The Abel Inversion of Noisy Data Using Discrete Integral Transforms

71 Pages·2016·0.97 MB·English

by  Subramania I. Sudharsanan

UUnniivveerrssiittyy ooff TTeennnneesssseeee,, KKnnooxxvviillllee
TTRRAACCEE:: TTeennnneesssseeee RReesseeaarrcchh aanndd CCrreeaattiivvee
EExxcchhaannggee
Masters Theses Graduate School
8-1986
TThhee AAbbeell IInnvveerrssiioonn ooff NNooiissyy DDaattaa UUssiinngg DDiissccrreettee IInntteeggrraall
TTrraannssffoorrmmss
Subramania I. Sudharsanan
University of Tennessee - Knoxville
Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes
Part of the Electrical and Computer Engineering Commons
RReeccoommmmeennddeedd CCiittaattiioonn
Sudharsanan, Subramania I., "The Abel Inversion of Noisy Data Using Discrete Integral Transforms. "
Master's Thesis, University of Tennessee, 1986.
https://trace.tennessee.edu/utk_gradthes/2922
This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and
Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE:
Tennessee Research and Creative Exchange. For more information, please contact [email protected].
To the Graduate Council:
I am submitting herewith a thesis written by Subramania I. Sudharsanan entitled "The Abel
Inversion of Noisy Data Using Discrete Integral Transforms." I have examined the final electronic
copy of this thesis for form and content and recommend that it be accepted in partial fulfillment
of the requirements for the degree of Master of Science, with a major in Electrical Engineering.
Bruce W. Bomar, Major Professor
We have read this thesis and recommend its acceptance:
Roy Joseph, Dennis Keefer
Accepted for the Council:
Carolyn R. Hodges
Vice Provost and Dean of the Graduate School
(Original signatures are on file with official student records.)
Tot hGer aduCaotuen cil:
Ia ms ubmihtetrienawtg ih tewhsr iistb tySe unb raImSa.dun hiraas aennatni tled
"ThAeb eiln vseiorofNn o iDsatyaU siDnsigc rIenttee Tgrarnaslf oIrh masv.e"
examitnhfieend a l ocfto htpihys e fsofirros m a ncdo ntenrte caonmtdm heantd
ibte a cceipnpt aerdft uifillallm oeftn htre e qeumiernftotsrh dee gorfMe aes ter
ofS ciewnictaemh ,a jionEr l ectErnigecieianrnlg .
/? .!'.,-}
£
L/,//./1J/L ..._, ... .k. .�lJ. .-r'c..: -..-..- ----
BruWc.eB omaMraj,o Pror efssor
Weh avree atdht ihse sis
anrde comimteasncd c eptance:
,'/
//,, / 1 <II / ;
/t(, f.?. t £/t,/'
r'
1
I
I . I
tt�fi·L
'
Accefpottreh cdeo uncil:
VicPer ovost
anDde aonTf heG raduSactheo ol
STATEMENT OF PERMISSION TO USE
Inp resetnhttiihsn egis npi rsat iflaufill lomfet nhrtee quirfeomare nts
Mastre'dse graetTe h eU niveorfTs eninteys Ksneoex,v Iia lglrete,hte at he
Librsahlramlya kieta vialatbobl oer rowerrsu loeuftsn h dLeei arbr ryB.r ief
quotaftorimto hntishs e asriaesl lowwaibtlhseoep uctip aelri msspiorovni,d ed
thtaa ccuarcaktneo wloeftd hgsemoe unirtmsc a ed e.
Permifsoserix otne nqsuiovteafo trmio orrn e prodouftc httiihsoe mnsa iys
beg ranbtyme ydm ajporro feosrs hoiiarnsb, s eeb,ny tc hHee adI notfe rlibrary
Servwihceeinsnt, h oep inoifeo int her, tuhseoe ft phmreao tpeoirsfsieo adrl
schopluarrployAs necyso .p yoirun sogef t hmea teirnti hatilhs e fsofiirns a ncial
gasianhl nlob tea lwleowdi thmoywu rti tpteernmo ins.s i
Signat_u� r_�e-'-·· --=
_ _ - """-= ==� -------
Date. 0_._/_;"8o_.g..+;;./-"9_.-''8"l,_,.b.__
____ ___ _ ___ _
THE ABEL INVERSION OF NOISY DATA USING
DISCRETE INTEGRAL TRANSFORMS
AT hesis
Presefnottrhe ed
MastoefSr c ience
Degree
ThUen iveorfTs einnteys sKneoex,v ille
SubramSanuidrahs aaIn.a n
Augu1s9t8 6
ACKN OWLEDGMEN TS
ThAeu thwoorul lidtk oee x prheiassps p rectioDa rtB.ir ouBnco em awrh,o
servetdhc eh airomfta hcneo mmiatntDder eR. o y Jwohssoee prhv ead
as as
memebrf,o r htehlepaifdruv alin cgdeu indcae.
Spelct aihnakasrd eu teoD rD.e nnKieeser af nMdr .M ontgoSmmeirtyh
fotrh eciorn siesntcreoanugte maennidtn ,v alsuuapbplaoenr gdtu indcaien
condutchtiiinsnv ge staingpdar teipoatnrh miean ngu scTrhietp htew.so iusl d
nobter ealwitizheotduh teh ierl p.
Thanakresex tetnoMdr eR.di chWaelrlfdeo n ru merhoeulisppn se o rmrfing
thdea taan alysis.
ii
ABSTRAC T
Thdee termoiafnr aatdliilsoyayn m mettwrdomii-ecn saiflou nncftorimio tns
ondemi enisolnp arojeickstn ioownn asi nrAvsbeieTolhn ifi.sn dasp pliciant ions
sevefirealoldfe s n ginaenesdrc iineignn ccel asutdrionyng,io mmapgreo cessing,
plasmdaig anosatniodcp st iOcnrsea. dl si laoiftc hefe nu ctiwohnic,co hm pletely
spectihfiteew so -diiomlnfe anuncstiiostn hi,en veHrasneeklt a rnsfooftr hme
FourTiraenrs fooftr hmpe r ojecWtiittohhn pe.r ojedcattaiavo ani alsaa b le
dirsecstieg ltn,ha Aeb eilne vrsciaobnne p erfoursmietndhgd e i scFroeutrei er
traonrsamfn tdh ien verset rHaoanrnsmTkf.he eelx istteicnuhgent sioc q ompute
inveHrasnekt erlao nrsmafsr dei scuasnsadem do difictaoot nioeof tn h ose
methiosdi sn troadnusdch eodwt noy ierledas onaacbcluyrr eastuewl ittsh
sgininfitcflaweye cra lculTahteei ffiocnisfeas.tnF to urtireaornr simfsu seidn
theevl auatoifbo ontt hht er aonrsmfs.
Expiemrelnltoyab tained dpta,raw ohjieiccsuht s iluolannyo iasnyod ff ­
centiedsre ,aw litti hnt hFeo urdioemra ina f uerusqienndgco ym afiilnta enrd
am aximluimk eleishtoiomddae troiforvr met dhas es umed chnaocirtisaesert ics.
Thmea ximlueimilk hoeosidtm atiuossr e tdoe stitmhaFeto eu rtiarenrsm f oofr
thper eorpcleyn tdetaraeR.de suolfnt usmi eclrea xperiamreen tgtsiov veyer nif
thmee thTohdep. r opmoestehdio csd o mputlalmtyoi roeenffi acitehnattnh e
exisctuirfinvtget -mientgh odpseo rmrffiontrgh A eb eiln vseiorofnn o idsay.t a
lll
TABLE OF CONTENTS
CHPATER PAGE
I.I NTORDUTCION. . .. 1
. . . .
IIN.U MERAILCI MPLEENMTATION
5
TheorBeatciokcuganrld
5
Numerical .P rocedure 8
IIII.N VERSOIFON NO ISDYAT A 20
IV. NUMERAILCE XPERIMENTS 31
ExperiwmietNnhot iss e-DFatrae e 33
ExperiwmietNnhot isDs atya 40
V.S UMMARAYN D CONCLUSI.O NS 49
BILBIOGARPHY. 52
APPENIXDES. 56
.
A.DF TS HIFTT HEOREFMO RN ON-INTESGHIEFRT S 57
B.IT ERATPIRVOECD EURET OF IND STHHFEIT 59
VITA. . . . . . . 61
. . . . . . . . . . . . . . . . . .
lV
LIST OF FIGURES
FIGURE PAGE
1.1.T hcer oss-asneidcn ttieopnnrs oiofitfaly ne a xisympmleastmrai2. c
2.1.G eomerterpirce seonftt hnaeet aironenes itg ihnbtoertr ipoonl1.a5
3.1.T yplii cnatepnrsoimfitelyae s uforrmea dna rgpolsnam a. 21
3.2.B utterfiwlotmreagtrnh i truedsep (ofinfstoehr dfec=r 3,2) . 24
3.3.F IBRl ackwmiannd ofiwletmdea rg nirteusdpe(op nasses band
edgfeerq ue=n 0c.y12 fi5,l ter = l43e).n gth 24
3.4.F IoRp tifimlatmlea rg nirteusdpeo nse. 26
4.1.D iscrreectgteua lanaprp roxiomfaac ntotiionnfu uocnutsi on.32
4.2.I nppurto JfiIli(e) . 34
4.3.I nverosffi( 1oin) . 35
4.4.I nppurto ffi(2lie) . 37
4.5.I nverosffi (2oin) . 38
4.6.G enernaotidesadyt l a!ni (). 42
4.7 . Inverosffiln io w)(ni thfiolutte orrsi ynmgm etgr.i zin 42
4.8.I nverosfli n{o1iun )s ing 1.F ilter 43
4.9.I nverosfli n(oliun )s iFnigl 2t. er 43
4.10.I nverosfIi n(oliun )s iFnigl 3t. er 44
4.11.I nverosffi ln(o in) .. . . . 45
4.12.G enernaotiedsadyt l an( 2i). 47
v
4.1I3n. verosfJi ino)2un s (iF nigl 1t.e. r . . . . . 7 . . . . . . . . 4
4.1I4.n verosfii 2nio(u)ns iFnigl2 t.e r . . . .4 8 . . . .
.
4.15I.n verosfii 2n(o iun)s iFnigl 3t.e r 48
5.1C.o ntpoluoorftt h ien teinmsaigtey. . . . .5 1 . . . .
5.2C.o ntpoluoorftt heem iscsoieoffinc iimeangt.e . . . .5 1 . . . .
VI