# Lecture-2 - PART I Measurement of Motion Contents Image...

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PART I Measurement of Motion Contents • Image Motion Models • Optical Flow Methods – Horn & Schunck – Lucas and Kanade – Anandan et al – Szeliski – Mann & Picard • Video Mosaics

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3-D Rigid Motion Y X Z Y X Z t1 t2 3-D Rigid Motion ! ! ! " # \$ \$ \$ % & ( " # \$ \$ \$ % & )( " # \$ \$ \$ % & " # \$ \$ \$ % & ) " # \$ \$ \$ % & X Y Z R X Y Z T r r r r r r r r r X Y Z T T T X Y Z 11 12 13 21 22 23 31 32 33 Translation (3 unknowns ) Rotation matrix (9 unknowns )
Rotation X * + Y R R Z ( , , ) X Y Z ( , , ) ! ! ! X Y Z sin cos R Y R X ( ( Y’ Y X’ X sin cos cos sin ) sin( sin sin cos cos ) cos( , ) , ( ) , ( ! , - , ( ) , ( ! R R R Y R R R X , ) , ( ! , - , ( ! cos sin sin cos Y X Y Y X X & % \$ \$ \$ # " & % \$ \$ \$ # " , , , - , ( & % \$ \$ \$ # " ! ! ! Z Y X Z Y X 1 0 0 0 cos sin 0 sin cos Rotation (continued) Y Z u’ v’ W , , X R ( - " # \$ \$ \$ % & cos sin sin cos ,, 0 0 0 0 1 Y Z u’ v W’ . X R ( - " # \$ \$ \$ % & cos sin sin cos .. 0 0 1 0 0 R ( " # \$ \$ \$ % & 100 010 001 Y v W Z X u

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Euler Angles & % \$ \$ \$ # " - - ) ) - ( ( / . 0 cos cos sin cos sin sin cos cos sin sin cos cos sin sin sin cos sin sin sin cos sin cos cos sin sin sin cos cos cos X Y Z R R R R & % \$ \$ \$ # " - - - ( 1 1 1 R cos , 1 1 sin ,, 1 if angles are small( Image Motion Models
Displacement Model Image Formation: Orthographic Projection (X,Y,Z) Image Plane image y World point X x Y y ( (

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Orthographic Projection (x,y)=image coordinates, (X,Y,Z)=world coordinates x X yY ( ( ! () ) ) ! ( ) ) ) x r x r y r Z T y r x r y r Z T X Y 11 12 13 21 22 23 2 4 3 1 2 1 b y a x a y b y a x a x ) ) ( !
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Lecture-2 - PART I Measurement of Motion Contents Image...

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