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A Fuzzy Logic Based Algorithm for Finding Astronomical Objects in Wide-Angle Frames

a r X i v :a s t r o -p h /0502548v 1 25 F e

b 2005

A Fuzzy Logic Based Algorithm For Finding Astronomical Objects in Wide-Angle Frames

Lior Shamir A ,B ,Robert J.Nemiro?A

A

Michigan Technological University,Department of Physics 1400Townsend Drive,Houghton,MI 49931B

Email:lshamir@https://www.doczj.com/doc/d811021374.html,

Abstract:Accurate automatic identi?cation of astronomical objects in an imperfect world of non-linear wide-angle optics,imperfect optics,inaccurately pointed telescopes,and defect-ridden cameras is not always a trivial ?rst step.In the past few years,this problem has been exacerbated by the rise of digital imaging,providing vast digital streams of astronomical images and data.In the modern age of increasing bandwidth,human identi?cations are many times impracticably slow.In order to perform an automatic computer-based analysis of astronomical frames,a quick and accurate identi?cation of astronomical objects is required.Such identi?cation must follow a rigorous transformation from topocentric celestial coordinates into image coordinates on a CCD frame.This paper presents a fuzzy logic based algorithm that estimates needed coordinate transformations in a practical https://www.doczj.com/doc/d811021374.html,ing a training set of reference stars,the algorithm statically builds a fuzzy logic model.At runtime,the algorithm uses this model to associate stellar objects visible in the frames to known-catalogued objects,and generates ?les that contain photometry information of objects visible in the https://www.doczj.com/doc/d811021374.html,e of this algorithm facilitates real-time monitoring of stars and bright transients,allowing identi?cations and alerts to be issued more reliably.The algorithm is being implemented by the Night Sky Live all-sky monitoring global network and has shown itself signi?cantly more reliable than the previously used non-fuzzy logic algorithm.

Keywords:data analysis -methods:statistical -techniques:astronomical image processing

1Introduction

Useful pipeline processing of astronomical images de-pends on accurate algorithmic decision making.For previously identi?ed objects,one of the ?rst steps in computer-based analysis of astronomical pictures is an association of each object with a known catalog entry.This necessary step enables such science as automat-ically detected transients and automated photometry of stars.Since computing the topocentric coordinates of a given known star at a given time is a simple and common task,transforming the celestial topocentric coordinates to image (x,y )coordinates might provide the expected location of any star in the frame.How-ever,in many cases slight shifts in the orientation,in-accuracy of the optics or imperfections in the CCD can make this seemingly simple task formidable.

Fuzzy logic (Zadeh 1965,1978,1983,1994)is an extension of Boolean logic that is useful for making complex decisions by computers.While in Boolean logic an item has only two levels of memberships to a set (false or true;0or 1),fuzzy logic supports any value within the interval [0,1]as a level of membership of an item to a fuzzy set.A fuzzy logic model consists of three di?erent types of entities:fuzzy sets,fuzzy variables and fuzzy rules.The membership of a fuzzy variable in a fuzzy set is determined by a function that produces values within the interval [0,1].These func-tions are called membership functions .Fuzzy variables are divided into two groups:antecedent variables that contain the input data of the fuzzy logic model,and

consequent variables that contain the results calculated by the fuzzy logic model.

In this paper,we present a fuzzy logic based algo-rithm that transforms celestial coordinates into image coordinates for even complex combinations of wide-angle non-linear optical distortions,slight optical im-perfections,and small unrecorded orientational per-turbations.In Section 2we describe the fuzzy logic algorithm,and in Section 3we present and discuss a practical implementation of the algorithm for auto-matic analysis of an operational astronomical project called “Night Sky Live”(Nemiro?et al.2005).

2

Fuzzy Logic Based Coordi-nate Transformations

The transformation of celestial topocentric spherical sky coordinates to local Cartesian image coordinates can be de?ned by a set of two functions.Mathematical conversion from right ascension,declination,latitude,longitude,and local time to altitude and azimuth is straightforward.Further transformation of azimuth and altitude sky coordinates to (x,y )image (CCD)coordinates is frequently harder in practice when faced with practical inaccuracies of knowledge.For practical application of fuzzy logic,this later transformation is broken up into two parts:

(altitude,azimuth )?→x (1)(altitude,azimuth )?→y

(2)

1

2Publications of the Astronomical Society of Australia

On a CCD image,pixel locations can be speci?ed in either Cartesian or polar coordinates.Let x z en be the x coordinate(in pixels)of the zenith in the image, and y z en be the y coordinate of the zenith.x and y coordinates of any given star visible in an astronomical image can be computed as follows:

x=x z en+sin(angle)·distance(3)

y=y z en+cos(angle)·distance(4) Where angle is a polar azimuthal angle and distance is a polar radial distance.

In order to use these equations it is necessary to compute a polar distance and angle for given objects. Given the observer’s latitude and longitude,this can be done by converting the celestial coordinates(azimuth and altitude)of a given stellar object at a given time to the corresponding angle and distance in the image. Since the azimuth and altitude of any given bright star or planet at any given time can be easily computed, the only missing link here is the transformation of the altitude and azimuth to image coordinates,so the ob-ject can be found in the image.

2.1Reference Stars

Each of the two models is built based on manually identi?ed reference stars.A reference star can be any star within an image that was correctly associated with the corresponding stellar object.Being familiar with the night sky,we can inspect the frame by eye and identify the stellar objects that appear in it.The im-age(x,y)coordinates of the object can be taken from the peak of its point spread function(PSF).Each ref-erence star provides a record with the following?elds: azimuth,altitude,angle,and distance.

Each identi?ed star contributes an azimuth and al-titude(by basic astronomy)and also an angle and dis-tance(by measurement from the image).These pro-vide the raw data for constructing a mapping between the two,using the fuzzy logic model that will be de-scribed later in the paper.In order to obtain an accu-rate fuzzy logic model that calculates the angle,it is necessary to select reference stars that uniformly cover the entire image.This assures that the calculation of one value will depend on reference points that are rel-atively close to it.This is also true for the model that calculates the distance.

2.2Building the Fuzzy Logic Model In order to transform celestial coordinates into image coordinates,two di?erent fuzzy logic models are being built based upon the two transformations:

f1:azimuth?→angle(5) f2:altitude,azimuth?→distance(6) Here,altitude,azimuth and angle are angular mea-sures,while distance is measured in pixels.Each trans-formation(f1and f2)is computed by a di?erent fuzzy logic model,thus one model calculates the angle and the other calculates the distance.The fuzzy logic model f1,which calculates the angle,is an approximation based on the assumption that the optical axis is per-fectly aligned,so the angle is not depended on the altitude.

The fuzzy logic model f1has one antecedent(in-put)fuzzy variable and one consequent(output)fuzzy variable,while the fuzzy logic model f2has two an-tecedent variables(altitude,azimuth)and one conse-quent fuzzy variable.

2.3Converting Azimuth to Polar An-

gle on the CCD(f1)

The?rst model(f1)is built according to the refer-ence stars such that each reference star adds to the model one fuzzy set and one fuzzy rule.Each fuzzy set is associated with a membership functions that is built in the form of a triangle(Zadeh1965).Each of these membership functions reaches its maximum at the reference value,and intersects with the x-axis at the reference values of its neighboring reference stars. For instance,suppose we would like to build the fuzzy logic model with a data set that contains the following four reference stars:

azimuth altitude angle distance

0?0θ0R0

α1?1θ1R1

α2?2θ2R2

α3?3θ3R3

The?rst reference star maps azimuth0o.Assum-ingα1<α2<α3,the membership functions that will be added to the model are described in Figure 1. Figure1:The membership functions of the fuzzy sets(F S0to F S4)created by the four ref-erence values(0,θ0),(α1,θ1),(α2,θ2),(α3,θ3)for f1(azimuth?→angle).

The membership of the fuzzy sets is determined by the following membership functions:

www.publish.csiro.au/journals/pasa 3

F 0(x )=

1?

1

α1

·x 0≤x <α1

1?

1

α2?α1

·(x ?α1)

α1≤x <α2

1?

1

α3?α2

·(x ?α2)

α2≤x <α3

1?

1

360?α3

·(x ?α3)

α3≤x ≤360

x <α3or x >360

F 0to F 4are the membership functions of the fuzzy sets F S 0to F S 4that were created according to the ref-erence stars (the function F m is the membership func-tion of the fuzzy set F S m ).The membership functions are built such that almost all azimuth values belong (with non-zero membership)to two fuzzy sets.Only the points of maximum (0,α1,α2,α3,360)have a non-zero membership to just one set.

The defuzzi?cation method used in this model is weighted average ,which is an e?cient defuzzi?cation method when the fuzzy logic model is built according to a set of signleton values (T.Takagi &M.Sugeno 1983,1985).Since weighted average is used,the consequent part of each rule is a crisp value and not a fuzzy set.There-fore,the fuzzy rules that will be added to the model are:

F S 0?→θ0F S 1?→θ1F S 2?→θ2F S 3?→θ3F S 4?→360

For instance,suppose that the ?rst three reference stars have azimuths of 0o ,10o ,and 20o ,and their po-lar angles are 3o ,12o ,and 22o respectively,such that α0=0o ,θ0=3o ,α1=10o ,θ1=12o ,α2=20o ,and θ2=22o .Each reference star adds one fuzzy set to the model such that the membership functions of the ?rst two fuzzy sets F S 0,F S 1are:F 0(x )=

1?

1

10

·x 0≤x <10

1?1

10

·6=0.4to the fuzzy

set F S 0and

1

0.4+0.6

=8.4.This computation practically provides the same results as a simple linear interpolation of the value 6with the two closest neighboring reference stars.

2.4

Converting Altitude and Azimuth to Radial Distance on the CCD (f 2)

Unlike the simpler f 1model used for transforming the azimuth to angle,the computation of the distance (in pixels)from (x z en ,y z en )should be computed based on two parameters,which are the altitude and the https://www.doczj.com/doc/d811021374.html,ing both the altitude and azimuth allows the model to deal with asymmetric behavior of the optics as well as inaccurate orientational information.In other words,using the assumption that the optics orientation is directly at the zenith and the distortion of the optics and hardware is completely symmetric,a reference point at a certain azimuth would allow calcu-lating the distance of a point at the same altitude but at a di?erent azimuth.However,this is not always the case.For instance,a stellar object with the azimuth of 0o (north)and altitude of 30o can be at distance of 150pixels from (x z en ,y z en ),while another stellar object at the same altitude (30o )but at azimuth of 60o will be at distance of 155pixels from (x z en ,y z en ).Moreover,(x z en ,y z en )does not necessarily appear in the center of the frame,and the frame is not necessarily central-ized.Therefore,when computing the distance of a stellar object from (x z en ,y z en ),it is required to take not only the altitude of the stellar object into consid-erations,but also the azimuth.In order to do that,the fuzzy logic model that calculates the distance is built according to reference stars that not only have di?erent altitudes,but also di?erent azimuths.

We build the f 2model that computes the distance using four di?erent sets of reference stars such that each set contains reference stars that share approxi-mately the same azimuth.For the sake of simplicity,the ?rst set contains reference stars that are near az-imuth 0o ,the second set contains reference stars near azimuth 90o ,and the other two sets contain stars near azimuth of 180o and 270o respectively.I.e.,all refer-ence stars used for this model should be fairly close to the azimuth of 0o ,90o ,180o or 270o .In order to use the four sets of reference stars,four new fuzzy sets are added to the model.Those fuzzy sets are “North”,“East”,“South”,and “West”.

Fuzzy logic can be viewed as a complex interpola-tion method.In two dimensions,our fuzzy logic re-alization allows f 2to perform the combined interpo-lation of linear and Gaussian functions in a straight-

4

Publications of the Astronomical Society of Australia

forward fashion,and to alter the model easily when needed.Note that this is quite di?erent than four sepa-rate linear and/or Gaussian interpolations for the four directions.The

membership functions of the fuzzy sets are described in Figure 2.

Figure 2:The fuzzy sets of the four directions (North,South,East,West).

The membership in the fuzzy sets is determined by the following Gaussian functions:North:F n (x )= e

?x

22σ2180

East:

F e (x )=

e

?

(x ?90)2

2σ2

270

South:

F s (x )=e

?

(x ?180)2

2σ2

90

e

?

(x +90)2

star azimuth altitude angle distance

North).Therefore,the fuzzy set Alt68N that will be added by S 1will have a triangle membership function that reaches its maximum at 68,and intersects with the x-axis at 62and 72.This membership function is described in Figure 3.

Figure 3:The membership function of the fuzzy set Alt68N .The ?gure also include parts of the membership functions of Alt62N and Alt72N that were added by the two neighboring reference stars S 2and S 3.

The membership function added by S 1is:F Alt 68N (x )=

x ?62

72?68

68

0x <62or x >72

Since the azimuth of S 1is approximately north,the fuzzy rule added to the model by S 1is:Alt68N ∧North ?→215

This rule is more signi?cant for stars that appear in the northern part of the sky (and in this case,also at an altitude of around 68o ).The practical a?ect of this rule will be stronger as the coordinates are closer to the 68o parallel.The fuzzy rules that will be added by the other reference stars are:Alt62N ∧North ?→224Alt72N ∧North ?→206Alt62E ∧North ?→188Alt66E ∧North ?→180

www.publish.csiro.au/journals/pasa5

Alt70E∧North?→172

Now suppose that we want to use this model in

order to compute the distance(in pixels)of a stel-

lar object with an azimuth of30o and altitude of64o.

The value64have a membership value of64?62

3

to the fuzzy set Alt68N(that was added to the model

by S1),and1?64?62

3

to the fuzzy set Alt62N

(that was added to the model by S2).Also,this value

would have a membership value of64?62

66?62=0.5to the fuzzy set added by S5.The

membership value of the azimuth30o to the fuzzy set North is e?30

2

2·452?0.412.The membership values to the fuzzy sets South and West are very close to0in this case,and therefore have very little e?ect on the output value.For the sake of the simplicity of the example, these membership values are assumed to be exactly0.

The computation process is based on product in-ferencing and weighted average defuzzi?cation.There-fore,the output value of the computation would be: 215·0.8·0.333+224·0.8·0.667+180·0.412·0.5+188·0.412·0.5

stars(image,date

PSFs←GetPSFs(image)3.for each image cords in image

distance←∞

6.for each star in catalog do

7.star cords←CelestialCoordinates(star, date

celestial

cords←F(star cords)

10.if distance(image PSF

distance then

11.min cords, image cords)

12.end if

13.end for

14.if min

PSF

cords and image cords are considered as referring to the same star.

In Figure

4taken by the Mauna Kea CONCAM, the names of bright stars and constellations were an-notated automatically using the above transformation. The coordinates of one bright point(appears at the lower right of the frame)could not be associated with any catalogued object so it was automatically marked with a yellow square.This object is believed to be a meteor.

The present fuzzy-logic algorithms allow practi-cally100percent chance of accurate identi?cation for NSL stars down to a magnitude of5.6.We are cur-rently unaware of any exceptions.A previously used NSL identi?cation algorithm that employed a straight-forward analytic transformation was only accurate to about magnitude3.5,although that was somewhat de-pendent on the NSL station.This dramatic improve-ment was the driving impetus for the creation of this paper.

A useful by-product of the newly accurate iden-ti?cations is the automatic generation of photometry ?les.The ability to associate each PSF with a cat-aloged star allows the system to provide continuous monitoring of many bright stars.This information is provided in the form of text(XML tagged)?les.Each

6Publications of the Astronomical Society of Australia frame produces one text?le that lists all PSFs that

were detected in the frame and the name and cata-

logue number of the stellar object associated with it.

It also lists some additional data about each detected

object such as the previously cataloged visual magni-

tude,spectral type,and celestial coordinates.Identi?-

cation allows other algorithms to process the frame and

match each star with real time photometric data such

as estimated counts of the background and the counts

of the PSF.For each bright star,the average of the

brightest1,5,9,16and25pixels are listed.For dim-

mer stars,however,the realized PSF is much smaller

so only the average of the top?ve pixels is listed.The

on-line database allows browsing the records for past

events.Since the photometry data itself is out of the

scope of this paper,more information about it can be

found at(Nemiro?et al.2005).

The inverse computation of the presented transfor-

mation formula(converting angle and distance to al-

titude and azimuth)can be built in the same method

described in this paper,with the exception of using

the angle and distance for de?ning the membership

functions,while the azimuth and altitude are used as

the crisp output values of the fuzzy rules.This trans-

formation formula is currently being used by NSL for

computing3D trajectories of meteors detected by the

twin CONCAMs located at Mauna Kea and Haleakala.

3.1Computation Accuracy

In order to check accuracy,the combined f1and f2

models were used to compute the image coordinates of

some150stars recorded by the NSL project(Nemiro?et al.

2005).The NSL project uses?sheye images with an

extreme viewable angle of180o.For each pair of im-

age(x,y)coordinates,we calculated the Euclidean dis-

tance from the location calculated by the model and

the location where the star appeared in the image.

We took into considerations the average of the150

Euclidean distances(the average error)and the worst

case error,which is the longest distance among all com-

puted150Euclidean distances.Table2below shows

the accuracy levels according to the number of refer-

ence stars that were used in order to build both fuzzy

logic models(the altitude f2model and the azimuth

f1model).

Table2:Accuracy Level of the Transformation as

a Factor of the Number of Reference Stars

5040 3.2 4.8

4240 3.2 4.8

4230 3.5 5.5

3630 3.6 5.7

3220 4.4 6.2

3020 5.28.0

2020 6.510.2

www.publish.csiro.au/journals/pasa7 Takagi T.,Sugeno M.,1985,IEEE Trans.Syst.Man

&Cybern.,Vol.20,No.2,p.116-132

Zadeh,L.A.,1965,Information and Control,Vol.8,p.

338-353

Zadeh,L.A.,1978,International Journal of Man-

Machine Studies,Vol.10,p.395-460

Zadeh,L.A.,1983,Memorandum No.UCB/ERL

M83/41,University of California,Berkeley

Zadeh,L.A.,1994,IEEE Software11(6),p.48-56

8Publications of the Astronomical Society of Australia

Figure4:A Night Sky Live picture with labeled stars,constellations,a planet and a meteor(boxed) processed using the fuzzy logic based transformation formula.

英语选修六课文翻译Unit5 The power of nature An exciting job的课文原文和翻译

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英语选修六课文翻译第五单元word版本

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人教版英语选修六Unit5 the power of nature(An exciting Job)

高二英语教学设计 Book6 Unit 5 Reading An Exciting Job 1.教学目标(Teaching Goals): a. To know how to read some words and phrases. b. To grasp and remember the detailed information of the reading material . c. To understand the general idea of the passage. d. To develop some basic reading skills. 2.教学重难点: a.. To understand the general idea of the passage. b. To develop some basic reading skills. Step I Lead-in and Pre-reading Let’s share a movie T: What’s happened in the movie? S: A volcano was erupting. All of them felt frightened/surprised/astonished/scared…… T: What do you think of volcano eruption and what can we do about it? S: A volcano eruption can do great damage to human beings. It seems that we human beings are powerless in front of these natural forces. But it can be predicted and damage can be reduced. T: Who will do this kind of job and what do you think of the job? S: volcanologist. It’s dangerous. T: I think it’s exciting. Ok, this class, let’s learn An Exciting Job. At first, I want to show you the goals of this class Step ⅡPre-reading Let the students take out their papers and check them in groups, and then write their answers on the blackboard (Self-learning) some words and phrases:volcano, erupt, alongside, appoint, equipment, volcanologist, database, evaluate, excite, fantastic, fountain, absolutely, unfortunately, potential, be compared with..., protect...from..., be appointed as, burn to the ground, be about to do sth., make one’s way. Check their answers and then let them lead the reading. Step III Fast-reading 这是一篇记叙文,一位火山学家的自述。作者首先介绍了他的工作性质,说明他热爱该项工作的主要原因是能帮助人们免遭火山袭击。然后,作者介绍了和另外二位科学家一道来到火山口的经历。最后,作者表达了他对自己工作的热情。许多年后,火山对他的吸引力依然不减。 Skimming Ⅰ.Read the passage and answer: (Group4) 1. Does the writer like his job?( Yes.) 2. Where is Mount Kilauea? (It is in Hawaii) 3. What is the volcanologist wearing when getting close to the crater? (He is wearing white protective suits that covered his whole body, helmets, big boots and

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Unit5 Reading An Exciting Job 说课稿 Liu Baowei Part 1 My understanding of this lesson The analysis of the teaching material:This lesson is a reading passage. It plays a very important part in the English teaching of this unit. It tells us the writer’s exciting job as a volcanologist. From studying the passage, students can know the basic knowledge of volcano, and enjoy the occupation as a volcanologist. So here are my teaching goals: volcanologist 1. Ability goal: Enable the students to learn about the powerful natural force-volcano and the work as a volcanologist. 2. Learning ability goal: Help the students learn how to analyze the way the writer describes his exciting job. 3. Emotional goal: Make the Students love the nature and love their jobs. Learn how to express fear and anxiety Teaching important points: sentence structures 1. I was about to go back to sleep when suddenly my bedroom became as bright as day. 2. Having studied volcanoes now for more than twenty years, I am still amazed at their beauty as well as their potential to cause great damage. Teaching difficult points: 1. Use your own words to retell the text. 2. Discuss the natural disasters and their love to future jobs. Something about the S tudents: 1. The Students have known something about volcano but they don’t know the detailed information. 2. They are lack of vocabulary. 3. They don’t often use English to express themselves and communicate with others.

an exciting job 翻译

我的工作是世界上最伟大的工作。我跑的地方是稀罕奇特的地方,我见到的是世界各地有趣味的人们,有时在室外工作,有时在办公室里,有时工作中要用科学仪器,有时要会见当地百姓和旅游人士。但是我从不感到厌烦。虽然我的工作偶尔也有危险,但是我并不在乎,因为危险能激励我,使我感到有活力。然而,最重要的是,通过我的工作能保护人们免遭世界最大的自然威力之一,也就是火山的威胁。 我是一名火山学家,在夏威夷火山观测站(HVO)工作。我的主要任务是收集有关基拉韦厄火山的信息,这是夏威夷最活跃的火山之一。收集和评估了这些信息之后,我就帮助其他科学家一起预测下次火山熔岩将往何处流,流速是多少。我们的工作拯救了许多人的生命,因为熔岩要流经之地,老百姓都可以得到离开家园的通知。遗憾的是,我们不可能把他们的家搬离岩浆流过的地方,因此,许多房屋被熔岩淹没,或者焚烧殆尽。当滚烫沸腾的岩石从火山喷发出来并撞回地面时,它所造成的损失比想象的要小些,这是因为在岩石下落的基拉韦厄火山顶附近无人居住。而顺着山坡下流的火山熔岩造成的损失却大得多,这是因为火山岩浆所流经的地方,一切东西都被掩埋在熔岩下面了。然而火山喷发本身的确是很壮观的,我永远也忘不了我第一次看见火山喷发时的情景。那是在我到达夏威夷后的第二个星期。那天辛辛苦苦地干了一整天,我很早就上床睡觉。我在熟睡中突然感到床铺在摇晃,接着我听到一阵奇怪的声音,就好像一列火车从我的窗外行驶一样。因为我在夏威夷曾经经历过多次地震,所以对这种声音我并不在意。我刚要再睡,突然我的卧室亮如白昼。我赶紧跑出房间,来到后花园,在那儿我能远远地看见基拉韦厄火山。在山坡上,火山爆发了,红色发烫的岩浆像喷泉一样,朝天上喷射达几百米高。真是绝妙的奇景! 就在这次火山喷发的第二天,我有幸做了一次近距离的观察。我和另外两位科学被送到山顶,在离火山爆发期间形成的火山口最靠近的地方才下车。早先从观测站出发时,就带了一些特制的安全服,于是我们穿上安全服再走近火山口。我们三个人看上去就像宇航员一样,我们都穿着白色的防护服遮住全身,戴上了头盔和特别的手套,还穿了一双大靴子。穿着这些衣服走起路来实在不容易,但我们还是缓缓往火山口的边缘走去,并且向下看到了红红的沸腾的中心。另外,两人攀下火山口,去收集供日后研究用的岩浆,我是第一次经历这样的事,所以留在山顶上观察他们

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