Advanced Computer Vision

1
Advanced Computer Vision

• Structure from Motion

2
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)

3
Two Types

• Affine structure from motion
• Projective structure from motion

4
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.
• Ignore the correspondence problem, assuming that
  the projections of n points have been matched
  across m pictures.

5
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.
• Active vision system robot probes.

6
Affine Structure from Motion
2mn equations in 8m3n unknowns
Overconstrained problem, that can be solved using
(non-linear) least squares!
7
Ambiguity of Solution

So are M and P where
i
j
and
Q is an affine transformation.
8

The Projective Structure-from-Motion Problem
Given m perspective images of n fixed points P
we can write
j
2mn equations in 11m3n unknowns
Overconstrained problem, that can be solved using
(non-linear) least squares!