Geomerc Algebra Ngel Boso /5/07 Geomerc Algebra Prof. Ngel Boso Camera Nework esearch Group Meeg Nov. 8 & 5, 007 Sraegy: Use Clfford algebra o develop varas for projecve rasformaos. (eferece: J. Laseby, e al., New Geomerc Mehods for Compuer Vso: a applcao o srucure ad moo esmao, 996) Idea: Bass-free geomery (sce s bass-free, varas wll aurally arse, herefore should be useful for our purposes.) Gve: vecors, a, b a b (grade 0) a x b (grade ) (oly vald for 3D) More geerally, roduce a /\ b (a wedge b) s a dreced area sweepg from a o b.(see Fg. ) (a parallelogram) (hs s a bvecor) Ad, b /\ a - a /\ b () I geeral, we have a /\ a /\ /\ a m- /\ a m (mul-vecor) (a parallelopped) (grade m) Defe: ab a b + a /\ b () follows ha: ab a /\ b f a, b are orhogoal, ad (3) so he, ab a b f a, b are parallel (4) ba b a + b /\ a a b - a /\ b (5) of 7
Geomerc Algebra Ngel Boso /5/07 a b ½ (ab + ba) (assocaed er produc) (6) a /\ b ½ (ab - ba) (assocaed ouer produc) (7) Defo: A geomerc algebra has a produc sasfyg: (ab)c a(bc) aλ λa, for scalar λ a(b + c) ab + ac a a (b + c)a ba + ca ad he assocaed er ad ouer produc are defed by (6) ad (7). If {σ,, σ } s ay orhoormal bass for, we ge a bass for he algebra: {, {σ }, {σ /\ σ j j}, {σ /\ σ j /\ σ k #{, j, k} 3}, ec.} whch wll have elemes. The hghes grade eleme s σ /\ /\ σ. Example: 3 The bass wll have 3 8 elemes: {, σ, σ, σ 3, σ σ, σ σ 3, σ σ 3, σ σ σ 3 } (Noe: σ σ 0 sce he bass s orhoormal, hece σ σ σ /\ σ.) (Also, sce he bass s orhoormal, σ σ.) 8 (Ths s wh a srage mulplcao.) I follows ha: (σ σ σ 3 ) σ σ σ 3 σ σ σ 3 -σ σ σ σ 3 σ σ 3 -σ σ σ σ 3 -σ σ σ 3 - Also, oe ca verfy ha σ σ σ 3 commues wh every eleme of he algebra (wll be rue for odd). Call σ σ σ 3 (.e., ) I follows ha: σ 3 σ σ, σ σ σ 3, σ σ 3 σ (8) These bvecors roae vecors her ow plae by 90. Noe: se I σ, J - σ, K σ 3, he we ge: I J K IJK - (Hamlo s quaeros). The algebra wh bass, I, J, K s of our geomerc algebra. 4 wh a srage mulplcao,.e., a sub-algebra of 7
Geomerc Algebra Ngel Boso /5/07 Geeral oaos: Frs, sudy reflecos. (easer) (Ay roao s he produc of wo reflecos.) Cosder a vecor, a refleced he plae orhogoal o he u vecor, o ge a (see Fg. ). The, we ca breakup a o compoes, a a +, (9) a he compoes perpedcular ad parallel o, ad he, a a. (0) a Sce s a u vecor, he, ad a a ( a + /\ a) ( a) + ( /\ a), whch mples ha a a ( /\ a) - ( a) a -( a) - ( /\ a) -( a + /\ a) -a. () Thus, reflecg he plae perpedcular o s he map (Noe: hs works ay dmeso) a a a Suppose you reflec he plae perpedcular o, ad he he plae perpedcular o m. (he produc of wo reflecos) Ths s a roao. The, a maps o -m(-a)m (m)a(m) a, where m, m. () s called a roor ( s also a mulvecor) ( ecapsulaes he formao abou he roao.) Noes: ) oly has eve grade elemes (scalar, bvecor, ec.) ad mm m m mm, ad ( s he mulplcave verse of ) ) aa a hadles roaos ay dmeso. 3) Ca roae elemes of ay grade, o jus vecors very geeral! 3 of 7
Geomerc Algebra Ngel Boso /5/07 Cosder he problem of roag o, say by agle θ (see Fg. 3). Wha s? Noe ha oe soluo s +, bu we also eed, so ry ( + ), he ( + )( + ) ( + + + ) ( + ) ( + ) So, + θ exp, (3) ( + ) where s orhogoal o he plae cu ou by ad (he axs of roao). /5/07 Example: Camera moo from wo scee projecos wh rage daa kow. (3D-o-3D correspodeces) Assume cameras wh opcal ceers a O ad O, wh respecve Axes {σ, σ, σ 3 } ad { σ, σ, σ 3 }, where σ 3 s orhogoal o, he mage plae for camera oe, ad smlarly for camera wo. (See Fg. 4.) Le O P, x O M, O O (4) The frame {σ, σ, σ 3 } s roaed o a frame { σ, σ σ } correspodg roor, we have a O, where for beg he, 3 σ σ σ σ (5) Le ( P O The, ( (6) 4 of 7
Geomerc Algebra Ngel Boso /5/07 5 of 7 The observables (measuremes) he {σ } ad { } σ frames are: σ ad σ ( Defe he vecor he {σ }frame as: ( ) ( ) ( ) ( σ σ σ (7) earragg: + (8) For smplcy, assume. Suppose we have po correspodeces he wo vews, ad he coordaes he vews, { } ad { }, ( ), are kow. We wa o recover he camera moo,.e., fd ad ha mmze he sum: ( ) [ ] S (9) To fd hs mmzao, we dffereae wr ad, he se o zero. Frs, wr : ( ) [ ] ( ) S (0) whch wll be zero whe, ( ) [ ] 0 Solve for : [ ] 0 + [ ] [ ] ()
Geomerc Algebra Ngel Boso /5/07 where ad () So, he opmal s he cerods of he pos. (Noe: hs s a kow resul ha we have recovered here.) (Noe: here wll be a ssue wh robusess sce oe ouler ca adversely affec he resuls a so-called black swa ) The, dffereag wr, ca be show ha we ge: [ ( ) ] 0 (3) Plug- he opmal ad we ge: [ u ] v 0 (4) where u ad v We ca fd he roor,, usg he Sgular Value Decomposo (SVD) o he marx F, defed erms of he u s ad v s as: F β σ f ( σ ) ( σ u )( σ v ) β β (5) he he SVD gves F USV T, ad he VU T. Paper for ex me (Ma): A geomerc approach for he heory ad applcaos of 3D projecve varas Bayro-Corrochao, Eduardo; Baarer, Vladmr Joural of Mahemacal Imagg ad Vso, v. 6,., March 00, pp. 3-54 6 of 7
Geomerc Algebra Ngel Boso /5/07 Appedx: Fg. - a /\ b Fg. Vecor a refleced he plae orhogoal o. Fg. 3 oao of o. Fg. 4 Objec po vewed from wo camera posos. 7 of 7