1. 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
5. 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
7. 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
8. 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 Digi-
tal 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 clas-
sification. 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 mis-
classification 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 fea-
ture 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 en-
hanced data sets for a performance comparison between the original and reduced features spaces.
Classification was performed by the Support Vector Machine learning algorithm.
v
15. 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 excla-
mation, 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 solu-
tion 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 dis-
tinguish 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 ro-
tating 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
16. 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/as-
tro1020/ch20/ch26_fig26_9.jpg.
of galaxy morphology assigns members to the categories of elliptical and spiral, the most promi-
nent 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
17. 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
1. Elliptical....................................................En
(n=1,2,...,7 indicates the ellipticity of the image)
3379
221
4621
2117
E0
E2
E5
E7
2. Spirals:
a) Normal spirals............................................S
(1) Early..........................................................Sa
(2) Intermediate..............................................Sb
(3) Late...........................................................Sc
b) Barred spirals.............................................SB
(1) Early..........................................................SBa
(2) Intermediate..............................................SBb
(3) Late...........................................................SBc
N.G.C.
4594
2841
5457
N.G.C.
2859
3351
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 lenticu-
lar 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 cri-
terion, 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
18. 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 origi-
nally 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
19. 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
20. Figure 1.3: Plate scan of Spiral and Barred Spiral Nebulae from Mount Wilson Observatory orig-
inally included in Hubble’s paper, Extra-galactic Nebulae.
6
21. 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
22. 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
23. 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 Fig-
ure 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 lentic-
ulars 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 no-
tation 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
24. 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 va-
10
25. rieties across the classification volume and between stages along each sequence; each point in the
classification volume represents potentially a possible combination of morphological characteris-
tics. 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 sys-
tem.
description.
11
26. 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 be-
tween 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) de-
tected 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 Obser-
vatory (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
27. 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
28. 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 con-
tained 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 mul-
tiple 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 perfor-
14
29. 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 limita-
tions 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 en-
hancement. 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 pa-
rameters 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 Ko-
honen 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 clas-
sification 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 artifi-
cial 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
30. de-noised and compressed by principal component analysis. A supervised artificial neural net-
work 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 lu-
minosity 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 com-
parison 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 robust-
ness 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 clas-
sification 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 corre-
lates 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 morphol-
ogy 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 con-
16
31. 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 sur-
veys 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 calcu-
lated 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 in-
cluding 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 mul-
tiple 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
32. 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 recog-
nition [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 trans-
form 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 trans-
form 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 transforma-
tions, negative and identity transformations, nth-power and nth-root transformations, histogram
matching and local histogram processing. Adaptive histogram equalization, which uses local con-
trast 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
33. matching for registration of infrared astronomical images. Astronomical images were logarithmi-
cally 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 en-
hancement 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.
19
34. 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).
20
35. 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 As-
trofísica, 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
36. 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 process-
ing 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 Quad-
core 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 Infini-
Band 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 to-
gether 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 de-
scribed 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 en-
hancement 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 k1k2 blocks of size L1 × L2,
22
37. 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} → { ˆfn,m}, is defined by
EMEa( ˆf) =
1
k1k2
k1
k=1
k2
l=1
20 log10
maxk,l( ˆf)
mink,l( ˆf)
,
where maxk,l( ˆf) and mink,l( ˆf) respectively are the maximum and minimum of the image ˆfn,m
inside the (k, l)th block, and a is a parameter, or a vector parameter of the enhancement algorithm.
23
38. Figure 2.4: Marked plate scan AC8431
EMEa( ˆf) is called a measure of enhancement, or measure of improvement of the image f. We
define a parameter a0 such that EME( ˆf) = EMEa0 ( ˆf) to be the best (or optimal) Φ-transform-
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 ˆf = 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
39. 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
40. 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
41. 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
42. 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.
0 50 100 150 200 250
0
50
100
150
200
250
identity
negative
46*log(1+r)
16*sqrt(1+r)
40*(1+r)(
1/3)
0.004*r2
c*r3
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) = c0log(1 + f(x, y))
28
43. 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
44. 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
45. 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.
0 50 100 150 200 250
0
50
100
150
200
250
original
0.05
0.85
1.65
2.45
3.25
4.05
4.85
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
46. (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
47. random variable
ξ → ξ = w(ξ) (w : r → s)
such that w is a monotonically increasing function
h(s) = fbξ(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
w(rmax) − w(rmin)
ds =
r
rmin
h(a)da
which yields s = w(r)
w(r) − w(rmin)
w(rmax) − w(rmin)
=
r
rmin
h(a)da = F(r).
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 =
⎧
⎪⎪⎨
⎪⎪⎩
M
r
k=1
h(k) if r = 1, 2, . . . , M − 1
0 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
48. So, independent of the image intensity probability density function, the intensity density func-
tion of the processed image is uniform,
fbξ(s) =
1
w(rmax) − w(rmin)
.
Histogram equalization applied to galaxy NGC 6070 is shown in Figure 2.13 with the correspond-
ing 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
49. 0 50 100 150 200 250 300
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 50 100 150 200 250 300
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
Figure 2.14: Top to Bottom: Histogram of original and enhanced image.
35
50. 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
k=1
|xk − m|.
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.
x
xx
x
x
mx
x
5
2
1
4
3
6
7
(a) median in the line
x x x x m x x x1 2 3 4 5 6 7 x8
(b) median in space
Figure 2.15: Illustration of the median of a set of points in different dimensions.
36
51. 2.4 Transform-based image enhancement
In parallel to directly processing image pixels in the spatial domain by contrast enhancement meth-
ods, 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 com-
plexity 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
Fp =
N−1
n=0
xncos
2πnp
N
− jxnsin
2πnp
N
xn =
1
N
N−1
p=0
Fpcos
2πnp
N
+ jFpsin
2πnp
N
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
52. 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
n=0
xn cos
2πnp
N
+ sin
2πnp
N
=
N−1
n=0
xncas
2πnp
N
where the basis function cas(t) = cos(t) + sin(t). The inverse transform is calculated by
xn =
1
N
N−1
p=0
Hpcas
2πnp
N
Cosine Transform
The cosine transform or cosine transform of type 2 is determined by the following basis functions:
φp(n) =
⎧
⎪⎪⎨
⎪⎪⎩
1
√
2N
, if p = 0
1
√
N
cos
π(n + 1/2)p
N
, if p = 0
for the p = 0 case as
Xc
0 =
1
√
2N
N−1
n=0
xn
and for the p = 0 case as
Xc
p =
1
√
N
N−1
n=0
xncos
π(n + 1/2)p
N
=
1
√
N
N−1
n=0
xn cos
πn
2N
cos
pπn
N
− sin
πn
2N
sin
pπn
N
where p = 1 : (N − 1).
38
53. Paired Transform
The one dimensional unitary discrete paired transform (DPT), also known as the Grigoryan trans-
form 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,2kt(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
54. The totality of the N paired functions
{χ2k,2kt; 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:
[HA2] =
⎡
⎢
⎣
1 1
1 −1
⎤
⎥
⎦
[HA4] =
⎡
⎢
⎣
[HA2] [HA2]
√
2I2 −
√
2I2
⎤
⎥
⎦ ,
where I2 is the unit matrix 2 × 2, and for k > 2
[HA2k+1 ] =
⎡
⎢
⎣
[HA2k] [HA2k]
√
2kI2k −
√
2kI2k
⎤
⎥
⎦ .
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
55. 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 vec-
tor 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 num-
ber 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
56. 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 + y2
+ arctan
y
x
.
The first pair to be processed is (x0, x1),
(x0, x1) → (x
(1)
0 , a),
42
57. the next is (y0, x2),
(x
(1)
0 , x2) → (x
(2)
0 , a),
with the new value of x0 = x
(2)
0 , 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).
Therefore, at the stage k, the first component of the transform is y0 = x
(k)
0 .
The complete transform of the signal-generator x is
T(x) = (y0, a1, a2, . . . , aN−1), (y0 = x
(N−1)
0 ).
The signal-flow graph of processing the five-point generator x is shown in Figure 2.16.
T
1
T
2
T
3
T
4
x
1
x
0
x
3
x
2
x
4
a
4
a
3
a
1
a
2
y
0
y
0
y
0
y
0
T
k
=T(φ
k
), k=1:4
φ
k
=r(y
0
,x
k
,a
k
)
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
Tϕ1 : (z0, z1) → (z
(1)
0 , z
(1)
1 ),
next
Tϕ2 : (z
(1)
0 , z2) → (z
(2)
0 , z
(1)
2 ),
43
58. φ
1
φ2 φ
N−1
z
0
(1)
z
0
(2)z
0
z
1
(1)
z
0
(N−1)
φ
1
,T
1
φ
2
,T
2
φ
N−1
,T
N−1
T
φ
1
T
φ
2
T
φ
N−1
z
1
z2 zN−1
x1
z
2
(1)
zN−1
(1)
z
0
(N−2)
x2
xN−1
x
0 x
0
(1)
x
0
(2)
x
0
(N−2)
y
0
...
...
...
...Level 1
Level 2
Figure 2.17: Network of the x-induced DsiHT of the signal z.
and so on. The result of the transform is
T[z] = (z
(n−1)
0 , z
(1)
1 , z
(1)
2 , . . ., z
(1)
N−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
stage of the flow-graph, the angle ϕk is calculated by inputs (x
(k−1)
0 , xk), where k ∈ {1, N − 1}
and x
(0)
0 = x0. This angle is used in the basic transform Tk = Tϕk
to define the next component
x
(k)
0 , 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
59. 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),
aα
p,s, bα
p,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
60. 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
61. (a) original image (b) low pass filtering
Figure 2.19: Butterworth lowpass filtering performed in the Fourier (frequency) domain.
(a) original image (b) enhancement by α = 0.02
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 prepro-
cessing 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
62. 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
63. 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
64. 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
pi =
ni
N
, pi ≥ 0,
L
i=1
pi = 1.
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
i=1
pi = w(k)
w1 = Pr(C1) =
L
i=k+1
pi = 1 − w(k)
and
μ0 =
k
i=1
iPr(i|C0) =
k
i=1
ipi
w0
=
μ(k)
w(k)
50
65. μ1 =
L
i=k+1
iPr(i|C1) =
L
i=k+1
ipi
w1
=
μT − μ(k)
1 − w(k)
where w(k) and μ(k) are the zeroth- and first-order moments up the the kth level, respectively, and
μT = μ(L) =
L
i=1
ipi
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
σ2
0 =
k
i=1
(i − μ0)2
Pr(i|C0) =
k
i=1
(i − μ0)2
pi
w0
σ2
1 =
L
i=k+1
(i − μ1)2
Pr(i|C1) =
L
i=k+1
(i − μ1)2
pi
w1
.
The following criteria to measure k as an effective threshold are introduced from discriminant
analysis
λ =
σ2
B
σ2
W
, κ =
σ2
T
σ2
W
, η =
σ2
B
σ2
T
,
where
σ2
W = w0σ2
0 + w1σ2
1
σ2
B = w0(μ0 − μT )2
+ w1(μ1 − μT )2
and from equation 2.2
σ2
T =
L
i=1
(i − μT )2
pi
are the within-class variance, the between-class variance, and the total variance of levels, respec-
tively.
51
66. Through relationships between the criteria, the problem becomes finding the k that maximizes
the criterion η or equivalently σ2
B by
η(k) =
σ2
B
σ2
T
or
σ2
B(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
σ2
B(k∗) = max
1≤k<L
σ2
B(k).
Figure 2.23 shows original images with subtracted backgrounds by different manual thresholds
and Otsu’s method.
(a) Original image (b) T = 60
(c) T = 74 (d) Otsu’s T = 85
Figure 2.23: Background subtraction of Galaxy NGC 4274 by manual and Otsu’s thresholding.
The average difference between single thresholds and thresholds by Otsu’s method for the
enhanced data set was 6.67 with a standard deviation of 11.21.
Mathematical morphology provides image processing with powerful nonlinear filters which
52
67. operate according to the Minkowski’s addition and subtraction. Given subsets X and B of Rn
,
Minkowski’s addition, X ⊕ B, of sets X and B is the set
X ⊕ B =
b∈B
{Xb = {x + b; x ∈ X}}.
For the set ˇB = {−b; b ∈ B} symmetric to B with respect to the origin, the set X ⊕ ˇB is called a
dilation of the set X by B. The set B is said to be a structuring element.
So, in the symmetric case, if ˇB = B, Minkowski’s addition of sets X and B and the dilation
of X by B are the same concepts.
The dual operation to Minkowski’s addition of sets X and B is the subtractions, X B, which
is defined as
X B = (Xc
⊕ B)c
=
b∈B
{Xb = {x + b; x ∈ X}}.
The set X ˇB dual to the dilation X ⊕ ˇB is called an erosion of the set X by B. By means of
dilation and erosion of sets, the corresponding operations of opening, X ◦ ˇB, and closing, X • ˇB,
can be defined as
X ◦ ˇB = (X ˇB) ⊕ ˇB = {x + ˇB; x + ˇB ⊂ X}
X • ˇB = (Xc
◦ ˇB)c
= (X ⊕ ˇB) ˇB.
Herewith, the operation of opening of X by B is dual to the operation of closing of X by B, i.e.,
X ◦ ˇB = (Xc
• ˇB)c
. Figure 2.24 shows star and artifact removal of Galaxy NGC 5813 with pixel
connectivity P = 64.
2.5.2 Rotation, Shifting and Resizing
To achieve invariance to orientation, position, and scale, the galaxies were shifted by their geomet-
rical center, rotated by the angle between their first principal component and the image x-axis, and
resized to a uniform size of 128x128 pixels, respectively.
53
68. (a) original image (b) thresholded image
(c) opened image
Figure 2.24: Morphological opening for star removal from Galaxy NGC 5813.
54
69. The geometrical center, or centroid, of an object in an image is the center of mass of the object.
The center is the point where one can concentrate the whole mass of the object without changing
the first moment relative to any axis. The first moment with respect to the x axis is defined by
μx
X
f(x, y)dxdy =
X
xf(x, y)dxdy.
The first moment with respect to the y axis is defined by
μy
X
f(x, y)dxdy =
X
yf(x, y)dxdy.
The coordinate of the object center is then (μx, μy).
In the discrete case, the first moment with respect to the axis x is defined by
μx
n m
fn,m =
n m
nfn,m =
n
n
m
fn,m
and with respect to the y axis
μy
n m
fn,m =
n m
mfn,m =
n
m
m
fn,m
where the summation is performed over all pixels (n, m) of the object O.
The center of the object is defined as
(μx, μy) =
⎛
⎜
⎜
⎝
n m
nfn,m
n m
fn,m
, n m
mfn,m
n m
fn,m
⎞
⎟
⎟
⎠ .
55
70. In the discrete binary case, the center is defined as
(μx, μy) =
⎛
⎜
⎜
⎜
⎝
(n,m)∈O
n
(n,m)∈O
1
,
(n,m)∈O
m
(n,m)∈O
1
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
(n,m)∈O
n
card(O)
,
(n,m)∈O
m
card(O)
⎞
⎟
⎟
⎟
⎠
where card(O) is the cardinality of the set O that defines the binary image.
To find the orientation of an object in an image, if possible or if such exists and is unique,
consider a line along which the second moment is minimum. In other words, consider the integral
E = μ2(l) =
l
r2
f(x, y)dxdy (2.3)
where r is the distance of point (x, y) from the line l, i.e., the length of the perpendicular emitted
from point (x, y) to the line l. The line l is described by the equation
l : xsinθ − ycosθ + p = 0
where p is the length of the perpendicular drawn from the origin (0, 0) to the line l. Therefore, 2.3
can be rewritten as
E = E(θ) =
l
(xsinθ − ycosθ + p)2
f(x, y)dxdy. (2.4)
The following two denotations are made to for the image coordinates shifted by the geometrical
center of the object
x = x − μx, y = y − μy,
and the second moments of the shifted object are denoted
a =
l
(x )2
f(x, y)dx dy , c =
l
(y )2
f(x, y)dx dy , b =
l
(x )2
(y )2
f(x, y)dx dy .
56
71. E(θ) can then be rewritten as
E(θ) = asin2
(θ) − bsin(θ)cos(θ) + ccos2
(θ)
or E(θ) =
1
2
(a + c) −
1
2
(a − c)cos(2θ) −
1
2
bsin(2θ).
Differentiating E by θ gives
E(θ) = 0 → tan(2θ) =
b
a − c
(a = c = b).
Therefore, the angle of the orientation line l(θ) is found by
sin(2θ) = ±
b
b2 + (a − c)2
, cos(2θ) = ±
a − c
b2 + (a − c)2
.
The angle of the orientation line l(θ) was calculated for each galaxy image, and the used to rotate
the image by the Matlab function imrotate. Figure 2.25 shows this rotation for galaxy image
NGC 4096 by angle −64 degrees. Note that the image x-axis of the image in Matlab is vertical, and
the desired orientation of the galaxy’s first principal component being collinear with the horizontal
axis of the image is achieved by rotating the galaxy an additional 90 degrees.
(a) segmented galaxy (b) rotated galaxy
Figure 2.25: Rotation of Galaxy image NGC 4096 by galaxy second moment defined angle.
57
72. Resizing an image involves either subsampling if the desired image size is less than the original
image size and resampling if the desired image size is greater than the original image. Subsampling
reduces the size of an image by creating a new image with pixel value a calculated from the values
of a neighborhood of pixels about a in the original image. Resampling from the image size of
128 × 128 into 256 × 256 is calculated by
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
· · · ·
· a b ·
· c d ·
· · · ·
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
→
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
· · · · · ·
· a a b b ·
· a a b b ·
· c c d d ·
· c c d d ·
· · · · · ·
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
Another process of subsampling is defined by calculation of means, as follows below for the
2 × 2 subsampling example where
a =
a1 + a2 + b1 + b2
4
, b =
a3 + a4 + b3 + b4
4
c =
c1 + c2 + d1 + d2
4
, d =
c3 + c4 + d3 + d4
4
.
Image resizing is peformed in Matlab by the function imresize. Figure 2.26 shows an example
of image resizing from size 138 × 197 into size 128 × 128 for galaxy image NGC 4220.
2.5.3 Canny Edge Detection
The Canny edge detection method was developed by John Canny in 1986. The Canny edge detector
was developed to satisfy the performance criteria: (1) Good detection (2) Good localization (3)
Only one response to a single edge. Good detection means reducing false positives (non edges
being detected as edges) and false negatives (edges not being detected). Good localization means
that minimal error exists between identified edge points and true edge points. Only one response
58
73. (a) cropped image size 138 × 197 (b) image resized to 128 × 128
Figure 2.26: Resizing of Galaxy NGC 4220.
to a single edge ensures that the operator eliminates the multiple maxima output from the filter
at step edges. Canny formulated each of these three criterion mathematically and found solutions
through numerical optimization. The result is that impulse response of the first derivative of a
Gaussian approximately the optimal edge detector which optimizes the signal-to-noise ratio and
localization, i.e., the first two criteria. The edge detected algorithm is presented below here. Let
f(x, y) denote the input image and G(x, y) denote the Gaussian function
G(x, y) = e
−
x2
+ y2
2σ2
.
The convolution of these two functions results in a smoothing of the input image and is written as
s(x, y) = f(x, y) ∗ G(x, y),
where σ controls the degree of smoothing of the image.
First order finite difference approximations are used to compute the gradient of s(x, y) which
59
74. is written as [sx, sy] where
sx =
δs
δx
, sy =
δs
δy
.
The gradient magnitude and orientation or angle are respectively computed by
M(x, y) = sx
2 + sy
2
and
α(x, y) = tan−1 sy
sx
.
The array of image magnitudes will contain large values in the directions of greatest change. The
array is then thinned so that only the magnitudes at the points of greatest local change remain.
This procedure is called non maxima suppression. An example presents this notion. Consider a
3 × 3 grid where 4 possible orientations are found through the center point in the grid: horizontal,
vertical, +45 degrees, and −45 degrees. All possible orientations have been discretized into these
4 orientations. A range of orientations is then specified to quantize the orientations. Edge direction
is determined by the edge normal, computed by 2.5.3.
Let dk, k = 1, 2, . . . , n represent the discrete orientations where n is the number of orientations.
Using the 3 × 3 grid, every nonmaxima suppression scheme at every point (x, y) in α(x, y) can be
formulated as where st(x, y) is the nonmaxima suppressed image.
Algorithm 2.2 Nonmaxima suppression algorithm
1. Find the orientation dk which is closest to α(x, y)
2. Set st(x, y) = 0ifM(x, y) is less than at least one of its two neighbors along dk, otherwise,
st(x, y) = M(x, y).
Finally, a hysteresis thresholding is applied to st(x, y) to reduce falsely detected edges. Two
thresholds are used here and are referred to as a weak (or low) threshold τ1 and a strong (or high)
threshold τ2. Too low of a threshold will retain false positives. Too high of a threshold will remove
60
75. correctly detected edges. The double threshold produces two new images written as
stw(x, y) = st(x, y) ≥ τ1
where stw(x, y) denotes the image created due to the weak threshold and
sts(x, y) = st(x, y) ≥ τ2
where sts(x, y) denotes the image created due to the strong threshold. Edges in sts(x, y) are
linked into contours by searching through an 8 pixel neighborhood in stw(x, y) for edges that can
be linked to the end of the current edge. The output of the algorithm is the image of all nonzero
points in stw(x, y) appended to sts(x, y). Canny edge detection was performed using the Matlab
function edge with τ1 = 0.3, τ2 = 0.9 and σ = 1.5. Figure 2.27 shows the Canny edge detector
for multiple galaxy images.
2.6 Data Mining and Classification
The canonical problem addressed in the field of data mining and classification is the following:
Given a very large family of vectors (signals, images, etc.) each of which lives in a high dimen-
sional space, how can the set be effectively represented this data for storage and retrieval, for
recognizing patterns within the images, and for classifying objects. In the subsequent sections, a
small subset of the tools used in statistics, data mining, and machine learning in astronomy will be
investigated to address the posed problem of the representation and classification of galaxy images.
2.6.1 Feature Extraction
A useful galaxy feature descriptor varies in value so that a classifier can discriminates between
input galaxies and place each galaxy into one of several classes. The shape, or morphologi-
cal, features used in this paper are described in [26, 31, 137] and are Elongation (E), Form Fac-
61
76. (a) NGC 6070 original (b) NGC 6070 canny edge
(c) NGC 4460 original (d) NGC 4460 canny edge
(e) NGC 4283 original (f) NGC 4283 canny edge
Figure 2.27: Canny edge detection.
62
77. tor (F), Convexity (C), Bounding-rectangle-to-fill-factor (BFF), Bounding-rectangle-to-perimeter
(BP), and Asymmetry Index (AI). Table ?? gives the average values of the original data for these
features.
Elongation has higher values for spiral and lenticular galaxies and lower values for irregular
and elliptical galaxies. This feature can be written as
E =
(a − b)
(a + b)
where a is the major axis and b is the minor axis.
Form factor is useful in dividing spiral galaxies from other classes. This feature can be written
as
F =
A
P2
where A is the number of pixels in the galaxy and P is the number of pixels in the galaxy edge
found by canny edge detection.
Convexity has larger for spirals with open winding arms and lower values for compact galaxies
such as are in the class elliptical. This feature can be written as
C =
P
(2H + 2W)
where P is as defined above and H and W are the height and width of minimum bounding rectangle
for the galaxy.
Bounding-rectangle-to-fill-factor is... This feature is defined as
BFF =
A
HW
where A, H, and W are as defined above.
Bounding-rectangle-to-perimeter shows a decreasing trend from compact and circular galaxies
63
78. Table 2.1: Morphological Feature Descriptions
Feature Formula
E (a−b)/(a+b) Has higher values for s
F A/P2 Form factor is useful in dividing spiral gala
C P/(2H+2W) Convexity has larger for spirals with open winding arms and lower v
BFF A/HW
BP HW/(2H+2W)2 Bounding-rectangle-to-perim
AI
P
i,j |I(i,j)−I180(i,j)|/P
i,j |I(i,j)| The asymmetry index tends towa
Table 2.2: Feature Values Per Class
Feature Elliptical Lenticular Simple Spiral Barred Spiral Irregular
E 0.071 0.382 0.547 0.485 0.214
F 0.059 0.049 0.025 0.029 0.044
C 0.888 0.872 1.05 1.01 0.953
BFF 0.744 0.699 0.609 0.583 0.634
BP 0.062 0.052 0.043 0.048 0.059
AI 0.274 0.375 0.510 0.464 0.354
to open and edge-on galaxies. This feature can be written as
BP =
HW
(2H + 2W)2
where H and W are as defined above.
The asymmetry index tends towards zero when the image is invariant under a 180 degree rota-
tion. This feature can be written as
AI =
i,j
|I(i, j) − I180(i, j)|
i,j
|I(i, j)|
where I is the original image and I180 is the image rotated by 180 degrees.
2.6.2 Principal Component Analysis
Data may be highly correlated, but represented such that its axes are not aligned with the directions
in which the data varies the most. A data set generated by N observations with K measurements
64
79. per observation lives in a K-Dimensional space, each dimension, or axis, representing a feature of
the data. To represent the data in a more compact form, the axes can be rotated to be collinear with
the directions of maximum variance in the data, thereby discriminating between the data points. In
other words, this rotation results in the first feature being collinear with the direction of maximum
variance, the second feature being orthogonal to the first and maximizing the residual variance,
and so on. This dimensionality reduction technique is called Principal Component Analysis (PCA),
also known as the Karhunen-Loéve transform or Hotelling transform, and is depicted in Figure 2.28
for a bivariate Gaussian distribution. Consider the data set xi with N observations and K features
Figure 2.28: PCA rotation of axes for a bivariate Gaussian distribution.
written as the N × K matrix X. The covariance matrix of zero mean data is estimated as
CX =
1
N − 1
XT
X
65
80. where N is the dimension of the matrix and division by N−1 is necessary for CX to be an un-biased
estimate of the covariance matrix. Nonzero components in the off diagonal entries represent corre-
lation between the features, whereas zero components represent uncorrelated data. PCA transform
the original data into equivalent uncorrelated data so that the covariance matrix of the new data is
diagonal and the diagonal entries decrease from top to bottom. To achieve this, PCA attempts to
find a nonsingular matrix R which transforms X into such an ideal matrix. The data transforms to
Y = XR and its covariance estimate to
CY = RT
XT
XR = RT
CXR
The first column r1 of R is the first principal component, and is along the direction of the data with
maximum variance. The columns of R which are called principal components form an orthonormal
basis of the data space. The first principal component r1 can therefore be derived using Lagrangian
multipliers and setting equal to zero the cost function φ(r1, λ) as
φ(r1, λ) = rT
1 CXr1 − λ1(rT
1 r1 − 1).
Setting
δφ(ri, λ)
δri
set
= 0 then gives
CXr1 − λ1r1 = 0 or CXr1 = λ1r1.
This shows that λ1 is an eigenvalue of the covariance matrix CX, i.e., a root of (CX − λ1I) = 0.
λ1 = rT
1 CXr1 being the largest eigenvalue in CX equates to maximizing the variance along the
first principal component. The remaining principal components are derived in the same manner.
CX The matrix CY is the transformation of CX in the basis consisting of the columns of R,
the eigenvectors of CX. This comes to have the new basis, i.e., the columns of R have a basis, of
eigenvectors of CX. Since CX is symmetric by definition, the Spectral Theorem guarantees that
the eigenvectors of CX are orthogonal. These eigenvectors can be listed in any order and CY will
66
81. remain diagonal. However, the requirement of PCA is to list them such that the diagonal entries of
CY be in decreasing order of their values, which comes to a unique order of the eigenvectors which
make the columns of R. The order of the components (or dimensions) is the so named rank-order
according to variance. With CX = RCY RT
and these eigenvectors in this order, the set of principal
components is defined.
The morphological feature data described in 2.6.1 was reduced in dimension from 6 to 2 by
keeping the first two principal components for both the comparison of classification performance
with compressed data and visualization. All classification figures in the following sections were
generated from the classification of PCA features.
2.6.3 Support Vector Machines
The Support Vector Machine (SVM) learning algorithm captures the structure of a multi-class
training data set towards predicting class membership of unknown data with correctness and high
decision confidence. Classes are divided by a decision boundary or hyperplane defined by with the
minimum distance between the boundary and nearest point in each class defining the margins of
the boundary, which the SVM optimizes. Points that lie on the margin are called support vectors.
Consider a linear classifier for a binary classification problem with labels y, y ∈ {−1, 1}, and
features x. The classifier is written as
hw,b(x) = g(wT
x + b),
and
g(z) =
⎧
⎪⎨
⎪⎩
1 if z ≥ 0
−1 otherwise
67
82. where w is the weight vector, and b is the bias of the hyperplane. Given a training example
(x(i)
, y(i)
), the functional margin of (w, b) is defined with respect to the training example as
γ(i)
= y(i)
(wT
x(i)
+ b).
If y(i)
= 1, then wT
(x(i)
+ b) need to be a large positive number for a large functional margin, and,
conversely, if y(i)
= −1, then wT
(x(i)
+ b) needs to be a large negative number. A large functional
margin represents a confident and correct prediction.
With the chosen g, if w and b are scaled by 2, the function margin is scaled by a factor of 2.
However, since g(wT
x + b) = g(2wT
x + 2b), no change would occur in hw,b(x). This shows that
hw,b(x) depends only on the sign, and not the magnitude, of g(wT
x + b).
Given a training set S = {(x(i)
, y(i)
); i = 1, 2, . . . , m}, the functional margin of (w, b) with
respect to S is defined as the smallest functional margin of the individual training examples and is
written as
γ = min
i=1,...,m
γ(i)
.
Another type of margin is the geometric margin. Consider the training set in Figure 2.6.3.
The hyperplane defined by (w, b) is shown, along with vector w, which is normal to the hyper-
plane. Point A represents positive training example x(i)
with label y(i)
= 1. The geometric margin
of point A, γ(i)
, has distance of line segment AB. Point B is defined by x(i)
− γ(i)
w/||w||. Since
point B is on the decision boundary, which satisfies the equation wT
x + b = 0, then
wT
x(i)
− γ(i) w
||w||
+ b = 0.
Solving for γ(i)
yields
γ(i)
=
wT
x(i)
+ b
||w||
=
w
||w||
T
x(i)
+
b
||w||
.
In general, the geometric margin of (w, b) with respect to any training example (x(i)
, y(i)
) is given
68
83. 6
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
-
e
e
e
e
e
e
e
e
e
w
u
uA
B γ(i)
u
u
u
u
u
u
Figure 2.29: Pictorial representation of the development of the geometric margin.
by
γ(i)
= y(i) w
||w||
T
x(i)
+
b
||w||
.
Note that if ||w|| = 1, then the geometric margin equals the functional margin. Additionally, the
geometric margin is invariant to scaling the parameters w and b.
Given a training set S = {(x(i)
, y(i)
); i = 1, 2, . . . , m}, the geometric margin of (w, b) with
respect to S is defined as the smallest geometric margin of the individual training examples and is
written as
γ = min
i=1,...,m
γ(i)
.
Assuming the training data is linearly separable, the problem of determining the boundary decision
that maximizes the geometric margin is posed as the follow optimization problem
max
γ,w,b
γ subject to y(i)
(wT
x(i)
+ b) ≥ γ, i = 1, 2, . . . , m and ||w|| = 1.
The ||w|| = 1 constraint is non-convex. To work towards recasting the optimization problem as
convex, first recall that γ = γ/||w||. With this relation, the problem can then be written as an
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