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3D Motion Estimation-1

# 3D Motion Estimation-1 - 3D Motion Estimation 3D model...

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3D Motion Estimation

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3D model construction
3D model construction

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Video Manipulation

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Visual Motion Allows us to compute useful properties of the 3D world, with very little knowledge. Example: Time to collision
Time to Collision f f L L v v L L D D o l(t) l(t) An object of height An object of height  L  L  moves with constant   moves with constant  velocity  velocity  v: v: At time  At time  t=0 t=0  the object is at:  the object is at:     D(0) = D D(0) = D o At time  At time  t t  it is at   it is at  D(t) = D D(t) = D o  – vt  – vt It will crash with the camera at time: It will crash with the camera at time:     D( D( τ τ ) = D ) = D o  – v  – v τ τ  = 0  = 0     τ τ     = D = D o /v /v t=0 t=0 t t D(t) D(t)

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Time to Collision f f L L v v L L D D o l(t) l(t) t=0 t=0 t t D(t) D(t) The image of the object has size l(t): The image of the object has size l(t): Taking derivative wrt time: Taking derivative wrt time:
Time to Collision f f L L v v L L D D o l(t) l(t) t=0 t=0 t t D(t) D(t) And their ratio is: And their ratio is:

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Time to Collision f f L L v v L L D D o l(t) l(t) t=0 t=0 t t D(t) D(t) And  And  time to collision time to collision : : Can be directly  Can be directly  measured from  measured from  image image Can be found, without knowing  Can be found, without knowing  L L  or   or  D D o  or   or  v v  !!  !!

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Structure from Motion = × × r t z u ˆ 1 tr Z [ ] ( 29 ( 29 ( 29 r r z u × × × = ω ˆ 1 rot F t, ϖ r Z rot tr u u u + = u
Passive Navigation and Structure from Motion ( 29 ( 29 . velocity rotational and , velocity ional h translat motion wit rigid a with moves system The , T z y x T z y x ω ω ω t t t , , ϖ = = t ( 29 ( 29 f y x Z Y X T , , points image onto project , , points Scene = = r R ( 29 image in the observed is point scene a of , , velocity 3D the and z y x V V V = R ( 29 . 0 . , velocity as v u = r

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Image Flow due to Rigid Motion The velocity of a point with respect to the XYZ coordinate system is X X W Z Z X V Y Y BZ U X β α α γ γ + - - = + - - = + - - = × - - = R t R ϖ y v x u Z Y y Z X x f = = = = = ; then , 1 Let + - - - + - - = - = = + - - - + - - = - = = x y Z W y x Z V Z Z Y Z Y Z Y v x y Z W x y Z U Z Z X Z X Z X u β α α γ β α γ β 2 2 ( 29 ( 29 rot tr 2 rot tr 2 1 1 v Z v x xy y Z yW V v u Z u y x xy Z xW U u + = - - + + + - = + = + + - + + - = γ β α γ β α ( 29 ( 29 ( 29 ( 29 r r z r t z z R r z R R z R r × × × + × × = = = ϖ 0 0 0 0 0 1 where , 1 : notation in vector Z Scaling ambiguity: We can compute the translation only up to a scale factor ( K t , KZ ) give the same flow as ( t , Z ).
Consider a 3D point P and its image: f f P P p p Z Z z z Using pinhole camera equation: Using pinhole camera equation:

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