The Algebraic Least Squares Fitting Of Ellipses

Transcription

1 IOSR Jourl of Mthets (IOSR-JM) e-issn: ISSN: Volue 4 Issue Ver II (Mr - Ar 8) PP wwwosrjourlsorg he Algebr Lest Squres Fttg Of Ellses Abdelltf Betteb Dertet of Geerl Studes Jubl Idustrl College Jubl Idustrl Ct Kgdo of Sud Arb Corresodg Author: Abdelltf Betteb Abstrt: Fttg ellses to set of gve ots the le s roble tht rses lto res eg outer grhs [9] [3] oordte etrolog [] etroleu egeerg [8] I ths er we reset severl lgorths whh the ellse for whh the su of the squres of the dstes to the gve ots s l hese lgorths re ored wth lssl d tertve ethods Ellses s rereseted lgebrll e b equto of the for F() = he lgorth outes otuous futo losel rotg the ellses for whh the su of the squres to the gve set of ots s zed We wll look rtulrl t oe ethod b gvg eles d usg Mtlb to solve these robles d the ores the effe of the Dte of Subsso: -4-8 Dte of ete: I Itroduto Let reltosh betwee vrbles d be gve b f( ;) = where reters For ele ths ould be ellse or o the le Let dt ots ( ) = be gve he dell we wsh to hoose so tht f ( ; ) R s vetor of However ths s ulkel to be ossble so we eed soe other ws of hoosg Dfferet ethods re vlble d re osdered the et setos Oe w of hoosg s lled "Algebr Fttg " uses the lt for of the ellse d tkes the for of ostred lest squre roble hs lgorth ws etesvel dsussed b Vrh [9] d Golub Gder d Strebel [3] I the other setos we trodues dfferet uerl dfferet uerl eles wth relevt fgures d results he we ore ll the ethods studed ths er usg dfferet eles II Algebr Fttg Gve the lt futo f ( ; ) d the dt ots ( ; ); = ; : : ; the le we wt to se the stdrd lest squre futol : ( ) f ( ; ) () Geerll ths results ole r sto roble but the ortt se tht f s ler we obt ler lest squres roble of the for ( ) S () suh tht the -th ooet of S s f ( ; ) where S s tr We ssue > d s vetor (reter)clerl = s trvl soluto herefore to olete the roble we ust dd soe orlzto odto o d here we hve four dfferet ossbltes hese gve rse to ostred otzto robles d therefore we eed to trodue soe lss of robles of ths kd hs s osdered et DOI: 979/ wwwosrjourlsorg 74 Pge

2 he Algebr Lest Squres Fttg of Ellses III he ostred otzto roble he struture of ost ostred otzto robles s essetll oted the followg: Mze f ( ) R (3) Subjet to ( ) E ( ) I where f() s objetve futo but there re ddtol ostrt futos () = k where k s the uber of des EUI d E s the de set of equtos or eqult ostrts the roble I s the set of eqult ostrts d both of these sets re fte More geerl ostrts be ut to ths for: for ele b ( ) If ot stsfes ll ostrts () b beoes (3) t s sd to be fesble ot d the set of ll suh ots s referred to s the fesble rego R (see [] ge 39-4) 3 SVD Costrt subjet to (3) where s the l or We hve the roble s S subjet to (33) he qutt S ssed t the se tht ses S d S ( S) ( S) S S ( S S) S s eser to stud Observe tht: Also S S s setr tr se (S S) = S S = S S So the roble ow s to ze the qudrt for (S S) subjet to the ostrt = Let A be setr tr d defe d M s: A: M A: he b heore 6 ge 4 fro [5] M s the lrgest egevlue of A d s the sllest egevlue of A he vlue of As M whe s ut egevetor v orresodg to M he vlue of A s whe s ut egevetor orresodg to hus the u vlue s the egevlue orresodg to the sllest egevetor v If S s tr the S S s setr s we sw before d be orthogoll dgolzed Let v v be orthogol bss for R osstg of egevetors of S S d let be the ssoted egevlues of S S he for S Se S ( S ) ( S ) S s egevetor of S S ( S S) (34) ( ) se s ut vetor So the egevlues of S S re ll oegtve B uberg f eessr we ssue tht the egevlues re rrged so tht DOI: 979/ wwwosrjourlsorg 75 Pge

3 he Algebr Lest Squres Fttg of Ellses he sgulr vlues of S re the squre roots of the egevlues of S S deotedb d the re rrged deresg order ht s for B the equto (34) the sgulr vlues of S re the legths of the vetors 43) For S the u vlue of S s We wrte S the for of sgulr vlue deoosto (SVD) of S s S U V Where S S (see [6] ge 48- D D r here s tr where the dgol etres D re the frst r sgulr vlues of S r d U s orthogol tr d orthogol tr V he we surse ths b sg the roble s equvlet to fdg the rght sgulr vetor V (:) ssoted wth the sllest sgulr vlue of S Rerk: he SVD ethod gves good ft whe he sgulr vetor V (:) orresodg to s ssued uque 3Qudrt ostrt S subjet to B (35) whereb s geerl k tr We fd the soluto usg the geerlsed SVD of S d B or b solvg geerlsed egevlue roble volvg S S d B B However we tke here the sel se whe B = [O I] d we show the followg dsusso how to fd the soluto For ste we tke the se of ellse wth reter b b ) wth ( If we defe vetors v b b ) w ( ) the ( v; w); d the oeffet tr S ( thebookste ostrt e the qudrt ostrt be wrtte s w d we hve the sste: v s w DOI: 979/ wwwosrjourlsorg 76 Pge

4 he Algebr Lest Squres Fttg of Ellses he QR deoosto of S leds to the equvlet sste R R v R3 w d t s equvlet to R v Rw R 3w Whh be solved the followg stes: R w w 3 Usg the sgulr vlue deoosto of R U V w s the sgulr vetor orresodg to the sllest sgulr vlue of R 3 3 Fll fd v fro: R v R w d ths s equvlet to R Rw R v R d the: v w hus v w 33 Ellse ostrt S subjet to C (38) Where C s 5 C 5 We redue thsroble to geerlsed egevlue roble: ( C S S) (39) For the ellse the soluto s the egevetor orresodg to the uque ostve egevlue the ellse se or the bggest bsolute vlue of the egevlue Assug S s osgulr ths geerlsed egevlue roble hs the se uber of ostve egevlues s C does el oe More geerll the tkes the vlue of the egevetor orresodg to the lrgest oegtve bsolute vlue of the revous roble s follows: If we l the frst odto of Lgrge ultler (see []) we get L( ) S S ( C ) DOI: 979/ wwwosrjourlsorg 77 Pge

5 he Algebr Lest Squres Fttg of Ellses Dfferette wrt to : Dfferette wrt to : S S C C he we get the roble s bove S S C ( S S C) S S C S S C C S he S S C S C C S S thus S S S S S d ust be ostve beuse S d to get otrvl soluto ust be strtl ostve Nuerll we foud lws orresods to the - olu of ( MALAB otto we ut D sted of ) d the soluto orresods to the - vetor v of the tr V (b MALAB) suh tht D eg( M ) V where M ( S S) C beuse ll the revous egevlues the se of ths tr C rezeros he we wt the sllest ostve vlue whh orresods to the lrgest bsolute vlue of of S S C Rerk: Usull for ll the ostrts gve the solutos obted re ver dfferet d led to ver dfferet fgures geerl I the se whe the dt re ver erl ftted b the lt futos (e where the resduls re sll d ) the solutos obted re qute slr Wth the SVD for ellse ths s ot well ftted beuse the tr A lthough ostve defte A ; s ver erl sgulr We eto here tht deedg o the dt the geerted tr A be defte or sgulr [9] Ad ow we gurtee the geerto of ostve defte tr A d hee of ellse (or rle) rther th other o seto b the fourth ostrt (Ellse ostrt) DOI: 979/ wwwosrjourlsorg 78 Pge

6 he Algebr Lest Squres Fttg of Ellses DOI: 979/ wwwosrjourlsorg 79 Pge IV Algebr Fttg of ellses Cosder the lt futo of geerl ellse the le whh eressed s ) ; ( C b A f where A b b d s defed s b b he ) ( S where S s 6 tr S we ut ) ( ) ( S S S S S he lgebr resdul be defed s S res If we dd to the roble the ellse ostrt we should hve ths for: C to subjet S S where C ws defed erler We gurtee A's ostve defteess b ths ethod e ellse ostrt beuse A s ostve defte f

7 he Algebr Lest Squres Fttg of Ellses () (3) () C hs roble be redued to the geerlsed egevlue roble: ( C S S) d ths s equvlet to C ( S S) where ( C S S) C d S S hve the se order he egevlues of C ( S S) re the egevlues of ( S S) C Ad the re lso the egevlues of CB B( B C) B d B C ( S S) C where (see [7] d [6]) If B s osgulr e (det( A) ) B C he soluto s the egevetor orresodg to the uque ostve egevlue the se of ellses After solvg the lgebrfttg for we ust ostrut the ellses b overtg ts lgebr for to the retr reresetto: Q z Where z os s q B substtutg the equto A b We get ( Q AQ) (z A b ) Q ( z Az b z ) We ut: A ( Q AQ) b (z Ab ) Q ( z Azb z ) Ad the equto wll be A b Choose Q so tht: A dg( ) where re the egevlues of A Choose z so tht: b (z A b ) Q z A b z b / A If the o s ellse wth DOI: 979/ wwwosrjourlsorg 8 Pge

8 he Algebr Lest Squres Fttg of Ellses z d q / A d A hve the se (rel) egevlues / Choose Q Vl where Vl s the tr ots the egevetors of A orresodg to suh tht: Vl Vl I hus the retr for of the ellse s: os Q s q hs s the se of Ellse ostrt where vres fro to 4he Geerl lgorth Ste : Clulte S Ste : Clulte the egevlues of bggest bsolute vlue of B C where B S S d hoose the sllest ostve oe e () or the Ste 3: he soluto () would be the egevetor orresodg to ( ) whh s equvlet to the bggest ostve vlue of suh tht V Eles Wth dfferet set of ots we led here the lgebr ethod for ellse Mtlb ws used here beuse t s es to leet gst other kge or lguge lke Fortr d we sve lot of te too Also we used the lgebr t wth dfferet ostrts 5Fttg Ellses Cosder the S th dt set gve fro [4] ble (4) whh s used for ll eles of ellses Ele : Algebr ethod wth Ellse Algorth Fgure shows the ellse geerted fro the dt usg the ellse ostrt Ad the resdul s res = 7734e - - he u of = e he eter s z = [5:64 4:4887] he best ftobted s the ellse = os = s DOI: 979/ wwwosrjourlsorg 8 Pge

9 he Algebr Lest Squres Fttg of Ellses 53Ele : Algebr ethod wth SVDAlgorth Fgure shows the ellse geerted fro the dt usg SVD ethod he ellse obted s = os = s Fgure : Ellse fts wth SVD Algorth 54Ele 3: Algebr ethod wth Qudrt Algorth We hve Fgure 3 shows the ellse geerted fro the dt usg the Qudrt Algorth ethod - resdul = 3863e = 43776e he ellse geerted s = os = s DOI: 979/ wwwosrjourlsorg 8 Pge

10 he Algebr Lest Squres Fttg of Ellses Fgure 3: Ellse fts wth Qudrt Algorth Bblogrh [] Do L C WSedto E d Szego G P Noler Otzto: theor d lgorths Brkhuser Bosto 98 ISBN [] Flether R G Prtl Methods of Otzto Joh Wle d Sos 995 ISBN [3] Gder W Golub G H d Strebel R Fttg of rles d ellses: lest squre solutobi 34(994) [4] Huffel Sbe V Reet Adves totl lest squres tehques d errors vrbles odelg " Orthogol Lest Squres Fttg b Co Setos " b HeluthSth Lbrr of Cogress USA 997 ISBN [5] L D C Ler Algebr d ts ltos Addso-Wesle 994 ISBN [6] Orteg J M Mtr heor Pleu Press 987 ISBN [7] Stewrt G W Itroduto to Mtr Couttos Ade Press INC 973 [8] he MthWorks I MALAB REFERENCE GUIDE 99 PP 5 [9] Vrh J M Lest Squres Dt Fttg wth lt futos BI 36 (996) Abdelltf Betteb "he Algebr Lest Squres Fttg Of Ellses IOSR Jourl of Mthets (IOSR-JM) 4 (8) PP: DOI: 979/ wwwosrjourlsorg 83 Pge