A Mathematical Comparison of Point Detectors

Transcription

1 Mathematcal Comarson of Pont Detectors M. Zulan, C. Kenney and B. S. Manjunath Vson Research Lab Deartment of Electrcal and Comuter Engneerng Unversty of Calforna, Santa Barbara Santa Barbara, C 9306 Emal: {zulan, kenney, manj}@ece.ucsb.edu bstract Selectng salent onts from two or more mages for comutng corresondences s a fundamental roblem n mage analyss. hree methods orgnally roosed by Harrs et al. n [], by Noble et al. n [] and by Sh et al. n [3] roved to be qute effectve and robust and have been wdely used by the comuter vson communty. he goal of ths aer s to analyze these ont detectors startng from the algebrac and numercal roertes of the mage auto-correlaton matrx. o accomlsh ths task we wll frst ntroduce a natural constrant that needs to be satsfed by any ont detector based on the auto-correlaton matrx. hen, by castng the ont detecton roblem n a mathematcal framework based on condton theory [4], we wll show that under certan hyothess the ont detectors [], [3], [4] are equvalent modulo the choce of a secfc matrx norm. he results resented n ths aer wll rovde a novel unfyng descrton for the most commonly used ont detecton algorthms. I. INRODUCION In order to establsh corresondences among a collecton of mages t s frst necessary to dentfy a set of salent onts n each mage. Pont corresondences consttute the nut for more comlex algorthms that am at regsterng mages, at reconstructng the three dmensonal structure of a scene or at montorng actvtes n a certan area, just to lst a few. One of the aradgms to comute ont corresondences that s largely used by the comuter vson communty s comosed of two stes: ntally a set of te onts s detected n each nut mage and then ths set s ftted to a model that s suosed to descrbe the transformaton between the mages. tycal examle s the estmaton of the lanar homograhy between two vews of a lanar scene. Crucal for the success of ths aroach s a hgh reeatablty rate of the ont detector see [5]). hs means that f a ont q s detected n mage I then, wth hgh robablty, the corresondng ont q wll be detected n mage I. Snce the transformaton that relates the mages s not known a ror, a ont detector should exlot only the nformaton contaned n one sngle mage. Intensty based methods acheve ths goal by assocatng to each ont q a scalar value that defnes ts goodness as a te ont. hs evaluaton s done by erformng some oeratons on the mage ntensty values n a neghborhood of the ont q. he methods roosed by Harrs and Stehens [] and the modfed verson ntroduced by Noble []), by Sh and omas [3] and by Kenney et al. [4] can all be consdered ntensty based methods. he goal of ths work s to exlore n detal what s the quanttatve relaton between these detectors, startng from the algebrac and numercal roertes of the mage autocorrelaton matrx. hs s motvated by the fact that all the detecton rules are based on the sectral structure of ths matrx. Our frst contrbuton to ths analyss s to observe that any functon based on the auto-correlaton matrx) used to detect te onts should satsfy a monotoncty constrant that arses naturally from the hyscal nterretaton of the autocorrelaton matrx. hen, after havng notced that the ratonale behnd the ont detectors roosed by Harrs, Stehens and Noble and by Sh and omas ental a noton of comutatonal stablty, we shall see how these consderatons can be restated from frst rncles usng a mathematcal framework based on condton theory. hs aroach wll shed new lght on the rules used to detect te onts, demonstratng that the revously mentoned aroaches are equvalent modulo a sutable choce of a matrx norm. he aer s structured as follows: secton II ntroduces some of the notaton that wll be used throughout the aer, secton III summarzes the condton theory framework used to relate the dfferent ont detectors, whch are resented n secton IV together wth the nondecreasng constrant. he man results of the aer are stated, roved and dscussed n secton V, whereas the fnal conclusons can be found n secton VI. II. PRELIMINRIES. Geometrc ransformatons of Images sngle channel mage I can be thought as a bounded scalar functon defned over a comact subset of R : I : D [I mn, I max ] R q Iq) We are nterested n detectng te onts n mages related by a geometrc transformaton θ : R R that deends on the arameter vector θ. hese transformatons can be groued n transformaton classes defned as: def {f : θ such that f θ } In ths aer we wll focus our attenton on two classes of transformatons: ranslaton ): θ x ) def x tx + t y for whch the arameter vector s θ [ t x t y ]

2 Rotaton and [ translaton R ]): θ x ) def cos φ sn φ x sn φ cos φ q tx ) + t y for whch the arameter vector s θ φ t x t y and q s the ont about whch the mage atch rotates. For an examle of a transformaton n the class R see Fgure. Note that we dd not consder transformatons that nvolve scale, snce scale requres a secfc treatment. hs s because the ont detecton methods we wll be consderng are ntensty based methods. hs means that they rocess the mage ntenstes n a neghborhood centered about the ont of nterest and therefore such neghborhood should transform covarantly wth the mages. s an examle consder two mages related only by a change of scale: the neghborhood of corresondent onts wll contan the same mage orton only f t scales covarantly wth the mages. he only set of transformatons where the structure of the neghborhood s reserved s the class of rotatons and translaton the shae of a crcle whch s translated and rotated does not change). dscusson about automatc methods to retreve corresondent neghborhoods n mages related by transformaton that do not reserve crcular neghborhoods can be found n [6] and [7]. B. he uto-correlaton Matrx he smoothed verson of the mage I s gven by: Lq, σ, I) G σ I) q) where G σ s a Gaussan functon wth standard devaton σ. he gradent ndcated usng the symbol x ) of L s used to comute the auto-correlaton matrx assocated to mage I: µq, σ I, σ D, I) wσi x L, σ D, I) x L, σ D, I) ) q) w σi q x) x Lx, σ D, I) x Lx, σ D, I)dx ) R where w σ s a weghtng functon whose sze s deendent on the arameter σ I. We shall refer to σ I as the ntegraton scale t s the arameter that defnes sze of the wndow about the ont q ), and to σ D as the dfferentaton scale. he dscretzed verson of equaton ) s: µq, σ I, σ D, I) x w σi q x ) x Lx, σ D, I) x Lx, σ D, I) ) he vectors x are the onts that form the dscretzed suort set of the weghtng functon w σi. Exresson ) can be rewrtten comactly n matrx form as: µq, σ I, σ D, I) W 3) Fg.. he fgure shows the mang between the mage ar I and Î. he shaded crcle reresents the neghborhood of the ont q and the black dots nsde the shaded crcle n the left mage reresent the onts n the neghborhood of q that are used to construct the matrx see equaton 3)). where the matrx R N s formed by stackng the gradents x Lx, σ D, I) one on to of the other and W s a dagonal weghtng matrx such that W w σi q x ). he matrx q, σ D, I) deends both on the ont q and on the dfferentaton scale σ D. Smlarly the matrx W W σ I ) deends on the ntegraton scale σ I. However, to smlfy the notaton we wll henceforth avod to wrte exlctly such deendences. he auto-correlaton matrx rovdes an mortant nformaton about the mage neghborhood for whch the wndow functon s non zero. Each egenvalue µ measures the mage gradent strength along the drectons of the egenvectors of µ. he larger the gradent strength the stronger the corner-lke structure s. C. Matrx Norms In ths secton we wll ntroduce the notaton and the basc concets about matrx norms that are used n ths aer. For a more thorough dscusson of these tocs lease refer to [8], [9]. Let s frst ntroduce the vector -norm: x def x ) Vector -norms lead to the defnton of the nduced matrx -norm: def x su 4) x 0 x It can be shown that σ max ), where σ max ) s the maxmum sngular value of the matrx. he Schatten matrx -norm s defned as: Often tmes w σi s chosen to be a Gaussan functon, whose suort s the entre lane. In ths case the rght hand sde of equaton ) becomes an nfnte summaton. However, to deal wth fnte dmensonal matrces, we can consder the suort of w σi to be the comact set { x R : x Σ x c }, where c s some sutable constant. Dscretzng the revous set we obtan a fnte summaton. S, def σ ) ) where σ ) s the th sngular value of the matrx. 5)

3 III. CONDIION HEORY FOR POIN MCHING In ths secton we wll ntroduce the fundamental facts about condton theory for ont matchng. more thorough dscusson of ths theory can be found n [4]. he basc dea s to dentfy whch onts n an mage can be used to estmate robustly the arameters that defne a certan transformaton. Suose I and I are two mages related by the transformaton θ so that I θ q )) I q ), and let s defne the transformed mage lus nose as: Îq ) def I θ q )) + ηq ) Let s also ntroduce the error cost functon: J θ q ) def [ ] w σ q x ) I θ x )) Îx ) dx R 6) In the nose free case.e. η 0) the mnmzer for 6) s gven by θ whereas n resence of nose such mnmzer s moved to θ + θ. We would lke to quantfy the effect of the nose on the estmaton of the transformaton arameter θ. o acheve ths goal we defne a number that relates θ to η n the lmt for the nose tendng to zero. Defnton : he transformaton condton number CN) at ont q wth resect to the transformaton θ s defned as: K θ q ) def lm su δ 0 η δ θ η where η takes nto consderaton the nose over the suort of the wndow functon w σ. he nterretaton of exresson 7) s the followng: a large condton number means that small erturbatons n the lumnance of the mage can greatly affect the value of the mnmzer of 6). herefore t s reasonable to seek a set of te onts for whch the CN has a small value. Unfortunately for an arbtrary transformaton the CN deends on the arameters of the transformaton tself, so t s of lttle use f we want to assess the sutablty of certan ont to estmate θ by usng just one sngle mage: n fact at ths tme θ s unknown. hs consderaton leads us to a further defnton. Defnton : he class condton number CCN) at ont q wth resect to the class of transformatons s defned as: K q ) def max K θ q ) 8) θ In ths case the condton number s ndeendent from the transformaton arameters. he followng theorems rovde a mean to calculate the condton numbers. For the roofs see the aendces I and II. heorem : he CCN at ont q for the class of translatons s aroxmated by: K q ) W ) W 9) where the matrces and W are defned as n 3). 7) heorem : For W I, usng the nduced matrx -norm n 9), the CCN satsfy the followng nequalty: K q ) K R q ) Moreover, the scalar ) s a lower bound for K q ) and K R q ) for any weghtng matrx W. heorem rovdes a closed form exresson for the CCN wth resect to the class of translaton transforms. heorem states that f a ont s badly condtoned wth resect to the class of transformatons t wll also be bad condtoned wth resect to the class of transformatons R. hs result agrees wth the ntuton: f we have to estmate more arameters usng the same amount of data, then the nose wll have a stronger nfluence on the estmate, roducng a larger varaton θ. he lower bound ntroduced n heorem rovdes a comutatonally effcent method to dscard onts that are bad condtoned: ts calculaton requres the comutaton of a matrx roduct and the soluton of a second order equaton to detect the mnmum egenvalue. IV. POIN DEECORS ont detector s an algorthm that takes one mage as nut and then oututs a set of te onts that can be dentfed wth hgh reeatablty n mages that are related by a transformaton θ. Such an algorthm s comletely defned as long as we secfy the rule for detectng the te onts. In ths secton we wll brefly descrbe the Harrs-Stehens corner detector rule [] wth a artcular emhass on the modfcaton roosed by Noble []), the Sh-omas ont detector rule [3] and fnally the ont detector rule roosed by Kenney et al. n [4]. In all these works the detecton rule s strongly connected to the sectral structure of the autocorrelaton matrx µ. In general a ont q s consdered to be a te-ont as long as: M λ µq, σ I, σ D, I))) 0) where M s a functon of the egenvalues of the autocorrelaton matrx whch are ndcated usng the notaton λ )) and s some sutable threshold. s dscussed at the end of secton II, the strength of cornerlke mage structures s reflected n the magntude of the egenvalues of µ. herefore t s natural to requre M to be a non decreasng functon of the egenvalues of µ n other words the stronger s a corner-lke structure the stronger must be the resonse of the functon 0). We wll refer to ths condton as the nondecreasng constrant on M: Constrant : ny functon M used to measure the strength of a corner-lke mage structure based on the autocorrelaton matrx must be a non-decreasng functon of the egenvalues of. In other words, for any λ λ, λ λ : Mλ, λ ) Mλ, λ ) )

4 . Harrs-Stehens and Noble Corner Detector Harrs and Stehens s corner detector draws ts orgns n the corner detector roosed by Moravec [0], where the authors consder to be good corners those onts for whch the dfference between ther neghborhood and shfted verson of the same neghborhood roduces an error surface wth a well defned mnmum. hs dea encasulates the noton of comutatonal stablty that s the core of the condton theory resented n [4] and summarzed n secton III. Harrs and Stehens roosed the followng functon to measure the corner strength: M H q, σ I, σ D, I) det W ) α trace W )) ) he larger the value of α the less senstve s the detector to corner lke structures. he major drawback of ths corner detector s the resence of a arameter α that needs to be manually tuned. o overcome ths dffculty Noble [] roosed a modfed verson of the Harrs functon that does not contan any constant: M N q, σ I, σ D, I) det W ) trace 3) W ) + ε he small constant ε s used to avod a sngular denomnator n case of a rank zero auto-correlaton matrx. B. Sh-omas Pont Detector In [3] Sh and omas roosed a crteron to decde whch onts are sutable for trackng see also []). her dea s to select onts for whch the system that rovdes an estmate of the dslacement from one frame to the other s numercally well condtoned. lso n ths case we fnd a connecton wth the core deas of the condton theory summarzed n secton III. he Sh and omas functon s defned as: M S q, σ I, σ D, I) λ mn W ) 4) where λ mn W ) ndcates the smallest egenvalue of the auto-correlaton matrx. C. Kenney et al. Pont Detector s dscussed n secton III, heorem ) and heorem ) suggest the followng rule to detect te onts: select those onts that have a small condton number wth resect to the class. he detecton rule can be formalzed by defnng the Kenney s functon: M K, q, σ I, σ D, I) W ) W where ndcates whch matrx norm has been used. V. RELION BEWEEN POIN DEECORS. Equvalence of Pont Detectors s onted out n secton IV, both the Harrs-Stehens detector and consequently the Nobel detector) and the Sh- omas detector somehow encasulate a noton of comutatonal stablty. hs fact suggests that the functons 3) and 4) should reflect n some way ths noton. o suort ths observaton we wll rove a theorem that establshes a relaton between the ont detectors descrbed before, showng that the Noble functon, the Sh-omas functon and the Kenney functon are equvalent modulo the choce of a sutable matrx norm. heorem 3: For ε 0 and W I: M N M K, Schatten 5) M S M K, 6) Proof: Suose the sectrum of the matrx W s gven by λ λ 0. We wll frst show the equvalence relaton between the Harrs-Stehens-Nobel functon and Kenney s one when the -Schatten norm s used. Consder the matrx M ). Frst we wll rove that: σ M) σ ) ) 7) hs results follows drectly from the SVD decomoston of the matrces M and ) : ) V Σ U UΣV ) V Σ V M ) V Σ V V Σ U V [ Σ 0 ] U where Σ s the dagonal matrx contanng the sngular values of ). From the defnton of the Schatten norm 5) t follows that: ) S, σ M) ] σ ) ) ] σ M) ] M S, Gven that for a symmetrc matrx the sngular values concde wth the egenvalues we have that: M K, Schatten ) S, M S, because of 7)) λ + λ λ λ λ + λ for ε 0) M N whch roves the frst art of the theorem, snce the Nobel functon 3) can be rewrtten as: M N λ λ λ + λ + ε

5 Now let s consder the equvalence between the Sh and omas functon and Kenney s one. Snce the -norm of a matrx concdes wth the largest sngular value, usng the dentty 7) we can wrte: M ) herefore the followng equalty holds: M K, M ) λ M S 8) whch roves also the second equvalence. B. he Nondecreasng Constrant he nondecreasng constrant ) ntroduced at the end of Secton IV says that the detecton functon should be nondecreasng n λ and λ. he followng lemma shows that M N and M S satsfy such constrant, whereas for the Harrs- Stehens functon M H the constrant holds true only when α 4 ths fact rovdes a justfcaton of the common habt of settng α 0.04). Lemma : he functons M N, M S and consequently M K, Schatten and M K, ) all satsfy the nondecreasng constrant ). he functon M H satsfes ) rovded that α 4. Proof: he constrant ) s satsfed as long as the artal dervatves of the functon M are non negatve: M λ 0 M λ 0 For the Nobel functon we have that the artal dervatves are gven by: M N λ jλ j + ε) λ λ + λ j + ε) whch shows that they are always nonnegatve. Now assume that λ λ 0. For the Sh-omas functon we have that: { M N for λ λ λ 0 for λ λ herefore also n ths case the artal dervatves are always non negatve. From heorem 3 t follows that Kenney s ont detector satsfes the nondecreasng constrant when W I and ether the -norm or the Schatten -norm are used. Fnally let s consder the Harrs-Stehens functon for whch: M H λ λ j αλ + λ j ) he artal dervatves are non negatve f and only f: α λ λ + λ ) λ Snce λ +λ ) 4 for any λ λ 0, we conclude that M H wll satsfy ) f and only f α 4. VI. CONCLUSION In ths aer we reformulated the ont detecton rules roosed by Harrs, Stehens and Nobel, by Sh and omas and by Kenney usng a common framework based on condton theory ntroduced n [4] and summarzed n secton III. We restrcted our attenton to rotaton and translaton snce more comlex transformaton would requre an automatc rocedure to detect neghborhoods that vary covarantly wth the transformaton [7]. he central result s stated n heorem 3, where we showed that the ont detectors revously lsted are equvalent modulo a sutable choce of a matrx norm, as long as the weghtng functon s constant over ts suort). hs results tells that the Harrs, Stehens and Nobel corner detector and the feature detector roosed by Sh and omas both select onts that are sutable to estmate robustly the arameters of a translaton between an mage ar. We also ntroduced a constrant that should be satsfed by all the functons M used to detect te onts based on the auto-correlaton matrx. We showed that all the detectors studed n ths aer satsfy the revous constrant automatcally, wth the relevant exceton of the Harrs Stehens corner detector, for whch t s necessary to force α to be less than 4. Future research drectons am at consderng a more flexble defnton of condton numbers, n order to be able to coe wth a larger varety of transformatons ossbly wthout havng to solve exlctly the roblem of the detecton of a covarant neghborhood. We would also lke to study n more detal whch s the role of the weghtng functon w σ, n order to resent an extenson of these results that holds also n the case where W I. cknowledgments. hs roject was suorted n art by the followng grants from the Offce of Naval Research: ONR #N Zulan and Manjunath) and ONR #N Kenney). PPENDIX I PROOF OF HEOREM. Proof: Our goal s to exress the arameter vector θ that mnmzes the cost 6) n terms of the nose η. Suose η 0: then the mnmzer θ wll move to θ + θ. necessary condton for otmalty s that θ J θ + θ q ) 0,.e. : [ ] w σ q x ) I θ + θ x )) Îx ) R x I θ + θ x ))J θ θ + θ x )dx 0 9) where x I θ + θ x )) s the mage gradent comuted at the ont θ + θ x ) and J θ θ x ) s the Jacoban of the transformaton θ. he aylor truncated exanson of the last three factors n 9) s gven by: I θ + θ x )) Îx ) x I θ + θ x )) x I θ x ))J θ θ x ) θ ηx ) x I θ x )) + θ J θ θ x )H x I θ x ))

6 where H x I θ x )) s the mage Hessan. s far as the Jacoban s concerned we have that n the ure translaton case: J θ θ + θ x ) J θ θ x ) I 0) whereas n the case of rotaton and translaton: J θ θ + θ x ) J θ θ x ) + M θ [ Rφ + π ) q I ] + [ Rφ) q 0 ] θ he matrx Rα) s the matrx n SO) that roduces a rotaton of an angle α and q x q. By drong the second order terms.e. the terms contanng the roducts θ θ and θ ηx )), we obtan the new equaton: R w σ q x ) [ x I θ x ))J θ θ x ) θ ηx )] x I θ x ))J θ θ x ) 0 ) Let s now consder the dscretzed verson of equaton ): x w σ q x ) [ x L θ x ), σ D, I)J θ θ x ) θ ηx )] x L θ x ), σ D, I)J θ θ x ) 0 ) σ )vx )ηx ) 3) where the contnuous dervatves are relaced by convolutons wth the dervatves of a gaussan kernel wth standard devaton σ D. By settng v x ) L θ x ), σ D, I)J θ θ x ), ) can be wrtten as: ] x w q x vx )w σ q x )v x ) θ x By defnng W as n 3) and: 3) can be wrtten as: η [ ηx )... ηx n) ] θ q )W θ q ) θ θ q )W η 4) he exresson for the CN s obtaned by solvng n a least square sense the normal equaton 4). For a ure translaton does not deend on θ because J θ θ x ) I) and therefore: K q ) K θ q ) W ) W rovdes also an exresson for the CCN for translaton. he matrx s the same as n 3). For the rotaton and translaton case the Jacoban J θ θ x ) unfortunately deends on θ and therefore only an exresson for the CN can be rovded: K θ q ) θ q )W θ q ) ) θ q )W PPENDIX II PROOF OF HEOREM. We frst show the followng lemmas. Lemma : Let be defned as n 3) and let w R N. def Let s also defne the matrx J [ w ]. hen the followng nequalty holds: J J ) J ) 5) Proof: s shown n the roof of heorem 3, f 5) holds, then such nequalty can be rewrtten as: or equvalently as: J J ) ) λ mn J J ) λmn ) 6) herefore the lemma wll be roved f we are able to rove 6). For our uroses t s more convenent to consder the transose of the matrces J J and ; n artcular we have that: λ J J ) λ J J ) {0,, 0} {λ }{{} 3, λ, λ, 0,, 0} }{{} N 3 N 3 where λ 3 λ λ 0 and analogously: λ ) λ ) {0,, 0} {λ }{{}, λ, 0,, 0} }{{} N N where λ λ 0. If we rewrte the matrx J J as ww + we can exlot the nterlacng roerty of the egenvalues see n artcular heorem n [9]) accordng to whch λ [0, λ ]. herefore we can conclude that: λ mn J J ) λ λ λ mn ) whch roves the lemma. Lemma 3: For any matrces and W wth comatble dmensons the followng nequalty holds true: W ) W ) 7) Proof: s showed n more detal n the roof of heorem 3, the SVD decomoston of the matrx s: Σ U V 0 and the SVD decomoston of M ) s: M V [ Σ 0 ] U Because the matrx -norm s nvarant wth resect to orthogonal rotatons we have: M [ Σ 0 ] Σ. Introducng the matrx: m m U W U m m

7 comatbly dmensoned wth U, t s ossble to show that : W ) W V [ Σ Σ m m ] U and consequently: W ) W [ Σ Σ m m t ths ont the lemma s roved, snce: [ Σ Σ m m ] [ Σ 0 ] ] [] W. Förstner and E. Gülch, fast oerator for detecton and recse locaton of dstnct onts, corners and center of crcular features, n Proc. of ISPRS Intercommsson Conference on Fast Processng of Photogrammetrc Data, Interlaken, Swtzerland, June , Usng these results we can now rove heorem. Proof: In the roof of heorem we showed that for rotaton and translaton the rows of the matrx θ q ) can be wrtten as v x ) L θ x ), σ D, I)J θ θ x ), where the Jacoban J θ θ x ) has the form [ u I ]. herefore we can wrte: θ q ) [ w ] If we rename θ q ) wth J then we can aly Lemma, showng drectly that the CN for rotaton and translaton s always greater or equal than the CCN for translaton rovded W I). Snce Lemma holds for any vector w, then the result can be extended to the CCN for rotaton and translaton, yeldng the desred nequalty: K q ) K R q ) he roof of the last art of the theorem s a straghtforward alcaton of Lemma 3. REFERENCES [] C. Harrs and M. Stehens, combned corner and edge detector, n Proc. of the 4th LVEY vson conference, M. M. Matthews, Ed., Unversty of Manchester, England, Setemeber 988, []. Noble, Descrtons of mage surfaces, Ph.D. dssertaton, Deartment of Engneerng Scence, Oxford Unversty, 989. [3] J. Sh and C. omas, Good features to track, n Proc. of IEEE Conference on Comuter Vson and Pattern Recognton CVPR 94), Seattle, Washngton, June 994, [4] C. Kenney, B. Manjunath, M. Zulan, G. Hewer, and. Van Nevel, condton number for ont matchng wth alcaton to regstraton and ost-regstraton error estmaton, IEEE ransactons on Pattern nalyss and Machne Intellgence, vol. 5, no., , November 003. [5] C. Schmd, R. Mohr, and C. Bauckhage, Comarng and evaluatng nterest onts, n Proc. of IEEE 6th Internatonal Conference on Comuter Vson, Bombay, Inda, January 998, [Onlne]. valable: htt:// [6]. Lndeberg, Scale-Sace heory n Comuter Vson. Dordrecht, Netherlands: Kluwer cademc, 994. [7] K. Mkolajczyk and C. Schmd, n affne nvarant nterest ont detector, n Euroean Conference on Comuter Vson. Coenhagen, Denmark: Srnger, 00, [Onlne]. valable: htt:// [8] G. H. Golub and C. F. Van Loan, Matrx Comutatons. he John Hokns Unversty Press, 996. [9] C. D. Meyer, Matrx nalyss and led Lnear lgebra. SIM, 00. [0] H. Moravec, Obstacle avodance and navgaton n the real world by a seeng robot rover, Carnege-Mellon Unversty, Robotcs Insttute, ech. Re. CMU-RI-R-3, Setember 980. We assume m to be nvertble: f t s not a smle lmtng argument gves the same result.