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```					Advanced Computer Vision

Structure from Motion
Structure from Motion
   Geometric structure-from-motion problem:
using image matches to estimate:
   The 3D positions of the corresponding scene
points in some fixed coordinate (the scene
structure)
   The projection matrices associated with the
camera observing them (the motion of the
points relative to the cameras)
Two Types
   Affine structure from motion (Chapter 12)
   Projective structure from motion (Chapter 13)
Basic Assumption
   The cameras’ positions and possibly their
intrinsic parameters are a priori unknown and
may change over time.
   Different from,say, stereopsis, where the
camera parameters are known so we can
focus on the correspondence problem.
   Chapters 12 and 13 ignore the
correspondence problem, assuming that the
projections of n points have been matched
across m pictures.
Applications
   Image-based rendering: a video clip recorded
by a hand-held camcorder, possibly zooming
during the shoot, is used to capture the
shape of an object and render it under new
viewing conditions. (Chapter 26).
   Active vision system: robot probes.
Affine Structure from Motion

Problem: estimate the m 2x4 matrices Mi i and
the n positions Pjj from the mxn correspondences pij .
ij

2mn equations in 8m+3n unknowns

Overconstrained problem, that can be solved
using (non-linear) least squares!
Ambiguity of Solution
If M i and P are solutions,
j

So are M’i and P’ where
j

and

Q is an affine
transformation.
The Projective Structure-from-Motion Problem
Given m perspective images of n fixed points P we can write
j

Problem: estimate the m 3x4 matrices M i and
the n positions P j from the mn correspondences p .
ij
2mn equations in 11m+3n unknowns

Overconstrained problem, that can be solved
using (non-linear) least squares!

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