HOMOGENEOUS LEAST SQUARES PROBLEM

Transcription

1 the Photogrammetrc Jornal of nl, Vol. 9, No., 005 HOMOGENEOU LE QURE PROLEM Kejo Inklä Helsnk Unversty of echnology Laboratory of Photogrammetry Remote ensng P.O.ox 00, IN-005 KK RC he homogeneos Least qares Problem s frst defned dscssed. hen two methods for solvng the L-problem are presented. hese methods are based on the se of the generalzed egenvale decomposton the generalzed snglar vale decomposton, respectvely. nally, a nmercal example s gven.. INRODUCION he solton of a (non-homogeneos) lnear system x b s a typcal problem n photogrammetry. Usally, the system s, n addton, overdetermned (.e. x b ) the exstence of the solton s ensred by the Least qares condton. he solton of a homogeneos lnear system x 0 s a less common problem. However, some nterestng applcatons exst, where homogeneos systems natrally arse. rst, the eqaton ax + by + c 0 represents a straght lne n xy-plane. If the ponts ( x, y ),,..., n, le (exactly) on the lne, we have x x M x n y y M y n 0 a 0 b M M c 0.e. a homogeneos system x 0 wth a n 3-matrx. mlarly, makng the sbstttons nto the eqaton ax + by + cxy + dx + ey + f 0 of a conc secton (e.g. crcle or ellpse) leads also to a homogeneos system. In photogrammetry compter vson there are two mportant examples, where solton of a homogeneos system s needed (see e.g. (Mkhal et al., 00) (orsyth et al., 003)): he determnaton of the parameters of the Drect Lnear ransformaton (DL). he determnaton of the essental or fndamental matrx from the gven (measred) coordnates of the correspondng ponts n two mages. 34

2 the Photogrammetrc Jornal of nl, Vol. 9, No., 005. HE HOMOGENEOU L-PROLEM he solton of any homogeneos system x 0 has the followng specal featres: here s always the trval solton x 0, bt t s not nterestng. If x 0 s a solton, then kx, where k s a arbtrary scalar, s also a solton (only the ratos of the nkowns can be determned). n addtonal condton s ths reqred to fx the length of the solton vector (to exclde the trval solton). smple way s to fx some component of the solton vector, say x, to solve the rest of the nknowns x from the non-homogeneos system however, lead to dffcltes: If the (tre) vale of k 0 sch that kx 0. It s ths better to se a more general condton j x j j. hs may, x j happens to be zero, there exsts no scalar x where s a p -matrx (for smplcty, we choose x nstead of x α, becase the scalng s not crtcal). Note that the condton x x x contans the followng specal cases: If I, then x x x + x + L + x If [ I 0] p, then x x + x + L + x p If dag b, b, K, b ), then x b x + b x + L + b x ( We trn now to the second dffclty. It s obvos, that the homogeneos system x 0 has no solton (other than x 0), f n the rank of s fll,.e. rank ( ) (typcal case n practcal problems!). It s then natral to search for a solton that makes the contradcton small,.e. x 0. poplar condton s the Least qares crteron x mn ased on the precedng consderatons we state now the followng problem: Gven a n -matrx : n rank ( ) () fnd the L-solton xˆ, that flfls the condtons: ˆ x mn () xˆ (3) hs problem here s called the homogeneos least sqares problem. 35

3 the Photogrammetrc Jornal of nl, Vol. 9, No., HE OLUION O HE HOMOGENEOU L-PROLEM In ths chapter we present two solton methods for the homogeneos L-problem. he core of these methods s the comptaton of the generalzed egenvale decomposton or the generalzed snglar vale decomposton. hese decompostons are brefly prescrbed n the ppendx. Mathematcal detals can be fond e.g. n (Golb et al., 99). It s mportant from practcal pont of vew that many mathematcal software packages sch as ML contan these decompostons (Mathworks, 005). 3. olton sng the generalzed egenvale decomposton he solton that flfls the condtons method,.e. by mnmzng x mn x can be fond sng the Lagrange L( x, λ ) x + λ ( x ) (4) where λ s the Lagrange mltpler. ettng the partal dervatves of L wth respect to x λ to zero gves the eqatons x λ x (5) x (6) It can be seen mmedately that the frst condton (5) s satsfed by the generalzed egenvectors of the symmetrc matrces. Usng x λ x we frther have L x + λ ( x ) x x + λ ( x x) λ (7) hs, the solton that mnmzes L s, n partclar, the generalzed egenvector e mn related to the smallest generalzed egenvale λ mn,.e. x ˆ emn. In general, e α, therefore, n order to flfll the condton xˆ, a smple scalng s reqred: mn x ˆ (/ e e (8) mn ) mn 36

4 the Photogrammetrc Jornal of nl, Vol. 9, No., 005 In smmary, we have the followng solton algorthm:. Compte the matrces M N. Compte the generalzed egenvale decomposton ME NED of M N. 3. earch for the smallest generalzed egenvale λ mn choose the related egenvector e mn as the L-solton, that s x ˆ emn. 4. Make an approprate scalng, f necessary (e.g. xˆ scaled (/ emn ) emn ). here are two mportant specal cases, where t s sffcent to compte the (ordnary) egenvale decomposton nstead of the generalzed egenvale decomposton. rst, f I (fnd xˆ, sch that xˆ mn x ˆ ) the condton (5) redces to the more smple condton x λ x (9) he egenvectors of satsfy ths eqaton. t, sng x λ x x + λ ( x ) x x + λ ( x x) x ( x λx) + λ λ hs, the L-solton s the egenvector correspondng to the smallest egenvale of alternatve dervaton for ths reslt s gven n (orsyth et al., 003). he second nterestng specal case s [ 0 I ] p. n,.e. the constrant contans only a part of the parameters (the last p parameters here). y parttonng [ ] have the condtons: [ x x ] x x + x mn x (0) he L-solton can be fond n ths case by performng the followng steps: () Compte the coeffcent matrx of the redced normal eqaton ( x elmnated!) N ) ( ) ( ) () ( () Compte the egenvale decomposton of N. x we (3) Choose the egenvector correspondng to the smallest egenvale as the L-solton,.e. x ˆ e () mn (4) olve ˆx from the eqaton xˆ ˆ x (3) 37

5 the Photogrammetrc Jornal of nl, Vol. 9, No., olton sng the generalzed snglar vale decomposton Let s assme that matrces have the generalzed snglar vale decomposton U (4) V where U s an orthogonal nxn-matrx ( U U I), V s an orthogonal pxp-matrx ( V V I ), s a dagonal nx-matrx, s a dagonal px-matrx s a reglar (non-snglar) xmatrx (see the ppendx for more nformaton on the generalzed snglar vale decomposton). y notng that x x U x x V x x by defnng y x, the orgnal problem ( x mn sbject to x ) can be redced to the eqvalent problem y mn sbject to y (5) ssmng that the dagonal elements of are ordered as α + α the dagonal elements of are ordered as β + β, the solton of ths problem s smply the vector [ / β 0 0 ] yˆ K 0 (6) he L-solton xˆ can then be solved from the system xˆ yˆ (7) Note that the L-solton xˆ s the frst row of the nverse matrx Note also that the sqared resdal sm s smply α / β dvded by the scalar β. s (8).e. the sqare of the frst generalzed snglar vale of. We ths have the followng algorthm for the L-solton:. Compte the generalzed snglar vale decomposton of (, ). olve xˆ from the system xˆ yˆ, where yˆ [ / β 0 0 0] K. Compared to the frst algorthm, a favorable featre of ths algorthm s that the comptaton of matrces s avoded. 38

6 the Photogrammetrc Jornal of nl, Vol. 9, No., 005 In the specal case I the solton smplfes agan essentally. Let UV be the snglar vale decomposton of. hen x mn UV x mn frthermore UV the problem: x V x x V x. hs, by defnng y V x, the orgnal problem redces to nd ŷ sch that yˆ mn y ˆ. ecase s a dagonal matrx s > s +, the L-solton of ths problem s smply yˆ 0 0 K 0. hs the L-solton xˆ of the orgnal problem s [ ] 0 0 x ˆ Vyˆ V M V (9) 0.e. the last colmn V of the matrx V or the snglar vector related to the smallest snglar vale of. 4. N EXMPLE We demonstrate the homogeneos L-problem by fttng a straght lne to gven measred ponts x, y ) sng the model ( ax + by + c 0 or [ x y a ] b 0 c We consder the followng constrants for the parameters: b a + b + c a + b he frst constrant leads to snglar case, f the lne s parallel to y-axs. he second constrant has no partclar advantages apart from ts smplcty. hrd constrant on the contrary trns ot very nterestng (Hessan Normal orm). Note that the dstance of a pont ( x, y) from the lne ax + by + c 0 s ( ax + by + c) / a b. hs, f + b d + a, then d ax + by + c. herefore, when choosng ths constrant, the resdals are orthogonal to the ftted lne! 39

7 the Photogrammetrc Jornal of nl, Vol. 9, No., 005 y defnng [ a b c] x we fnd easly that the constrant a + b s eqvalent to x, where Gven the ponts {(,), (,6), (6,)} we also have 6 6 ˆ. fter he comptatons sng ML gve the L-solton x [ ] approprate scalng the ftted lne can be wrtten e.g. as y x + 6 ˆ ˆ. he resdal vector s v x [ ] When compared to other methods for lne fttng the followng featres of the presented methods are worth emphaszng: No snglar cases (orentaton of the lne can be arbtrary). No need for approxmate vales teratons. Orthogonal resdals are mnmzed. 5. REERENCE orsyth, D.., Ponce, J., 003. Compter Vson Modern pproach. Pearson Edcaton. Golb, G.H., van Loan, C.., 989. Matrx Comptatons, nd ed., he Johns Hopkns Unversty Press. Hartley, R., Zsserman,., 00. Mltple Vew Geometry, Cambrdge Unversty Press. Mkhal, E.M., ethel, J.., McGlone, J.C., 00. Introdcton to Modern Photogrammetry, John Wley & ons. Mathworks, 005. Docmentaton for Mathworks Prodcts, Release 4 wth ervce Pack, ( ). 40

8 the Photogrammetrc Jornal of nl, Vol. 9, No., 005 PPENDIX Egenvale decomposton Let M be a symmetrc x-matrx. hen there exst an orthogonal matrx E a dagonal matrx D dag( λ, K, λ ) sch that ME ED Dagonal element λ s the th egenvale of M the colmn e s the correspondng egenvector. Generalzed egenvale decomposton Let M N be symmetrc x-matrces. hen there exsts a reglar x-matrx G a dagonal matrx D dag( λ, K, λ ) sch that MG NGD Dagonal element λ s th generalzed egenvale of M N. he colmn correspondng egenvector. g s the nglar vale decomposton Let be a nx-matrx ( n > ). hen can be decomposed as UV or U V 0 or U V 0 where U s an orthogonal nxn-matrx, V s an orthogonal x-matrx s a dagonal nxmatrx. Generalzed snglar vale decomposton Let be a nx-matrs a px-matrx. hen can be decomposed as (Mathworks, 005) U V where U s an orthogonal nxn-matrx, V s an orthogonal pxp-matrx, s a xq-matrx ( q mn( n + p, ) ), s a dagonal nx-matrx (dagonal elements α + α ) s a dagonal px-matrx (dagonal elements β β + ). 4

9 the Photogrammetrc Jornal of nl, Vol. 9, No., 005 Matrces satsfy n ths formlaton the condton + he strctre of depends on n, p. or example, f n > p <, then, dag( α,..., α ), α + α 0 [ 0], dag( β,..., β p ), β + β n >, then s a sqare matrx (x). In addton, f the rank of [ ] Note that, f then s a reglar matrx s fll, where U V dag α / β,..., α / β ) ( Dagonal elements s are the generalzed snglar vales of. 4