# Lecture-3 - Lecture 3 Displacement Models(contd Affine...

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Lecture 3 Displacement Models (contd)

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Affine Mosaic Projective Mosaic
Instantaneous Velocity Model 3-D Rigid Motion ! ! ! " # \$ \$ \$ % & ! ! ! " # \$ \$ \$ % & ! ! ! " # \$ \$ \$ % & ( ( ( ) ! ! ! " # \$ \$ \$ % & * * * Z Y X T T T Z Y X Z Y X 1 1 1 + , - ! ! ! " # \$ \$ \$ % & ! ! ! " # \$ \$ \$ % & ! ! ! " # \$ \$ \$ % & ( ( ( ) ! ! ! " # \$ \$ \$ % & ( * ( * ( * Z Y X T T T Z Y X Z Z Y Y X X 0 0 0 ! ! ! " # \$ \$ \$ % & ! ! ! " # \$ \$ \$ % & . . . / 0 1 1 1 2 3 ! ! ! " # \$ \$ \$ % & ! ! ! " # \$ \$ \$ % & ( ( ( ) ! ! ! " # \$ \$ \$ % & * * * Z Y X T T T Z Y X Z Y X 1 0 0 0 1 0 0 0 1 0 0 0 ! ! ! " # \$ \$ \$ % & ! ! ! " # \$ \$ \$ % & ! ! ! " # \$ \$ \$ % & 4 4 ( 4 ( 4 4 4 ( ) ! ! ! " # \$ \$ \$ % & 3 2 1 1 2 1 3 2 3 0 0 0 V V V Z Y X Z Y X ! ! ! 3 2 1 2 1 3 1 3 2 V X Y Z V Z X Y V Y Z X 4 ( 4 ) 4 ( 4 ) 4 ( 4 ) ! ! !

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3-D Rigid Motion 3 2 1 2 1 3 1 3 2 V X Y Z V Z X Y V Y Z X 4 ( 4 ) 4 ( 4 ) 4 ( 4 ) ! ! ! V X X 5 4 ) ! Cross Product ! ! ! " # \$ \$ \$ % & 4 4 4 ) 4 ! ! ! " # \$ \$ \$ % & ) 3 2 1 , Z Y X X Orthographic Projection (u,v) is optical flow 2 1 3 1 3 2 3 2 1 2 1 3 1 3 2 V Z x y v V y Z x u V X Y Z V Z X Y V Y Z X 4 ( 4 ) ) 4 ( 4 ) ) 4 ( 4 ) 4 ( 4 ) 4 ( 4 ) ! ! ! ! ! X x Y y ) )
Plane+orthographic(Affine) cY bX a Z ) 1 4 1 3 3 1 2 2 3 2 2 2 1 2 1 1 4 ( ) 4 ( 4 ) 4 ( ) 4 ( 4 ) 4 ) 4 ) c a b a a V b c a b a a V b b Ax u ) Z x V v y Z V u 1 3 2 3 2 1 4 ( 4 ) 4 ( 4 ) y a x a b v y a x a b u 4 3 2 2 1 1 ) ) (1) Show this Perspective Projection (arbitrary flow) Z fY y Z fX x ) ) Z Z y Z Y f Z Z fY Y fZ y v Z Z x Z X f Z Z fX X fZ x u ! ! ! ! ! ! ! ! ! ! ( ) ( ) ) ( ) ( ) ) 2 2 2 1 2 3 3 1 2 2 2 1 3 3 2 1 ) ( ) ( y f xy f y Z V x Z V f v x f xy f y x Z V Z V f u 4 ( 4 ( 4 4 ( ) 4 4 ( 4 ( ( 4 ) 3 2 1 2 1 3 1 3 2 V X Y Z V Z X Y V Y Z X 4 ( 4 ) 4 ( 4 ) 4 ( 4 ) ! ! !

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Plane+Perspective (pseudo perspective) y a c x a b a Z cY bX a Z ( ( ) ) 1 1 uf V Z V Z xy f f x vf V Z x V Z y f f y ) ( ( ( ) ( ( ( () 1 2 3 3 12 2 2 13 3 2 1 2 44 4 4 2 5 4 8 7 6 5 2 4 3 2 1 y a xy a y a x a a v xy a x a y a x a a u ) ) (2) Show this 3D Scenes When the scene is neither planar not distant, the relation between two consecutive frames is described by the fundamental matrix .
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## This note was uploaded on 10/04/2011 for the course CAP 6411 taught by Professor Shah during the Spring '09 term at University of Central Florida.

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Lecture-3 - Lecture 3 Displacement Models(contd Affine...

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