative size of the nucleus decreases as the arms of the spiral more widely open. The stages of this transition of spiral galaxies are designed as "a" for early types, "b" for intermediate types, and "c" for late types. Nebulae intermediate between E7 and Sa are occasionally designated as S0, or lenticular. Barred spirals is a class of spirals which have a bar of nebulosity extending diametrically across the nucleus. This class is designated by the symbol "SB", with a sequence which parallels that of normal spirals, leading to the subdivision of barred spirals designated by "SBa", "SBb", and "SBc" for early, intermediate and late type barred spirals, respectively. Examples of normal and barred spirals along with their subclasses are shown in Figure 1.3. Irregular nebulae are extra-galactic nebulae that lack both discriminating nuclei and rotational symmetry. Individual stars may emerge from an unresolved background in these galaxies. For any given imaging system, there is a limiting resolution beyond which classification cannot be made with any confidence. Hubble designed galaxies within this category by the letter "Q." On the usage of nebulae versus galaxy, the astronomical term nebulae has come down through the centuries as the name for permanent, cloudy patches in the sky that are beyond the limits of the solar system. In 1958, the term nebulae was used for two types of astronomical bodies: clouds of dust and gas which are scattered among the stars of the galactic system (galactic nebulae), and the remaining objects, which are now recognized as independent stellar systems scattered through space beyond the limits of the galactic system (extra-galactic nebulae). Some astronomers considered that since nebulae are now considered stellar systems they should be designated by some other name, which does not carry the connotation of clouds or mist. Today, those who adopt this consideration refer to other stellar systems as external galaxies. Since this paper only considers external galaxies we will drop the adjective and employ the term galaxies for whole external stellar systems [4].
5
Figure 1.3: Plate scan of Spiral and Barred Spiral Nebulae from Mount Wilson Observatory originally included in Hubble’s paper, Extra-galactic Nebulae.
6
1.1.2
de Vaucouleurs Scheme
The de Vaucouleurs Classification system is an extension of the Hubble Classification system, and is the most commonly used system. For this reason it is noted in this paper. About 1935, Hubble undertook a systematic morphological study of the approximately 1000 brighter galaxies listed in the Shipely Ames Catalogue, north of -30° declination, with a view of refining his original classification scheme. The main revisions include a) the introduction of the S0 and SB0 types regarded as transition stages between ellipticals and spirals at the branching off point of the tuning fork. S0, or lenticular galaxies resemble spiral galaxies in luminosity, but do not contain visible spiral arms. A visible lens surrounds these galaxies bordered by a faint ring of nebulosity. Characteristics of lenticular galaxies are a bright nucleus in the center of a disc or lens. Near the perimeter of the galaxy, there exists a faint rim or envelope with unresolved edges. Hubble separated the lenticulars into two groups, S0(1) and S0(2). These groups have a smooth lens and envelope, and some structure in the envelope in the form of a dark zone and ring, respectively. S0/a is the transition stage between S0 and Sa and shows apparent developing spiral structure in the envelope. SB0 objects are characterized by a bar through the central lens. Hubble distinguished three groups of SB0 objects: group SB0(1) have a bright lens, with broad, hazy bar and no ring, surrounded by a larger, fainter envelopes some being circular, group SB0(2) have a broad, weak bar across a primary ring, with faint outer secondary rings, and group SB0(3) have a well developed bar and ring pattern, with the bar stronger than the ring. c) Harlow Shapely proposed an extension to the normal spiral sequence beyond Sc designating galaxies showing a very small, bright nucleus and many knotty irregular arms by Sd. A parallel extension of the barred spiral sequence beyond the stage SBc was introduced by de Vaucouleurs in 1955 which may be denoted SBd or SBm [5, 6]. For Irregular type galaxies related to Magenellic Clouds, I(m), an important characteristic is their small diameter and low luminosity which marks them as dwarf galaxies. d) Shapely discovered the existence of dwarf ellipticals (dE) by observation of ellipticals with
7
very low surface brightness. de Vaucouleurs noted that after all such types or variants have been assigned into categories, there remains a hard core of "irregular" objects which do not seem to fit into any of the recognized types. These outliers are presently discarded, and only isolated galaxies are considered in the present article. The coherent classification scheme proposed by de Vaucouleurs which included most of the current revision and additions to the standard classification is described here. Classification and notation of the scheme are illustrated in Figure 1.4, which may be considered as a plane projection of the three dimensional representation in Figure 1.5. Four Hubble classes are retained: ellipticals E, lenticulars S0, spirals S, irregulars I. Lenticulars and spirals, were re-designated "ordinary" SA and "barred" SB, respectively, to allow for the use of the compound symbol SAB for the transition stage between these two classes. The symbol S alone is used when a spiral object cannot be more accurately classified as either SA or SB because of poor resolution, unfavorable tilt, etc. Lenticulars were divided into two subclasses, denoted SA0 and SB0, where SB0 galaxies have a bar structure across the lens and SA0 galaxies do not. SAB0 denotes objects with a very weak bar. The symbol S0 is now used for a lenticular object which cannot be more precisely classified as either SA0 or SB0; this is often the case for edgewise objects. Two main varieties are recognized in each of the lenticular and spiral families, the" annular" or "ringed" type, denoted (r), and the" spiral" or " S-shaped" type, denoted (s). Intermediate types are noted (rs). In the "ringed" variety the structure includes circular (sometimes elliptical) arcs or rings (SO) or consists of spiral arms or branches emerging tangentially from an inner circular ring (5). In the "spiral" variety two main arms start at right angles from a globular or little elongated nucleus (5 A) or from an axial bar (5 B). The distinction between the two families A and B and between the two varieties (r) and (s) is most clearly marked at the transition stage SO/a between the SO and 5 classes. It vanishes at the transition stage between E and SO on the one hand, and at the transition stage between 5 and I on the other (d. Fig. 3). 8
Four sub-divisions or stages are distinguished along each of the four spiral sequences SA(r), SA (s), SB(r), SB(s), viz. "early", "intermediate" and "late" denoted a, b, e as in the standard classification, with the addition of a "very late" stage, denoted d. Intermediate stages are noted 5 ab, 5 be, 5 cd. The transition stage towards the magellanic irregulars (whether barred or not) is noted 5 m, e.g. the Large Magellanic Cloud is 5 B (s) m. Along each of the non-spiral sequences the signs + and - are used to denote " early" and "late" subdivisions; thus E+ denotes a "late" E, the first stage of the transition towards the SO class 2. In both the SAO and S BO sub-classes three stages, noted SO-, 50°, 50+ are thus distinguished; the transition stage between SO and Sa, noted SO/a by HUBBLE, may also be noted Sa-. Notations such as S a+, S b-, etc. may be used occasionally in the spiral sequences, but the distinction is so slight between, say, 5 a+ and S b-, that for statistical purposes it is convenient to group them together as 5 a b, etc. Experience shows that this makes the transition subdivisions, Sab, Sbe, etc. as wide as the main sub-divisions, Sa, Sb, etc. 3. The classification of irregulars which do not show clearly the characteristic spiral structure are noted I(m). Figure 1.4 shows a plane projection of the revised classification scheme.Compare with Figure 1.5. The ordinary spirals SA are in the upper half of the figure, the barred spirals SB in the lower half. The ring types (r) are the the left, the spiral types (s) to the right. Ellipticals and lenticulars are near the center, magellanic irregulars near the rim. The main stages of the classification sequence from E to Im through S0-, S0, S0+, Sa, Sb, Sc, Sd, Sm are illustrated, approximately on the same scale, along each of the four main morphological series SA(r), SA(s), SB(r), SB(s). A few mixed or "intermediate" types SAB and S(rs) are shown along the horizontal and vertical diameters respectively. This scheme is superseded by the slightly revised and improved system illustrated in Figure 1.5. Figure 1.5 shows a 3-Dimensional representation of the revised classification volume and notation system. From left to right are the four main classes: ellipticals E, lenticulars S0, spirals S, and Irregulars I. Above are ordinary families SA, below the barred families SB; on the near side 9
Figure 1.4: A plane projection of the revised classification scheme. are the S-shaped varieties s(s), on the far side the ringed varieties S(r). The shape of the volume indicated that the separation between the various sequences SA(s), SA(r), SB(r), SB(s) is greatest at the transition stage S0/a between lenticulars and spirals and vanishes at E and Im. A central cross-section of the classification volume illustrates the relative location of the main types and the notation system. There is a continuous transition of mixed types between the main families and va10
rieties across the classification volume and between stages along each sequence; each point in the classification volume represents potentially a possible combination of morphological characteristics. For classification purposes this infinite continuum of types is represented by a finite number of discrete "cells" [5, 6, 7]. The classification scheme included here defers to [5, 6] for a complete
Figure 1.5: A 3-Dimensional representation of the revised classification volume and notation system. description.
11
1.2 Digital Data Volumes in Modern Astronomy 1.2.1
Digitized Sky Surveys
Modern astronomy has produced massive volumes of data relative to that produced at the start of the 20th century. Digitized sky surveys attempt to construct a virtual photographic atlas of the universe through the identification and cataloging of observed celestial phenomena for the purpose of understanding the large-scale structure of the universe, the origin and evolution of galaxies, the relationship between dark and luminous matter, and many other topics of research interest in astronomy. This idea is being realized through the efforts of multiple organizations and all sky surveys. Notable surveys and their night sky coverage contribution and data collection are mentioned here. The Sloan Digital Sky Survey (SDSS) is the most prominent on going all sky survey, in its seventh data release almost 1 billion objects have been identified in approximately 35% of the night sky. Comprehensive data collection for the survey which uses electronic light detectors for imaging is projected at 15 terabytes [8]. An image from the SDSS displaying the current coverage of the sky in orange with selected regions displayed in higher resolution is shown in Figure 1.6. The Galaxy Evolution Explorer (GALEX), a NASA mission led by Caltech, has used micro channel plate detectors in two bands to image 2/3 of the night sky from the GALEX satellite between 2003 and the present in its survey [9]. In 1969, the two micro sky survey (TMSS) scanned 70% of the sky and detected approximately 5,700 celestial sources of infrared radiation [10]. With the advancement of infrared sensing technology, the Two micron "all-sky" survey (2MASS) detected an 80,000 fold increase over the TMSS between 1997 and 2001. The 2MASS was conducted by two separate observatories at Mount Hopkins Arizona and Cerro Tololo Inter-American Observatory (CITO), Chile, using 1.3 meter telescopes equipped with a 3 channel camera and a 256x256 electronic light detector. Each night of released data consisted of 250,000 point sources, 2,000 galaxies, and 5,000 images weighing about 13.8 Gigabytes per facility. The compiled catalog has over 1,000,000 galaxies, extracted from 99.998% sky coverage and 4,121,439 atlas images [11]. 12
Figure 1.6: Sloan Digital Sky Survey coverage map. http://www.sdss.org/sdss-surveys/. Sky coverage by the Space Telescope Science Institute’s Guide Star Catalog 2 (GSC-2) survey which occurred from 2000 to 2009 was 100%. The optical catalog produced by this survey used 1" resolution scans of 6.5x6.5 square degrees photographic plates from the Palomar and UK Schmidt telescopes. Almost 1 billion point sources were imaged. Each plate was digitized using a modified microdensitometer with a pixel size of either 25 or 15 microns (1.7 or 1.0 arcsec respectively). The digital images are 14000x14000 (0.4 GB) or 23040x23040 (1.1 GB) in size [12]. The second Palomar Observatory Sky Survey (POSS2) images 897 plates between the early 1980’s and 1999 which covered the entire southern celestial hemisphere using the Oschin Schmidt telescope [13]. One of the main objectives of the ROSAT All-sky survey was to conduct the first all-sky survey in X-ray with an imaging telescope leading to a major increase in sensitivity and source location
13
accuracy. ROSAT was conducted between 1990-1991 covering 99.7% of the sky [14]. The Faint Images of the Radio Sky at Twenty-centimeters (FIRST) project was designed to produce the radio equivalent of the Palomer Observatory Sky Survey 10,000 square degrees of the North and South Galactic Caps. The survey began in 1993 and is currently active [15, 16]. The Deep Near Infrared Survey (DENIS) is a survey of the southern sky in two infrared and one optical band conducted at the La Silla European Space Observatory in Chile. The survey ran from 1996 through 2001 and cataloged 355 million point sources [17]. The present work is part of the Tonantzintla Digital Sky Survey which is discussed in Chapter 2. 1.2.2
Problem Motivation
The image quantity and data volume produced by digital sky surveys presents human analysis with an impossible task. Therefore, source detection and classification in modern astronomy necessitate automation in the image processing and analysis, providing the motivation for the present work. To address this problem, an algorithm for processing astronomical images to classify galaxies contained therein is presented and implemented using followed by class discrimination of the detected galaxies according to the scheme mentioned in section 1.1.1. Class discrimination is performed using extracted galaxy feature values which experience varying accuracy with different methods of segmentation. Faint regions of galaxies can be lost during segmentation, leading to increased error during feature extraction and subsequent classification. Enhancement of the galaxy image by multiple methods is proposed and implemented to reduce data loss during segmentation and improve the accuracy of feature extraction implied through the increase of classification performance.
1.3 Problem Description and Proposed Solution This project is part of the on going work within the Tonantzintla Digital Sky Survey. The present work focuses on automated astronomical image processing and classification. Final performance criterion is 100% classification in categories E0,. . . ,E7, S0, Sa, Sb, Sc, SBa, SBb, SBc, Irr, while the present work builds towards that goal by incremental improvement of classification perfor14
mance with categories elliptical "E," spiral "S," lenticular "S0," barred spiral "SB," and irregular "Irr." The intent in this work is to partially or fully resolve the classification performance limitations within the galaxy segmentation, edge detection and feature extraction stages of the image processing pipeline by enhancing the galaxy images by method of the Heap transform to preserve the faint regions of the galaxies which may be lost during the processing of images without enhancement. Classification is performed by the supervised machine learning algorithm Support Vector Machines (SVM).
1.4 Previous Work 1.4.1
Survey of Automated Galaxy Classification
Morphological classification of galaxies into 5 broad categories was performed by the artificial neural network (ANN) machine learning algorithm with back propagation trained using 13 parameters by Storrie-Lombardi in [18]. Odewahn classified galaxies from large sky surveys using ANNs in [35, 36, 37]. The development progress of an automatic star/galaxy classifier using Kohonen Self-Organizing Maps was presented in [38, 39] and using learning vector quantization and fuzzy classified with back-propogation based neural networks in [39]. An automatic system to classify images of varying resolution based on morphology was presented in [40]. Owens, in [19], shows comparable performance between the machine learning algorithms of oblique decision trees induced with different impurity measures to the artificial neural network used in [18] and that classification of the original data could be performed with less well-defined categories. In [20] an artificial neural network was trained on the features of galaxies that were defines as a galaxy class mean by 6 independent experts. The network performed comparable to the overall root mean square dispersion between the experts. A comparison of the classification performance of an artificial neural network machine learning algorithm to that of human experts for 456 galaxies with their source being the SDSS in [20] was detailed in [21]. Lahav showed the classification performance of galaxy images and spectra an unsupervised artificial neural network trained with galaxy spectra
15
de-noised and compressed by principal component analysis. A supervised artificial neural network was also trained with classes determined by human experts [22]. Folkes, Lahav and Maddox trained an artificial neural network using a small number of principal components selected from galaxy spectra with low signal-to-noise ratios characteristic of redshift surveys. Classification was the performed into 5 broad morphological classes. It was shown that artificial neural networks are useful in discriminating normal and unusual galaxy spectra [23]. The use of galaxy parameters luminosity and color and the image-structure parameters: size, image concentration, asymmetry and surface brightness to classify galaxy images into three classes was performed by Bershady, Jangren and Conselice. It was determined that the essential features for discrimination were a combination of spectral index, e.g., color, and concentration, asymmetry, and surface brightness [24]. A comparison using ensembles of classifiers for the classification methods Naive bayes, back propagation artificial neural network, and a decision-tree induction algorithm with pruning was performed by Bazell which resulted in the artificial neural network producing the best results, and ensemble methods improving the performance of all classification methods [30]. A computational scheme to develop an automatic galaxy classifier using galaxy morphology was shown to provide robustness for classification using artificial neural networks in [26, 34]. Bazell derived 22 morphological features, including asymmetry, which were used to train an artificial neural network for the classification of galaxy images to determine which features were most important [27]. Strateva used visual morphology and spectral classification to show that two peaks correspond roughly to early (E, S0, Sa) and late-type (Sb, Sc, Irr) galaxies. It was also shown that the color of galaxies correlates with their radial profile [28]. The Gini coefficient, a statistic commonly used in econometrics to measure the distribution of wealth among a population, was used to quantify galaxy morphology based on galaxy light distribution in [29]. In [31], an algorithm for preprocessing galaxy images for morphological classification was proposed. In addition, the classification performance between an artificial neural network, locally weighted regression and homogeneous ensembles of classifiers was performed for 2 and 3 galaxy classes. Lastly, compression and discrimination by principal component analysis was performed. The artificial network performed best under all con16
ditions. In [32], principal component analysis was applied to galaxy images and a structural type estimator names "ZEST" used a 5 nonparametric diagnosis to classify galaxy structure. Finally, Banerji presented morphological classification by artificial neural networks for 3 classes yielding 90% accuracy in comparison to human classifications [33]. 1.4.2
Survey of Support Vector Machines
This method of class segregation is performed by hyperplanes which can be defined by a variety of functions, both linear and non linear. The development of this method is presented in Chapter 2. Support vector machines (SVMs) have been employed widely in the areas of pattern recognition and prediction. Here a limited survey of SVM applications is presented, which includes two surveys conducted by researchers in the field. Romano applied SVMs to photometric and geometric features computed from astronomical imagery for the identification of possible supernovae in [42]. M. Huertas-Company applied SVM to 5 morphological features, luminosity and redshift calculated from galaxy images in [43]. Freed and Lee classified galaxies by morphological features into 3 classes using a SVM in [44]. Saybani conducted a survey of SVMs used in oil refineries in [45]. Xie proposed a method for predicting crude oil prices using a SVM in [90]. Petkovi used a SVM to predict the power level consumption of an oil refinery in [47]. Balabin performed near infrared spectroscopy for gasoline classification using nine different multivariate classification methods including SVMs in [48]. Byun and Lee conducted a comprehensive survey on applications of SVMs for pattern recognition and prediction in [41]. References contained therein are included here in support of the present survey. For classification with q classes (q>2), classes are trained pairwise. The pairwise classifiers are arranged in trees where each tree node represents a SVM. A bottom up tree originally proposed for recognition of 2D objects was applied to face recognition in [49, 50]. In contrast, an interesting approach was the top down tree published in [51]. SVMs applied to improve classification speed of face detection was presented in [63, 53]. Face detection from multiple views was presented in [56, 55, 54]. A SVM was applied to coarse eigenface detection for a fine detection in [57]. Frontal face detection using SVMs was discussed in [58]. [59] presented 17
SVMs for face and eye detection. Independent component analysis for face features were input to the SVM in [60], orthogonal Fourier-Mellin Moments in [61], and an overcomplete wavelet decomposition as input in [62]. A myriad of other applications have been ventured using SVMs including but not limited to 2-D and 3-D object recognition [64, 65, 66], texture recognition [66], people and pose recognition [67, 68, 69, 70, 71], moving vehicle detection [72], radar target recognition [73, 76], hand written character and digit recognition [74, 75, 71, 77], speaker or speech recognition [78, 79, 80, 81], image retrieval [82, 83, 84, 85], prediction of financial time series [86], bankruptcy [87], and other classifications such as gender [88], fingerprints [89], bullet-holes for auto scoring [90], white blood cells [91], spam categorization [92], hyperspectral data [93], storm cells [94], and image classification [95]. 1.4.3
Survey of Enhancement Methods
Image enhancement is the process of visually improving the quality of a region of or the entire image with respect to some measure of quality, e.g., the Image Enhancement Measure (EME) introduced in Chapter 2. Enhancement methods can be classified as either spatial domain or transform domain methods depending on whether the manipulation of the image is performed directly on the pixels or on the spectral coefficients, respectively. Here, a survey of both spatial and transform domain methods is presented for the enhancement of astronomical images and images in general. Spatial domain methods are commonly referred to as contrast enhancement methods. The core of these methods are histogram equalization, logarithmic and inverse log transformations, negative and identity transformations, nth-power and nth-root transformations, histogram matching and local histogram processing. Adaptive histogram equalization, which uses local contrast stretching to calculate several histograms corresponding to distinct sections of the image, was applied after denoising to improve the contrast of astronomical images in [96, 99, 100, 34] and generic images in [106]. Traditional histogram equalization was applied to the Hale-Bopp comet image for enhancement in [98] and other astronomical images in [97, 101, 103, 104, 105]. [102] included histogram equalization in the development of two algorithms for point extraction and 18
matching for registration of infrared astronomical images. Astronomical images were logarithmically transformed for visualization in [108] and likewise for generic images in [127]. Inverse log transformations, negative and identity transformations, nth-power and nth-root transformations, histogram matching and local histogram processing are introduced and applied to generic images in [107, 126, 127, 129]. At the core of transform domain methods for image enhancement exist the discrete Fourier, Heap, α-rooting, Tensor, and Wavelet transforms. Astronomical image enhancement performed by the discrete Fourier transform was presented in [109, 111, 112], by the Wavelet transform in [110] and by the Heap and α-rooting transform in [113], and the Curvelet transform in [114, 98]. The enhancement of generic images can be seen in [115, 127, 128, 129] by the discrete Fourier and Cosine transforms, in [116] by the Heap transform, in [117, 118, 127, 128] by the α-rooting, in [119, 120, 121, 122] by the Tensor or Paried transform, in [123, 98, 124] by the Wavelet transform, and in [124, 125] by other methods of transform domain processing.
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Chapter 2: MORPHOLOGICAL CLASSIFICATION AND IMAGE ANALYSIS
2.1 Astronomical Data Collection
Figure 2.1: Schmidt Camera of Tonantzintla. Permission to use image from the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE).
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The Tonantzintla Schmidt camera was constructed in the Harvard Observatory shop under the guidance of Dr. Harlow Shapley, and started operation in 1942. The spherical mirror is 762 mm in diameter and coupled to a 660.4 mm correcting plate. The camera is shown in figure 2.1. The 8x8 inch2 photographic plates cover a 5ºx5º field with a plate-scale of 95 arcsec/mm. The existing collection consists of a total of 14565 glass plates: 10445 taken in direct image mode; and 4120 through a 3.96° objective prism. Figure 2.2 shows the sky covered by the complete plate collection, marking the center of each observed field [130].
Figure 2.2: Plate Sky Coverage. Permission to use image from the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE). The plates are first digitized at the maximum optical resolution of the scanner, 4800 dots per inch (dpi), and then rebinned by a factor 3 for a final pixel size of ˜ 15 μm (1.51 arcsec/pixel) and transformed to the transparency (positive) mode. Each image has 12470 x 12470 pixels (about 350 Mb in 16-bit mode) and is stored in FITS format. The images in this project were received from the collection of digitzed photographic plates at 21
the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE). The present data set consists of 6 plate scans. All 6 plates were marked to indicate the galaxies contained within the image. The goal is the process the digitized plates automatically, i.e., segmenting galaxies within the image, calculating their features and performing classification. In initial attempts of processing the plate scans in Matlab on an Alienware M14x with an Intel Core i7-3840QM 2.80GHz CPU and 12.0GB DDRAM5, e.g, applying the watershed algorithm for segmentation, memory consumption errors were experienced. Consequently, the galaxies within each plate scan were cropped and processing individually. Figures 2.3, 2.4, 2.5, 2.6, and 2.7 show the original digitized plates AC841 and AC8409, their marked versions indicating captured galaxies, and the cropped galaxies from both plates. Upon performing automatic classification with the cropped images, one of the University of Texas at San Antonio’s (UTSA) high performance computing clusters SHAMU, will be used for the automatic classification of whole plate scans. SHAMU consists of twenty-two computational nodes and two high-end visualization nodes. Each computational node is powered by dual Quadcore Intel Xeon E5345 2.33GHz processors (8M Cache). SHAMU consists of twenty-three Sun Fire X4150 servers, four Penguin Relion 1800E servers, a DELL Precision R5400 and a DELL PowerEdge R5400. SHAMU utilizes GlusterFS open-source file system over high speed InfiniBand connection. A Sun StorageTek 2530 SAS array, fully populated with twelve 500GB hard drives, acts as SHAMU’s physical storage in a RAID 5 configuration. SHAMU is networked together with two DELL PowerConnect Ethernet switches and one QLogic Silverstorm InfiniBand switch.
2.2 Image enhancement measure (EME) To measure the quality of images and select optimal processing parameters, we consider the described in [131, 128] quantitative measure of image enhancement that relates to Weber’s law of human visual system. This measure can be used for selecting the best parameters for image enhancement by the Fourier transform, as well as other unitary transforms. The measure is defined as follows. A discrete image {fn,m } of size N1 × N2 is divided by k1 k2 blocks of size L1 × L2, 22
Figure 2.3: Digitized plate AC8431 where integers Li = [Ni /ki ], i = 1, 2. The quantitative measure of enhancement of the processed image, Ma : {fn,m } → {fˆn,m }, is defined by k1 k2 ˆ) 1 ( f max k,l ˆ = EMEa (f) 20 log10 , ˆ k1 k2 k=1 l=1 mink,l (f) ˆ and mink,l (f) ˆ respectively are the maximum and minimum of the image fˆn,m where maxk,l (f) inside the (k, l)th block, and a is a parameter, or a vector parameter of the enhancement algorithm. 23
Figure 2.4: Marked plate scan AC8431 EMEa (fˆ) is called a measure of enhancement, or measure of improvement of the image f. We ˆ = EMEa (f) ˆ to be the best (or optimal) Φ-transformdefine a parameter a0 such that EME(f) 0 based image enhancement vector parameter. Experimental results show that the discrete Fourier transform can be considered as the optimal, when compared with the cosine, Hartley, Hadamard, and other transforms. When Φ is the identity transformation, I, the EME of fˆ = f is called the enhancement measure of the image f, i.e., EME(f) = EMEI (f). EME values of the enhanced galaxy images are presented in subsequent subsections. 24
Figure 2.5: Plate scan AC8409
2.3 Spatial domain image enhancement Contrast enhancement is the process of improving image quality by manipulating the values of single pixels in an image. This processing is said to occur in the spatial domain, meaning that the image involved in processing is represented as a plane in 2-Dimensional Euclidean space, which coined contrast enhancement methods as spatial domain methods. Contrast enhancement in the spatial domain is paralleled by transform based methods which operate in the frequency domain as
25
Figure 2.6: Marked plate scan AC8409 is shown in following subsections. The image enhancement is described by a transformation T T : f(x, y) → g(x, y) = T[f(x, y)]
where f(x, y) is the original image, g(x, y) is the processed image, and T is the enhancement operator. As a rule, T is considered to be a monotonic and invertible transformation.
26
Figure 2.7: Cropped galaxies from plate scans AC8431 and AC8409 read left to right and top to bottom: NGC 4251, 4274, 4278, 4283, 4308, 4310, 4314, 4393, 4414, 4448, 4559, 3985, 4085, 4088, 4096, 4100, 4144, 4157, 4217, 4232, 4218, 4220, 4346, 4258. 2.3.1
Negative Image
This transformation is especially useful for processing binary images, e.g., text-document images, and is described as Tn : f(x, y) → g(x, y) = M − f(x, y)
27
for every pixel (x, y) in the image plane. M is the maximum intensity in the image f(x, y). Figure 2.8 shows this transformation for the image 0 ≤ f(x, y) ≤ L − 1, where L is the intensity value in the image. In the discrete, M is the maximum level, M = L − 1, and Tn : r → s = L − 1 − r, where r is the original image intensity and s is the intensity mapped by the transformation. The example of an image negative is given in Figure 2.9.
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identity negative 46*log(1+r) 16*sqrt(1+r)
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40*(1+r)(1/3) 0.004*r2 c*r3
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Figure 2.8: Negative, log and power transformations.
2.3.2
Logarithmic Transformation
The logarithmic function is used in image enhancement, because it is a monotonically increasing function. The transformation is described as Tl : f(x, y) → g(x, y) = c0 log(1 + f(x, y))
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Figure 2.9: Top to bottom: Galaxy NGC4258 and its Negative Image. where c0 is a constant and is calculated as c0 = M/log(1 + M) in order to preserve the resolution of the enhanced image by gray scale. For example, for the 256-gray level scale image, c0 ≈ 46. Other versions of this transform are based on the use of the nth roots instead of the log function as 29
shown in Figure2.8. For example, T2 : f(x, y) → g(x, y) = c0
1 + f(x, y).
where the constant c0 = 16, when processing a 256-level gray scale image. Examples of image enhancement by such transformations are given in Figure 2.10.
(a) Original image
(b) log transformation
(c) square root transformation
(d) 3rd root transformation
Figure 2.10: Logarithmic and nth root transformations.
2.3.3
Power Law Transformation
These transformations are parameterized by γ and described as Tγ : f(x, y) → g(x, y) = cγ (1 + f(x, y))γ
30
where γ > 0 is a constant which is selected by the user. The constant cγ is used to normalize the gray scale levels within [0,M]. For 0 ≤ γ ≤ 1, the transform maps a narrow range of dark samples of the image into a wide range of bright samples, and it smoothes the difference between intensities of bright samples of the original image. The Power law transformation is shown with γ = 0.0500, 0.8500, 1.6500, 2.4500, 3.2500, 4.0500, and 4.8500 in Figure 2.11.
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original 0.05 0.85 1.65 2.45 3.25 4.05 4.85
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Figure 2.11: γ-power transformation. Examples of image enhancement by power log transformations are given in Figure 2.12. 2.3.4
Histogram Equalization
Consider an image of size N × N as a random realization ξ takes values r from a range [rmin, rmax], and let h(r) = fξ (r) be the probability density function of ξ. It is desirable to transform the image in such a way that the new image will have the uniform distribution. The equates to a change of 31
(a) Original image
(b) γ = 0.005
(c) γ = 0.3
(d) γ = 0.9
Figure 2.12: Galaxy NGC 4217 power law transformations.
32
random variable ξ → ξ = w(ξ)
(w : r → s)
such that w is a monotonically increasing function h(s) = fξb(s) =
1 . w(rmax ) − w(rmin )
The following fact is well-known: h(s) = h(r)
dr ds
or h(r)dr = h(s)ds. Integrating this equality yields r rmin
1 ds = w(rmax) − w(rmin )
which yields s = w(r) w(r) − w(rmin ) = w(rmax) − w(rmin )
r h(a)da rmin
r h(a)da = F (r). rmin
In the particular case, when rmin = 0 and w(rmin ) = 0, the following result is obtained w(r) = w(rmax )F (r).
In the case of digital image, where the image has been sampled and quantized, the discrete version of this transform has the representation
r→s=
⎧ r ⎪ ⎪ ⎨ M h(k) if r = 1, 2, . . . , M − 1 ⎪ ⎪ ⎩ 0
k=1
if r = 0
where r is the integer value of the original image, s is the quantized value of the transformed image, and h(k) is the histogram of the image.
33
So, independent of the image intensity probability density function, the intensity density function of the processed image is uniform, fξb(s) =
1 . w(rmax) − w(rmin )
Histogram equalization applied to galaxy NGC 6070 is shown in Figure 2.13 with the corresponding original and enhanced image histograms shown in Figure 2.14. The histogram equalization destroys the details of the galaxy image, indicating that spatial methods of enhancement are not suitable for all images. This is part of the motivation for using α-rooting, Heap transform, and other transform based which are described in the next section.
(a) Original image
(b) Histogram equalization
Figure 2.13: Histogram processing to enhance Galaxy NGC 6070.
2.3.5
Median Filter
A noteworthy spatial domain filter is the Median filter. This filter is based on order statistics. Given a set of numbers S = {1, 2, 1, 4, 2, 5, 6, 7}, the values in S are rearranged in order of descending value, i.e., 7, 6, 5, 4, 2, 2, 1, 1, and labeled as order statistics in ascending order, i.e., 7 is the 1st order statistic and the second 1 is the 7th order statistic. The 4 and adjacent 2 can both be considered 34
9000 8000 7000 6000 5000 4000 3000 2000 1000 0
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Figure 2.14: Top to Bottom: Histogram of original and enhanced image.
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as the median here, and the selection is made at the discretion of the user. In general, the highest order statistics is regarded as the nth order statistic. The Median filter comes from the follow problem in probability. Given a set of points S = {x1, x2 , . . . , x7} containing the Median point m, i.e., m ∈ S, which point in the set closest to every other point in the set. Figures 2.15 illustrate this in two different ways. The Median m is found by minimization of the following function |m − x1 | + |m − x2| + |m − x3| + · · · + |m − xn | =
n
|xk − m|.
k=1
In signal filtration, the Median filter preserves the range and edges of the original signal in contrast to the mean filter which destroys the signal edges. For signals with many consecutive noisy points, the length of the median filter must be extended to retain this behavior. The Median filter has the root property where the output of the filtration will be identical to the previous output after a certain number of filtration iterations. The Median filter is effective in removing salt and pepper noise.
x6
x1 x2
x5
m
x7 x 1 x 2x 3 x 4
x3
m
x 5 x6 x 7
x8
x4 (a) median in the line
(b) median in space
Figure 2.15: Illustration of the median of a set of points in different dimensions.
36
2.4 Transform-based image enhancement In parallel to directly processing image pixels in the spatial domain by contrast enhancement methods, transform based methods of enhancement manipulate the spectral coefficients of an image in the domain of the transform. The primary benefits of these methods are low computational complexity and the usefulness of unitary transforms for filtering, coding, recognition, and restoration analysis in signal and image processing. First the operators that transform the domain of the image are introduced followed by methods of enhancement in the transform domain. 2.4.1
Transforms
Each of the following transforms presented here in one dimension can easily be extended into two dimensions which is where the transforms are useful for image processing. Fourier Transform The one dimensional discrete Fourier transform (1-D DFT) maps the real line in the time domain to the complex domain resulting in time domain signals being transformed into the frequency domain. The direct transform and inverse transform pair are defined, for a discrete function xn , as
2πnp 2πnp − jxn sin xn cos Fp = N N n=0
N −1 1 2πnp 2πnp xn = Fpcos + jFpsin N p=0 N N N −1
where n = 0, 1, . . . , N − 1 represents discrete time points and p = 0, 1, . . . , N − 1 represents discrete frequency points. The basis functions for this transform are complex exponentials. The "real" and "imaginary" parts of this sum are considered as the sum of the cosine terms and the sum of the sine terms, respectively, and are computed by the fast Fourier transform.
37
Hartley Transform Similar to the Fourier transform is the Hartley transform, but only generates real coefficients. This transform is defined in the one dimensional case as Hp =
N −1
xn
n=0
2πnp cos N
+ sin
2πnp N
N −1
2πnp = xn cas N n=0
where the basis function cas(t) = cos(t) + sin(t). The inverse transform is calculated by
N −1 1 2πnp Hp cas xn = N p=0 N Cosine Transform The cosine transform or cosine transform of type 2 is determined by the following basis functions: ⎧ 1 ⎪ ⎪ if p = 0 ⎨ √ , 2N
φp (n) = 1 π(n + 1/2)p ⎪ ⎪ , if p = 0 ⎩ √ cos N N for the p = 0 case as X0c = √
N −1 1 xn 2N n=0
and for the p = 0 case as Xpc
N −1 1 π(n + 1/2)p =√ xn cos N N n=0
N −1 pπn πn pπn 1 πn √ xn cos = cos − sin sin 2N N 2N N N n=0
where p = 1 : (N − 1).
38
Paired Transform The one dimensional unitary discrete paired transform (DPT), also known as the Grigoryan transform is described in the following way. The transform describes a frequency-time representation of the signal by a set of short signals which are called the splitting-signals. Each such signal is generated by a frequency and carries the spectral information of the original signal in a certain set of frequencies. These sets are disjoint. Therefore, the paired transform transfers the signal into a space with frequency and time, or space which represents a source "bridge" between the time and frequency. Consider the most interesting case, when the length of signals is N = 2r , r > 1. Let p, t ∈ XN = {0, 1, . . . , N − 1}, and let χp,t(n) be the binary function
χp,t(n) =
⎧ ⎪ ⎨ 1, if
np = tmodN
⎪ ⎩ 0, otherwise
n = 0 : (N − 1).
Given a sample p ∈ XN and integer t ∈ [0, N/2], the function χp,t(n) = χp,t(n) − χp,t+n/2(n) is called the 2-paired, or shortly the paired function. The complete set of these functions is defined for frequency points p = 2k , k = 0, . . . , r − 1 and p = 0, and time points 2k t. The binary paired functions can also be written as the following transformation of the consine function: χ2k ,2k t(n) = M(cos(2π(n − t)/2r−k )),
(χ0,0(n) ≡ 1),
where t = 0 : (2r−k−1 − 1). M(x) is the real function which is not zero only on the bounds of the interval [−1, 1] and takes values M(−1) = −1 and M(1) = 1. The paired functions are determined by the extremal values of the consine functions, when they run through the interval with different frequencies. 39
The totality of the N paired functions {χ2k ,2k t ; n = 0 : (r − 1), t = 0 : (2r−n−1 − 1, 1} is the complete and orthogonal set of functions [132, 134]. Haar Transform The Haar transform is the first orthogonal transform found after the Fourier transform, which is now widely used in wavelets theory and in applications in image processing, in the N = 2r , r > 1 the transform is defined without normalization by the following matrix: ⎡
⎤
⎢ 1 1 ⎥ [HA2] = ⎣ ⎦ 1 −1 ⎤
⎡
⎢ [HA2] [HA2 ] ⎥ [HA4] = ⎣ √ ⎦, √ 2I2 − 2I2 where I2 is the unit matrix 2 × 2, and for k > 2 ⎤
⎡
⎢ [HA2k ] [HA2k ] ⎥ [HA2k+1 ] = ⎣ √ ⎦. √ 2k I2k − 2k I2k Heap Transform The discrete Heap transform is a new concept which was introduced by Artyom Grigoryan in 2006 [135]. The basis functions of the transformation represent certain waves which are propagated in the “field" which is associated with the signal generator. The composition of the N-point discrete heap transform, T, is based on the special selection of a set of parameters ϕ1 , ..., ϕm, or angles from the signal generator and given rules, where m ≥ N − 1. The transformation T is considered
40
separable, which means there exist such transformations Tϕ1 , Tϕ2 , ..., Tϕm that T = Tϕ1,...,ϕm = Tϕi(m) . . . Tϕi(2) Tϕi(1) where i(k) is a permutation of numbers k = 1, 2, ..., m. Consider the case when each transformation Tϕk changes only two components of the vector z = (z1, ..., zN −1). These two components may be chosen arbitrarily and such a selection is defined by a path of the transform. Thus, Tϕk is represented as Tϕk : z → (z1, ..., zk1−1 , fk1 (z, ϕk ), zk1 +1 , ..., zk2−1 , fk2 (z, ϕk ), zk2 +1 , ..., zm).
(2.1)
Here the pair of numbers (k1 , k2 ) is uniquely defined by k, and 1 ≤ k1 < k2 ≤ m. For simplicity of calculations, we assume that all first functions fk1 (z, ϕ) in (2.1) are equal to a function f(z, ϕ), as well as all functions fk2 (z, ϕ) equal to a function g(z, ϕ). The n-dimensional transformation T = Tϕ1 ,...,ϕm is composed by the transformations Tk1 ,k2 (ϕk ) : (zk1 , zk2 ) → (f(zk1 , zk2 , ϕk ), g(zk1 , zk2 , ϕk )). The selection of parameters ϕk , k = 1 : m, is based on specified signal generators x, the number of which is defined through the given decision equations, to achieve a uniqueness of parameters and desired properties of the transformation T. Consider the case of two decision equations with one signal-generator. Let f(x, y, ϕ) and g(x, y, ϕ) be functions of three variables; ϕ is referred to as the rotation parameter such as the angle, and x and y as the coordinates of a point (x, y) on the plane. It is assumed that, for a specified set of numbers a, the equation g(x, y, ϕ) = a has a unique solution with respect to ϕ, for each point (x, y) on the plane or its chosen subset.
41
The system of equations
⎧ ⎪ ⎨ f(x, y, ϕ) = y0 ⎪ ⎩ g(x, y, ϕ) = a
is called the system of decision equations [135]. First the value of ϕ is calculated from the second equation which we call the angular equation. Then, the value of y0 is calculated from the given input (x, y) as y0 = f(x, y, ϕ). It is also assumed that the two-point transformation Tϕ : (z0, z1 ) → (z0 , z1 ) = (f(z0 , z1, ϕ), g(z0, z1, ϕ)), which is derived from the given decision equations by Tϕ : (x, y) → (f(x, y, ϕ), a), is unitary. We call Tϕ the basic transformation. Example 1: Consider the following functions that describe the elementary rotation: f(x, y, ϕ) = x cos ϕ − y sin ϕ, g(x, y, ϕ) = x sin ϕ + y cos ϕ. Given a real number, the basic transformation is defined as the rotation of the point (x, y) to the horizontal Y = a, Tϕ : (x, y) → (x cos ϕ − y sin ϕ, a). The rotation angle ϕ is calculated by ϕ = arccos
a x2 + y 2
+ arctan
The first pair to be processed is (x0 , x1), (1)
(x0, x1 ) → (x0 , a),
42
y x
.
the next is (y0 , x2), (1)
(2)
(x0 , x2) → (x0 , a), (2)
with the new value of x0 = x0 , and so on. The first component of the signal is renewed and participates in calculation of all (N − 1) basic transformations Tk = Tϕk , k = 1 : (N − 1). (k)
Therefore, at the stage k, the first component of the transform is y0 = x0 . The complete transform of the signal-generator x is (N −1)
T (x) = (y0, a1, a2, . . . , aN −1),
(y0 = x0
).
The signal-flow graph of processing the five-point generator x is shown in Figure 2.16. y0
x0
y0
y0
y0
T1 T2
a1
T3
x1
a2 x2
a3 x3
Tk=T(φk), k=1:4 φ =r(y ,x ,a ) k
0
T4
a4
k k
x4
Figure 2.16: Signal-flow graph of determination of the five-point transformation by a vector x = (x0, x1 , x2, x3, x4 ). This transform is applied the the input signal zn in the same order, or path P , as the generator x. In the first stage the first two components are processed (1)
(1)
Tϕ1 : (z0, z1) → (z0 , z1 ), next (1)
(2)
(1)
Tϕ2 : (z0 , z2) → (z0 , z2 ),
43
z(1)
z
0
T
T
φ
Level 2
z
1
z(1) 2
z2
φ
Level 1
x(1) 0
φ ,T 1
(N−1) 0
z T
φ
N−1
φ ,T
1
2
x1
(1)
zN−1
zN−1
φ2
1
0
... ...
2
z(1) 1
(N−2) 0
z
φ
1
x
(2) 0
z
0
φ
N−1
(N−2) x 0
x(2) 0
... ...
2
x2
φ
y ,T
N−1
0
N−1
xN−1
Figure 2.17: Network of the x-induced DsiHT of the signal z. and so on. The result of the transform is (n−1)
T [z] = (z0
(1)
(1)
(1)
, z1 , z2 , . . . , zN −1 ),
a = 0.
Now consider the case when all parameters ak = 0, i.e., when the whole energy of the vector x is collected in one heap, and then transfered to the first component. In other words, we consider the Givens rotations of vectors, or points (y0 , xk ) on the horizontal Y = 0. Figure 2.16 shows the transform-network of the transform of the signal z = (z0 , z1, z2, ..., zN −1) . The parameters (angles) of the transformation are generated by the signal-generator x. In the 1st level and the kth (k−1)
stage of the flow-graph, the angle ϕk is calculated by inputs (x0
, xk ), where k ∈ {1, N − 1}
(0)
and x0 = x0. This angle is used in the basic transform Tk = Tϕk to define the next component (k)
x0 , as well as to perform the transform of the input signal z, in the 2nd level. The full graph itself represents a co-ordinated network of transformation of the vector z, under the action on x. 2.4.2
Enhancement methods
The common algorithm for image enhancement via a 2-D invertible transform consists of: The frequency ordered system-based method can be represented as . x → X = T(x) → O · X → T−1 [O(X)] = x 44
Algorithm 2.1 Transform based image enhancement 1. Perform the 2-D unitary transform 2. Multiply the transform coefficients, X(p, s) by some factor, O(p, s) 3. Perform the 2-D inverse unitary transform
O is an operator which could be applied on the coefficients X(p, s) of the transform or its real and imaginary parts ap,s and bp,s if the transform is complex. For instance, they could be X(p, s), aαp,s , bαp,s , or logα ap,s, logα bp,s . The cases of greatest interest are when O(X)p,s is an operator of magnitude and when O(x)p,s is performed separately on the coefficients. Let X(p, s) be the transform coefficients and let the enhancement operator O be of the form X(p, s) · C(p, s), where the latter is a real function of the magnitude of the coefficients, i.e., C(p, s) = f(|X|)(p, s). C(p, s) must be real since only modification of the magnitude and not phase information is desired. The following possibilities are a subset of methods for modifying the magnitude coefficients within this framework. 1. C1 (p, s) = C(p, s)γ |X(p, s)|α−1 , 0 ≤ α < 1 (which is the so-called modified α-rooting); 2. C2 (p, s) = logβ |X(p, s)|λ + 1 , 0 ≤ β, 0 < λ; 3. C3 (p, s) = C1 (p, s) · C2 (p, s). α, λ, and β are the parameters of the enhancement which are selected by the user to achieve the desired enhancement. Denoting by θ(p, s) ≥ 0 the phase of the transform coefficient X(p, s), the transform coefficient can be expressed as X(p, s) = |X(p, s)|ejθ(p,s) where |X(p, s)| is the magnitude of the coefficients. The investigation of the operator O applied to the modules of the transform coefficients instead of directly to the transform coefficients X(p, s)
45
will be performed as O(X)(p, s) = O(|X|)(p, s)|e[jθ(p,s)] . The assumption that the enhancement operator O(|X|) takes one of the forms Ci (p, s)|X(p, s)|, i = 1, 2, 3 at every frequency point (p, s) is made. Figure 2.18 shows Galaxy NGC 4242 in the time domain (pixel intensity values) and frequency domain (spectral coefficients).
(a) intensity image
(b) spectral coefficients
Figure 2.18: Intensity values and spectral coefficients of Galaxy NGC 4242. Figure 2.19 shows Butterworth lowpass filtering for Galaxy UGC 7617 for n = 2 and D0 = 120. The transfer function of the filter of order n with cutoff frequency at a distance D0 from the origin is defined as X(p, s) =
1 . 1 + [D(p, s)/D0 ]2n
α-rooting Figure 2.20 shows the enhancement of Galaxy NGC 4242 by method C1(p, s) with α = 0.02. Heap transform Figure 2.21 shows the results of enhancing galaxy images PIA 14402 and NGC 5194 by the Heap transform.
46
(a) original image
(b) low pass filtering
Figure 2.19: Butterworth lowpass filtering performed in the Fourier (frequency) domain.
(b) enhancement by α = 0.02
(a) original image
Figure 2.20: α-rooting enhancement of Galaxy NGC 4242.
2.5 Image Preprocessing The steps taken to prepare the galaxy images for feature extraction are detailed in this section. The position, size, and orientation of the galaxy varies from image to image. Therefore, the preprocessing steps will produce a training set that is invariant to galaxy position, scale and orientation. Individual galaxies were cropped from the digitized photographic plates and processed manually by adjusting parameters at several stages in the pipeline. Automatic selection of these parameters if part of future work. Figure 2.5 shows the computational scheme for the classification pipeline.
47
Figure 2.21: Top: Galaxy PIA 14402, Bottom: NGC 5194, both processed by Heap transform. 2.5.1
Segmentation
Other than the object of interest, galaxy images contain stars, gast, dust, and artifacts induced during the imaging and scanning process. For a galaxy to be recognized, such contents not included in the galaxy need to be removed. In general, this process involves denoising and inpainting. Here, the background is subtracted via a single threshold or Otsu’s method. Otsu’s method is calculated in Matlab by the command graythresh. Otsu’s method automatically selects a good threshold for images where there are few stars and the galaxy intensity varies greatly from the background. As the quantity and size of stars increase in the image, or when the background is close in intensity to the galaxy, Otsu’s method is not performing well. After background subtraction by thresholding, stars and other artifacts are removed by the morphological opening operation by different values of pixel connectivity using the Matlab function bwareaopen. A grayscale image relates to a function f(x, y) that takes values from a finite interval [0, M]. In the discrete case, M is considered to be a positive integer. Consider an image with only one
48
Galaxy Images
Segmentation: Thresholding Morphological Opening
Feature Invariance: Rotation, Centering, Resizing
Canny Edge Detection
Feature Extraction: Elongation Form Factor Convexity Bounding-rectangle-to-fill-factor Bounding-rectangle-to-perimeter Asymmetry Index
Support Vector Machine
Galaxy Classes Figure 2.22: Computational scheme for galaxy classification.
49
object f(x, y) =
⎧ ⎪ ⎨ 1 (x, y) ∈ O ⊂ X ⎪ ⎩ 0
otherwise
where O is the set of pixels in the object, and X is the whole domain of the image. The function f(x, y) represents a binary image. Any number can be used instead of 1, e.g., 255. Thresholding is defined as the following procedure
g(x, y) = gT (x, y) =
⎧ ⎪ ⎨ 1 f(x, y) ≥ T ⎪ ⎩ 0
otherwise
where T is a positive number from the interval [0, M]. This number is called a threshold. Otsu’s method begins by representing a grayscale image by L gray levels. ni represents the number of pixels at level i, and the total number of pixels N = n1 + n2 + . . . + nL . The image histogram is then described by a probability distribution ni pi = 1. , pi ≥ 0, N i=1 L
pi =
The intensity values are then separated into two classes C0 and C1 by a threshold k, where C0 represents the intensities [0, . . . , k] and C1 , [k + 1, . . . , L]. The occurrence, mean levels for each class are respectively given by
w0 = Pr(C0 ) =
k
pi = w(k)
i=1
w1 = Pr(C1) =
L
pi = 1 − w(k)
i=k+1
and μ0 =
k
iPr(i|C0) =
i=1
k ipi i=1
50
w0
=
μ(k) w(k)
L
μ1 =
iPr(i|C1) =
i=k+1
L ipi μT − μ(k) = w1 1 − w(k)
i=k+1
where w(k) and μ(k) are the zeroth- and first-order moments up the the kth level, respectively, and
μT = μ(L) =
L
ipi
i=1
is the total mean level of the original image. The following relationships are easily verified for any k w0μ0 + w1μ1 = μT ,
w0 + w1 = 1.
(2.2)
The class variances are given by σ02
=
k
2
(i − μ0 ) Pr(i|C0) =
i=1
σ12
k (i − μ0 )2 pi i=1
L
w0
L (i − μ1 )2 pi = (i − μ1 ) Pr(i|C1) = . w 1 i=k+1 i=k+1 2
The following criteria to measure k as an effective threshold are introduced from discriminant analysis λ=
σB2 , 2 σW
κ=
σT2 , 2 σW
η=
σB2 , σT2
where 2 σW = w0 σ02 + w1σ12
σB2 = w0(μ0 − μT )2 + w1(μ1 − μT )2 and from equation 2.2 σT2 =
L
(i − μT )2 pi
i=1
are the within-class variance, the between-class variance, and the total variance of levels, respectively. 51
Through relationships between the criteria, the problem becomes finding the k that maximizes the criterion η or equivalently σB2 by η(k) =
σB2 σT2
or σB2 (k) =
[μT w(k) − μ(k)]2 , w(k)[1 − w(k)]
and, as shown in [136], the optimal threshold k∗, restricted to the range S∗ = {k; 0 < w(k) < 1} is σB2 (k∗) = max σB2 (k). 1≤k
