Homogeneous Representations of Points, Lines and Planes

Transcription

1 Chapter 5 Homogeneou Repreentation of Point, Line and Pane 5 Homogeneou Vector and Matrice Homogeneou Repreentation of Point and Line in 2D Homogeneou Repreentation in IP n Homogeneou Repreentation of 3D Line n Pücker Coordinate for Point, Line and Pane The Principe of Duait Conic and Quadric Normaization of Homogeneou Vector Canonica Eement of Coordinate Stem Eercie 245 Thi chapter motivate and introduce homogeneou coordinate for repreenting geometric entitie Their name i derived from the homogeneit of the equation the induce Homogeneou coordinate repreent geometric eement in a projective pace, a inhomogeneou coordinate repreent geometric entitie in Eucidean pace Throughout thi book, we wi ue Carteian coordinate: inhomogeneou in Eucidean pace and homogeneou in projective pace A hort coure in the pane demontrate the uefune of homogeneou coordinate for contruction, tranformation, etimation, and variance propagation A characteritic feature of projective geometr i the mmetr of reationhip between point and ine, caed duait In thi chapter we aim at epoiting the agebraic propertie of the repreentation of geometric entitie and at giving geometrica intuitive interpretation 5 Homogeneou Vector and Matrice 5 Definition and Notation A Short Coure 97 5 Definition and Notation Definition 52: Homogeneou coordinate (J Pücker 829) Homogeneou coordinate of a geometric entit are invariant with repect to mutipication b a caar λ 0:thu and λ repreent the ame entit We wi find homogeneou repreentation for geometric entitie, uch a point, ine and pane, but ao for tranformation The homogeneou repreentation i not unique, a λ 0can be choen arbitrari; and repreent the ame entit Uniquene of the entit i guaranteed a ong a not a coordinate vanih, thu 0

2 96 5 Homogeneou Repreentation of Point, Line and Pane In certain appication it wi be uefu to retrict the freedom of caing and to ditinguih between oppoite direction, eg, when reaoning about the eft or the right ide of an entit or when modeing a rea camera: point awa have to be ocated in front of a camera, and thi need to be refected in the modeing Thi ead to oriented entitie whoe repreentation i on invariant to the mutipication with a poitive caar It wi occaiona be uefu to reduce the ambiguit of the caing and normaize homogeneou entitie We wi ditinguih between pherica normaization, which we denote b the inde, eg where =, and imiar the Eucidean normaization e where ome of the eement of e can be interpreted a eement in Eucidean pace Due to thee repreentationa propertie, we need to carif the uage of the equa ign = in the contet of homogeneou entitie It ha three ue: The equa ign i ued to indicate equait, foowing the convention in mathematic 2 The equa ign i ued to indicate a vaue aignment a in ome computer anguage For eampe, = i read a the vector i to be determined a the cro product of and Thi i ometime written a := 3 The equa ign i ued to tate that the repreentation on the eft and the right-hand ide refer to the ame object Thu the two repreentation are equa up to caing The equation above = (a homogeneou reation), thu can be read a a condition for the ine parameter to be equa to the parameter of the ine connecting the point () and () Thi ometime i written a =, oraλ = with ome λ 0, making the cae factor epicit We wi ue the impe equa ign and on pecif the reation 2 and 3 if the contet require In contrat to inhomogeneou entitie uch a, X, and R, homogeneou entitie are deignated with upright etter, uch a, X, and P Pane are deignated with etter from the beginning of the aphabet, ine with etter from the midde of the aphabet and point with etter from the end of the aphabet Point and ine in the pane wi be caed 2D point and 2D ine, in contrat to 3D point and 3D ine in pace We ditinguih between name and repreentation of geometric entitie The mbo X denote the name of the point wherea it coordinate are denoted b X or X; thu, we can write X (X) or X (X) depending on our aumption about the repreentation of the point X The notation ued are coected in Tabe 5 and 52 Tabe 5 Name of baic geometric entitie in 2D and 3D eement 2D 3D pane A, B, ine, m, L, M, point,, X, Y, Tabe 52 Notation for inhomogeneou and homogeneou vector and matrice 2D 3D tranformation inhomogeneou X R homogeneou, A, L, X H Homogeneou coordinate have a number of advantage which make them indipenabe in our contet: The aow u to repreent entitie at infinit, which occur frequent, eg, when deaing with vanihing point Conceptua, homogeneou coordinate are the natura

3 Section 5 Homogeneou Vector and Matrice 97 repreentation of eement of a projective pace, b which we mean the correponding Eucidean pace together with the eement at infinit of a ine in that pane Homogeneou coordinate aow u to eai repreent traight ine-preerving tranformation, thu not on tranation, rotation or affine tranformation but ao projective tranformation, eg, when repreenting the mapping from 3D object pace to 2D image pace in a pinhoe camera The impif concatenation and inverion of traight ine-preerving tranformation, ince a tranformation are repreented a a matri vector product The impif the contruction of geometric eement from given one a we a the epreion of geometric contraint a et of homogeneou equation A geometric operation, contruction, and tranformation are biinear form A a conequence, the uncertaint of vector and matrice uing covariance matrice can eai be propagated, a the necear Jacobian are derived without effort We wi firt introduce the baic idea in a hort coure, motivating the content of the chapter, and then dicu the individua concept in detai 52 A Short Coure Thi ubection i meant to give an intuitive introduction to the ue of homogeneou coordinate in 2D pace and to eempif their advantage for point and ine and their reation and tranformation 52 Repreentation with Homogeneou Coordinate The Heian norma form of a traight ine in the -pane i given b Heian norma form co φ + in φ d =0, (5) ee Fig 5 Wherea the point i repreented with it inhomogeneou coordinate = [, ] T,thu (, ), the ine i repreented with the Heian coordinate h =[φ, d] T, name the direction φ of it norma in mathematica poitive, ie, countercockwie, direction counted from the -ai, and it ditance d from the origin, thu (φ, d) n d φ Fig 5 Straight ine with parameter of Heian norma form The norma direction n of the ine point to the eft wrt the direction (or orientation, cf Sect 93, p 346) of the ine Equation (5) ma be written in different form and aow different interpretation: The equation repreent the incidence of the point () with the ine (h) Thi mmetric incidence reation ι(, ) i equivaent to the dua reation: The point ie on the ine and The ine pae through the point The equation ma be written a We aume the ditance i meaured in the direction of the norma

4 98 5 Homogeneou Repreentation of Point, Line and Pane n T = d with n = [ n ] = n [ ] co φ in φ (52) if we ue the norma vector n It ugget the ine i to be repreented b three parameter, [n,n,d] T However, the atif one contraint, name n = The repreentation with n doe not have a inguarit when etimating the direction n, unike the ange repreentation of the direction with φ (ee the dicuion on uing quaternion for repreenting rotation, Sect 852, p 335) Thi ha ignificant advantage The equation ma be written a et e =0 with e = e = co φ in φ (53) d Thi ugget that both the point and the ine are to be repreented with 3-vector, thu ( e ) and ( e ) The are homogeneou vector, a mutiping them with an arbitrar caar 0doe not change the incidence reation But the are normaized in a we-defined wa, name uch that the inhomogeneou parameter (, ) and d can be direct inferred We wi dicu normaization beow Moreover, the equation i mmetric in and a T = T =0, which agebraica refect the mmetr of the incidence propert ι(, ) The equation ma more genera be written a T =0 (54) with the vector u = 2 = v = w, (55) 3 w = = a b = ± a 2 + b 2 co φ in φ (56) c d 2 3 homogeneou coordinate The factor w 0and [, 2 ] = a 2 + b 2 0can be choen arbitrari Therefore, point and ine can be repreented b near arbitrar 3-vector, name b retricting the aboute vaue of w = 3 and the aboute vaue of [a, b] T =[, 2 ] T not to be zero A the reation (54) i a homogeneou equation, the correponding repreentation of the point are homogeneou, and the 3-vector and are caed the homogeneou coordinate of the point and the ine repective We can eai determine the Eucidean repreentation of the point and the ine from = u w = v w, φ = atan2 (b, a) d = c (57) a2 + b 2 or = [ ] 2, φ = atan2 ( 2, ) d = 3 [ ] 3 (58) Normaization Homogeneou coordinate of a point or a ine are not unique Uniquene ma be achieved b normaization, ie, b fiing the cae factor Two tpe of normaization are common, Eucidean and pherica

5 Section 5 Homogeneou Vector and Matrice 99 Eucidean Normaization B Eucidean normaization the vector i tranformed uch that the Eucidean propertie become viibe (Fig 52) We obtain e = N e () = u [ ] v =, e = N e () = a [ ] b n = (59) w w a2 + b 2 d c Therefore, foowing Brand (966), we introduce the foowing notation for point and ine to pecif the Eucidean part and the homogeneou part of a homogeneou vector The Eucidean part, indeed b 0, impicit contain the Eucidean propertie: for point the two coordinate, and for ine the ditance to the origin, = [ 0 Eucidean normaization then read a h ], = [ h 0 ] (50) e = h, e = h (5) Spherica Normaization B pherica normaization a coordinate of a homogeneou vector are proceed the ame wa and the compete vector i normaized to (Fig 53) The pherica normaized homogeneou coordinate of a 2D point and of a 2D ine are = N() = u v, = N() = a b (52) u2 + v 2 + w 2 w a2 + b 2 + c 2 c Thu the pherica normaized homogeneou coordinate of a 2D point and 2D ine buid the unit phere S 2 in IR 3 We wi frequent ue pherica normaized homogeneou vector The have evera advantage: The ie on a phere, which i a coed manifod without an border Thu geometrica, ie, if we do not refer to a pecia coordinate tem, there are no pecia point in the projective pane 2 The redundanc in the repreentation we ue three coordinate for a 3D entit require care in iterative etimation procedure, a the ength contraint need to be introduced Iterative correcting pherica normaized vector can be reaized in the tangent pace which for 2D point i a tangent pane at the pherica normaized vector 3 A the point and the point repreent the ame 2D point, the repreentation i not unique Taking thee two point a two different one ead to the concept of oriented projective geometr, which among other thing can ditinguih between ine with different orientation (Chap 9, p 343) 523 Geometric Interpretation of Homogeneou Coordinate and the Projective Pane The at two paragraph ugget an intuitive and important geometric interpretation of homogeneou coordinate a embedding the rea pane IR 2 with origin 2 and ae and into the 3D Eucidean pace IR 3 with origin 3 and ae u, v and w, cf Fig 52, eft The Eucidean normaized coordinate vector e =[u, v, w] T =[,, ] T ie in the pane w = The origin 2 ha coordinate 2 =[0, 0, ] T The u- and the v-ae are parae to the - and the -ae repective Thu, adding the third coordinate,, to an inhomogeneou coordinate vector to obtain e can be interpreted a embedding the rea Eucidean pane Eucidean normaized vector

6 200 5 Homogeneou Repreentation of Point, Line and Pane IR 2 w 2 3 ( e ) v u IR 2 e c,w 2 3 n z b,v a,u Fig 52 Repreentation with homogeneou coordinate, Eucidean normaization Left: 2D point The rea pane IR 2 i embedded into the 3D pace IR 3 with coordinate u, v and w An vector on the ine joining the origin 3 of the (uvw)-coordinate tem and the point, ecept the origin itef, can repreent the point on the Eucidean pane IR 2 The interection of the ine 3 with the pane w = ied the Eucidean normaized homogeneou coordinate vector e of Right: 2D ine The rea pane IR 2 i embedded into the 3D pace IR 3 Coordinate a, b and c are ued to repreent 2D ine The 2D ine i repreented b the norma of the pane paing through the origin 3 and the ine When the Eucidean homogeneou coordinate e (59), p 99 are ued, their firt two eement are normaized to, and the vector e ie on the unit cinder (ditance 3 =) parae to the w-ai The ditance of the ine from the origin 2, which i in the direction of the norma n, i identica to the c-component of e Eercie 53 pherica normaized vector point at infinit projective pane IR 2 into IR 3 Point with coordinate = λ e are on a traight ine through the origin 3 The repreent the ame point, name You can ao a: the traight ine 3, taking, which i embedded in the 3D (uvw)-pace, repreent the homogeneou point A imiar geometric interpretation can be given for ine Here, we embed the rea pane IR 2 into IR 3, but with an (a, b, c)-coordinate tem at 3 The vector e := [a, b, c] T =[coφ, in φ, d] T ie on the vertica cinder a 2 + b 2 = with unit radiu, ee Fig 52, right The vector e i the norma of the pane through 3 and,a T e =0for a point on The coordinate d of thi vector i equa to the ditance of the ine from the origin 2, a can be proven geometrica b invetigating the copanar triange ( 2, z, 3 ) and (, e, 3 ) The pherica normaized homogeneou coordinate can be geometrica interpreted in a imiar wa The point ie on the unit phere S 2 in the three-dimeniona (uvw)-pace IR 3, ee Fig 53, eft bviou, the negative vector, ao repreenting the point,ieon the unit phere A point on the unit phere S 2, ecept thoe on the equator u 2 + v 2 =, repreent point of IR 2 The point on the equator have a we-defined meaning: when a point move awa from the origin 2 toward infinit, it pherica normaized homogeneou vector move toward the equator Thu, point on the equator of S 2 repreent point at infinit The are repreented b homogeneou coordinate vector with w =0, independent of their normaization If we take the union of a point in the Eucidean pane IR 2 and a point at infinit, we obtain what i caed the projective pane IP 2 Both can be repreented b the unit phere, with oppoite point identified The point ao ie on the unit phere in the three-dimeniona (abc)-pace IR 3, ee Fig 53, right It i the unit norma of the pane through 3 and the ine Thi pane interect the unit phere in a great circe poarit on the The reation between thi circe and the norma i caed poarit on the phere: the phere point i what i caed the poe of the circe; the circe i the poar of the point If a ine move toward infinit, it homogeneou vector move toward the c-ai Therefore, the origin 2 or it antipode repreent the ine at infinit Since ine are dua to point, cf beow, thi unit phere S 2 repreent the dua projective pane Thi viuaization of the projective pane i hepfu for undertanding certain contruction and wi be ued throughout

7 Section 5 Homogeneou Vector and Matrice 20 2 IR w 2 IR 2 v 2 c,w b, 3 - u 3 - a, Fig 53 Spherica normaization Left: 2D point The pherica normaized point ie on the upper hemiphere of the unit phere S 2 The antipode ao repreent the point Point on the equator u 2 + v 2 = repreent point at infinit Right: 2D ine ing in IR 2 The pherica normaized homogeneou vector i the unit norma of the pane through 3 and When eeing the pane ( 3 ) from, the origin 3 i on the eft ide of the ine Therefore, the antipode point repreent the ame ine, however, with oppoite direction 524 Line Joining Two Point, Interection of Two Line, and Eement at Infinit Now, et u determine the ine = (53) joining two point and The mbo (read: wedge) indicate the join If the two point are given with their homogeneou coordinate, thu () and (), the joining ine i given b = : = = S(), (54) a then the vector i perpendicuar to and, thu T =0and T =0; thu, the Eercie 5 ine pae through both point Matri S() i the kew mmetric matri induced b the 3-vector, S() = 3 0 (55) 2 0 A firt remark on notation: The mbo for the join of two geometric entitie i not unique in the iterature The wedge ign often i ued for the cro product in phic Thi i the reaon for uing it here for the join of two point, a the homogeneou coordinate of the reuting ine i obtained b the cro product berve: ome author ue the ign for the join of two point We wi overoad the mbo in two wa: () We wi ue it ao for the join of geometric entitie in 3D, name 3D point and 3D ine (2) We wi ao ue it for the correponding agebraic entitie Thu we coud have written in (54) the epreion =, keeping in mind how the operation i performed agebraica Apping the wedge to two 3-vector therefore i identica to determining their cro product, independent of what the two vector repreent A imiar reaoning ead to the homogeneou coordinate of the interection point, = m : = m = S()m, (56) of two ine () and m(m) given with homogeneou coordinate, where the ign (read: cap) indicate the interection A econd remark on notation: It woud be more conitent to ue the ign for the interection We found that in onger epreion it i difficut to ditinguih viua between the -mbo and the -mbo Therefore, we intentiona ue the ign for the interection, which correpond to the ign for the interection of et Thi appear intuitive, a the interection point i the et-interection of the two ine, taken a the et overoading of and

8 202 5 Homogeneou Repreentation of Point, Line and Pane of infinite man point Again we wi overoad the mbo both for 3D entitie, name 3D ine and pane, and for agebraic entitie m oo { m n } oo Fig 54 Interection = m (eft) and join = (centre) of two geometric entitie The interection of two parae ine (right) ead to the point at infinit The figure indicate thi point to be the reut of two different imiting procee Given the direction [u, v] T of the ine, we ma end up with = im w 0 [u, v, w] T =[u, v, 0] T But due to homogeneit, we ao have = im w 0 [u, v, w] T = im w 0 [ u, v, w] T =[ u, v, 0] T, the vector pointing in the oppoite direction In Sect 9, p343, oriented projective geometr, we wi ditinguih between thee two direction If a vector are pherica normaized, we arrive at a ver intuitive interpretation of the contruction equation, ee Fig 55 2D point correpond to and are repreented a point on the unit phere, wherea 2D ine correpond to great circe on the unit phere and are repreented a unit norma, thu ao a point on the unit phere The two contruction read a = N( ) and = N( m ), (57) which can be derived geometrica from the two graph in Fig 55 c,w c,w m a,u b,v a,u b,v Fig 55 Join of two point and interection of two ine on the projective pane IP 2 and it dua pane IP 2 uperimpoed on the ame unit phere (for the definition of IP 2 cf (538), p 209) Left: The 2D ine joining an two 2D point, on the projective pane i the great circe through thee point The norma of the pane containing the great circe i determined b the normaized cro product of the two homogeneou coordinate vector Right: The 2D interection point of an two 2D ine on the projective pane i the interection of the two great circe defined b their norma and m The direction of the interection point i the normaized cro product of the two norma of the pane containing the two great circe If the interection ie on the equator, it at coordinate i zero, indicating the point i at infinit; thu, the two ine are parae berve, the cro product are unique, a propert which we wi epoit when dicuing oriented eement Two parae ine do not interect in a point in the rea pane but at infinit, which cannot be repreented with inhomogeneou coordinate However, the cro product of their homogeneou coordinate eit Thi aow u to epicit repreent point at infinit Let the two ine, ee Fig 54, right, have the common norma n and two different ditance d and d 2 from the origin, with

9 Section 5 Homogeneou Vector and Matrice 203 n = [ ] 0 n (58) 0 perpendicuar to the norma n of the ine Then the homogeneou coordinate of their interection point are obtained from Eercie 54 [ ] [ ] [ ] [ ] n n (d2 d = = )n n = (59) d d Thu, the firt two component [u, v] T of the 3-vector of a point at infinit, u : = v, (520) 0 point at infinit repreent the direction toward the point at infinit, wherea the third component i zero Two point, ([u,v, 0] T ) and ([u,v, 0] T ), at infinit pan what i caed the ine at infinit, Eercie 52 ine at infinit : = 0 0, (52) a the cro product ied [0, 0,u v u v ] T, which i proportiona to [0, 0, ] T An other point at infinit ie on the ine at infinit A point with IR 3 \ 0, auming proportiona vector repreent the ame point, are eement of the projective pane IP 2 2 Reaoning with uch projective eement i at the heart of projective geometr A ine with IR 3 \ 0 are eement of the correponding dua projective pane Thi correpond to the notion of a vector pace for point and it dua for it inear form We wi epoit the concept of duait between point and ine and generaize it to both 3D point and the correponding tranformation berve, the coordinate of the ine = are not the ame a the coordinate of the ine =, but are their negative, a the cro product i anti-mmetric Thi aow u to ditinguih between ine with different direction if we foow certain ign convention For eampe, if we aume point to be repreented with poitive third component, we can ditinguih between the ign of the ine and, a their norma differ b 80 If we conitent conider the ign convention, we arrive at the oriented projective geometr, which i the topic of Chap 9 The 2D coordinate tem can be decribed b it origin 0 and it ae and, with coordinate identica to unit 3-vector e [3] 0 = 0 0 = e [3] 3, = i, 0 0 = e [3] 2, = 0 0 = e [3] (522) Note that the -ai een a a ine ha the Eucidean norma [0, ] T and pae through the origin, therefore = e 2, not = e We wi dicu the eement of coordinate tem in detai in Sect 59 projective pane dua projective pane oriented projective geometr 525 Duait of Point and Line Each geometric eement, operation, and reation ha what i caed a dua, indicated b () The concept of duait reut from the undering three-dimeniona vector pace IR 3 for 2 Mathematica, thi i the quotient pace IP 2 =(IR 3 \ 0)/(IR \ 0), indicating that a vector IR 3 \ 0 are taken a equivaent if the are mutipied with ome λ IR \ 0

10 204 5 Homogeneou Repreentation of Point, Line and Pane repreenting point, with the dua vector pace IR 3, which contain a inear form T, repreented b the vector A the two pace are iomorphic, there i a natura mapping D :IR 3 IR 3, name the identit mapping Given the point =[u, v, w] T, the ine which i dua to thi point ha the ame coordinate a the point u = = v (523) w and vice vera Therefore, a given 3-vector [r,, t] T can be interpreted a either a 2D point =[r,, t] T or a 2D ine =[r,, t] T ; the are dua to each other The point X and the dua ine are poitioned on oppoite ide of the origin, with ditance d and d to the origin mutiping to, thu d d =(Tabe 73, p298) The ine through X and perpendicuar to the ine pae through the origin, ee Fig 56 We wi ee that thi propert tranfer to 3D _ = - /r r - / Fig 56 Duait of point and ine in the pane Point and ine are dua wrt each other The have the ame homogeneou coordinate [r,, t =] T : ( =[r, ] T ) and (r + +=0), from which the interection point [ /r, 0] and [0, /] with the ai can be derived For pherica normaized homogeneou coordinate, we ee from Fig 53 that a point and it dua ine = are reated b poarit on the phere, cf Sect 523, p200 For more on duait cf Sect 56, p Tranformation of Point Linear mapping of homogeneou coordinate can be ued to repreent caica tranformation For eampe, we have the tranation T and the rotation R, = T ( ): = T with T([t,t ] T )= 0 t 0 t (524) 0 0 homogeneou matri matri repreentation for point and co α in α 0 = R ( ): = R with R(α) = in α co α 0, (525) 0 0 which can eai be verified berve, the two 3 3 matrice are homogeneou entitie: their mutipication with a caar μ 0doe not change the tranformation, a the reuting vector i mutipied with μ 0, eaving the reuting point unchanged Concatenation and inverion are obviou ea, ince the geometric tranformation are repreented a matri vector product berve, the join of two point in (54) i ao a matri vector mutipication, uggeting that the kew matri S() i a matri repreentation of the point We wi generaize thi

11 Section 52 Homogeneou Repreentation of Point and Line in 2D 205 propert for a baic geometric entitie and tranformation and derive a repreentation with homogeneou vector and matrice We wi ee that a genera inear mapping of homogeneou coordinate i traight inepreerving 527 Variance Propagation and Etimation A reation dicued o far are biinear in the eement of the coordinate invoved Therefore, we ma eai derive the Jacobian needed for variance propagation For eampe, from we immediate obtain the two Jacobian = = = S() = S() (526) ( ) = S() and ( ) = S() (527) The ine coordinate are noninear function of the point coordinate, name um of product Foowing Sect 276, p44, in a firt approimation if the two point are tochatica independent with covariance matrice Σ and Σ, we obtain the covariance matri of the joining ine, Σ = S(μ )Σ S T (μ )+S(μ )Σ S T (μ ) (528) f coure, we wi need to dicu the meaning of the covariance matri of a homogeneou entit and the degree of approimation reuting from the homogeneit of the repreentation Fina, we wi dicu etimation technique for homogeneou entitie: we ma ue the homogeneit of the contraint to advantage to obtain approimate vaue For eampe, et u aume N point n,n=,, N, are given and we want to determine a bet fitting traight ine Due to meaurement deviation, the point and the unknown ine wi not atif the contraint T n =0but wi reut in ome reidua T n = w n Minimizing the ength of the vector w =[w,, w n,, w N ] T, ie, minimizing the um of quared reidua w T w = N n= w2 n wrt the ine parameter under the contraint =, ead to minimizing the Raeigh ratio ( N ) T n= n T n r = T min, (529) which i known to be equivaent to oving an eigenvaue probem A thi method doe not take the poib different uncertaintie of the point into account, we ao need to dicu tatitica optima etimate The pecia tructure of the contraint wi impif the etup of the correponding etimation probem 52 Homogeneou Repreentation of Point and Line in 2D 52 2D Point D Line 207 Thi ection give forma definition of homogeneou coordinate of 2D point and 2D ine It compete the decription of the concept given o far for 2D pace: the 3-vector etabihing the projective pane for point and the dua projective pane for ine Both contain point and ine at infinit