# Lecture-11 - Problem Lecture-11 Structure from Motion Given...

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1 Copyright Mubarak Shah 2003 Lecture-11 Structure from Motion Copyright Mubarak Shah 2003 Problem ! Given optical flow or point correspondences, compute 3-D motion (translation and rotation) and shape (depth). Copyright Mubarak Shah 2003 Tomasi and Kanade Orthographic Projection Copyright Mubarak Shah 2003 Assumptions ! The camera model is orthographic. ! The positions of “p” points in “f” frames (f>=3), which are not all coplanar, have been tracked. ! The entire sequence has been acquired before starting (batch mode). ! Camera calibration not needed, if we accept 3D points up to a scale factor. Copyright Mubarak Shah 2003 Tomasi & Kanade ! ! ! " # \$ \$ \$ % & ( V U W W ( u 11 " u 1 P # u F 1 " u FP v 11 " v 1 P # v F 1 " v FP & % \$ \$ \$ \$ \$ \$ \$ # " ! ! ! ! ! ! ! * + P p F f v u fp fp , , 1 , , , 1 | ) , ( " " ( ( Image point Copyright Mubarak Shah 2003 Tomasi & Kanade fP fP fP fP fP fP b v v a u u ( ( ~ ~ , ( ( P p p f u P a 1 1 , ( ( P p p f v P b 1 1

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2 Copyright Mubarak Shah 2003 ) ( ) ( f P T f fP f P T f fP t s j v t s i u ( ( ) , , ( P P p p Z Y X s ( 3D world point Orthographic projection k f ( i f - j f Copyright Mubarak Shah 2003 Copyright Mubarak Shah 2003 f fP fp a u u ( ~ ) ( 1 ) ( 1 f q P q T f f p T f t s i P t s i ( , ( ! " # \$ % & ( , ( P q q P T f s P s i 1 1 P T f s i ( Origin of world is at the centroid of object points Copyright Mubarak Shah 2003 P T f fP P T f fP s j v s i u ( ( ~ ~ ! ! ! " # \$ \$ \$ % & ( V U W ~ ~ ~ Copyright Mubarak Shah 2003 ./ RS s s j j i i W P T f T T f T ( ! ! ! ! ! ! ! ! " # \$ \$ \$ \$ \$ \$ \$ \$ % & ( " # # 1 1 1 ~ 2FX3 3XP P T f fP P T f fP s j v s i u ( ( ~ ~ ! ! ! " # \$ \$ \$ % & ( V U W ~ ~ ~ Rank of S is 3, because points in 3D space are not Co-planar Copyright Mubarak Shah 2003 Rank Theorem Without noise, the registered measurement matrix is at most of rank three.
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Lecture-11 - Problem Lecture-11 Structure from Motion Given...

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