3D Motion Estimation

# 3D Motion Estimation - 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 L v L D o l(t) l(t) An object of height An object of height L L moves with constant velocity velocity v: v: At time At time t=0 t=0 the object is at: D(0) = D o At time t it is at D(t) = D o – vt It will crash with the camera at time: o t=0 t=0 t D(t)

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

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Time to Collision f L v L D o l(t) l(t) t=0 t=0 t D(t) And And time to collision time to collision : Can be directly measured from image Can be found, without knowing Can be found, without knowing L or or D o or 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 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 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 + = - - + + + - = + = + + - + + - = ( 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 P p Z z Using pinhole camera equation:

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Let things move: f P V p v Z z The The relative velocity relative velocity of P wrt camera: Translation velocity Rotation angular
3D Relative Velocity: f P V p v Z z The relative velocity of P wrt camera:

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Motion Field: the velocity of p f P V p v Z z Taking derivative wrt time:
Motion Field: the velocity of p f P V p v Z z

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Motion Field: the velocity of p f P V p v Z z
Motion Field: the velocity of p f P V p v Z z Translational component Scaling ambiguity (t and Z can only be derived up to a scale Factor)

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## This note was uploaded on 01/13/2012 for the course CMSC 426 taught by Professor Staff during the Winter '08 term at Maryland.

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3D Motion Estimation - 3D Motion Estimation 3D model...

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