Motion and Flow II. Structure from Motion. Passive Navigation and Structure from Motion. rot
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1 Moto ad Flow II Sce fom Moto Passve Navgato ad Sce fom Moto = + t, w F = zˆ t ( zˆ ( ([ ] =? hesystemmoveswth a gd moto wth aslat oal velocty t = ( U, V, W ad atoalvelocty w = ( α, β, γ. Scee pots R = ( X, Y, poject oto magepots = ( x, y, f ad the 3D velocty R& = ( U, V, W of a sceepot s obseved the mage asvelocty & = (, v..
2 Image Flow de to Rgd Moto he velocty of a pot wth espect to the XY coodate system s R & = t w R X & = U B + γy Y& = V γx + α & = W αx + βx X Y Let f =, the x = y = ; = x & v = y & X & X & X& U W = = = β + γy x αy+ βx Y& Y& Y & V W v = = = γx+ α y αy + βx U + xw = + αxy β ( x + + γy = + Slg ambgty: We V + yw v v = + α ( y + βxy γx = + v compte the aslato oly p to a sle facto vecto otato: = R, whee = R z R z (Kt, K gve the same flow as (t,. & = ( z ( t + z ( ( w R z aslatoal flow feld Rotatoal flow feld W W = ( x x, ( y y U V whee( x, y = f, f s the focsof expaso ( W W o focsof coacto (FOC. α β f, f s the pot whe e the ato axs peces the mage plae (. γ γ Classl Sce fom Moto Establshed appoach s the eppola mmzato: he deated flow shold be paallel to the aslatoal flow. E - E - t Uqeess Let theebe two aslatost, t adtwosfaces, t = ( U, V, W t = ( U, V, W U + xw V + yw = v = U + xw V + yw = v = U + xw U + xw = V + yw V + yw ( U + xw ( V + yw = ( U + xw ( V + yw U V xv W yu W + xyw W = U V xv W yu W + xyw W mst hold fo all x ad y UV = UV U : V : W = U : V : W VW = VW = K U W = U W t = kt ad = k A aslatoal flow feld detemes the motos of the mea qely p to a slg facto. he aslatoal Case A least sqaes fomlato xw U yw V + v dx dy m Sbsttte a = U + xw b = V + yw a b + v m We mmze d Step :Mmze wth espect to. (Fd the legth of fo whch d wold be mmzed. a a b b + v = Sbsttte back ( b va dx dy m a + b a + b = a + vb Step : Dffeetate wth espect to U, V, W, set expesso to zeo. ( b va( a + vb Let K = a + b ( I ( V + yw K dx dy = II ( U + xw K dx dy = III ( yu + xv K dx dy = 3 lealy depedet eqatos ( U I + V II + W III =
3 he Rotatoal Case I max fom ( + ( v v m αxy+ β ( x + γy = v α( y + + βxy+ γx = xy ( y + ( x + xy A w = w = ( A A A α y β = x v γ Mmzato of eppola dstace v v o, vecto otato he Geeal Case v dxdy m (( t ( & w d m Moto Estmato echqes Leazato (sa Hag 984, Loget-Hggs 98 the dscete se x' Ex, wheee = R = x' Ex = a e wth a = ( x x, x y, x, y x, y y, y, x, y,. LSmmzato ( a e msolve fo E.. Obta fom E aslato ad ato sg SVD. Pazdy (98, Bge Bha (99, Nelso Alomoos (988, Heege Jepso (99: Decomposto of flow feld to aslatoal ad atoal compoets. aslatoal flow feld has a ceta sce: All vectos ae emaatg fom a pot. Ethe seach the space of atos o the space of aslatoal dectos. Loget-Hggs Pazdy (98, Waxma (987: Paamec model fo lol sface patches solve lolly fo moto paametes ad sce Optl flow dffcltes he apee poblem Depth dscottes aslatoal Nomal Flow = I the se of aslato each omal flow vecto cosas the loto of the to a half-plae. Itesecto of half-plaes povdes. Egoestmato fom omal flow pattes defed o the sg of omal flow alog patcla oetato felds postve depth cosat classes of oetato felds: copot vectos ad coaxs vectos 3
4 Copot vecto felds copot vectos Copot vectos : v cp( t pepedcla to aslatoalflow feld defedby t v ( t = zˆ ( t = zˆ ( zˆ cp ( t he compoets of flow alog v cp ( t amot to vcp & = ( t t + ( w ( t vcp vcp hs the aslatoal compoet s sepaated by a le to postve ad egatve vales ( t t = O he atoal compoet s sepaated by a secod-ode cve ( w ( t = to postve ad egatve vales Patte wth postve aeas, egatve aeas, ad some defed aeas aslatoal compoet (, s atoal compoet (, s Coaxs vecto felds Coaxs vectos: v ( w? pepedcla to ato v ( w? = zˆ ( w? = zˆ ( ( w? compoetsof flowalogv ( w amot to v & = ( w w + ( t ( w v v he hs the aslatoal compoet s sepaated by a secod-ode cve ( t ( w? = ad the atoal compoet s sepaated by a le ( w = w? Itesecto of pattes povdes the. (, s hee coaxs vecto felds (α (β (γ (a (b (c : Nega tve : Postve : Do't kow 4
5 Depth vaablty cosat Eos moto estmates lead to dstoto of the scee estmates. he dstoto s sch that the coect moto gves the smoothest (least vayg scee sce. Depth estmato Scee depth be estmated fom omal flow measemets: = = + (?ˆ = ˆ (ˆ t Vsal Space Dstoto ˆ = D, D = ( tˆ [ ( t ( d? ] Wog 3D moto gves se to a gged (smooth depth fcto (sface. he coect 3D moto leads to the smoothest estmated depth. he eo fcto A omal flow measemet: = + he eo fcto to be mmzed: Θ = W ( R ˆ Global paametes: ˆt,?ˆ Lol paamete: Ẑ Eo fcto evalato Gve a aslato ddate tˆ, each lol depth be compted as a lea fcto of the ato?ˆ. We obta a secod ode fcto of the ato; ts mmzato povdes both the ato ad the vale of the eo fcto. 5
6 Hadlg depth dscottes Gve a ddate moto, the scee depth be estmated ad fthe pocessed to fd depth dscottes. Splt a ego f t coespods to two depth vales sepaated space. he algothm Compte spato-tempoal mage devatves ad omal flow. Fd the decto of aslato that mmzes the depth-vaablty cteo. Heachl seach of the D space. Iteatve mmzato. Utlze cotty of the solto tme; sally the moto chages slowly ove tme. Soces: Ho (986 Femlle.ps.gz Femlle.ps.gz (pattes o omal flow bodsky.ps.gz (depth vaablty cosat 6
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