Geometry Reconstruction

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Geometry Reconstruction
March 22, 2007
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An important problem Determine the epipolar
geometry. That is, the correspondence between a
point on one camera and its epipolar line on the
other camera.
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Use eight-point algorithm, we can recover the
fundamental matrix F.
Knowing the fundamental matrix is lot easier.
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Knowing the fundamental matrix, and a pair of
corresponding pixels, we would like to obtain the
3D position of the corresponding scene point.

• There are three cases
• Calibrated cameras and extrinsic parameters are
  known.
• Calibrated cameras with unknown extrinsic
  parameters
• Uncalibrated cameras.

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The geometric reconstruction is absolute (without
ambiguity).
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The geometric reconstruction is only up to a
scale.
Main point we dont know T (the baseline of the
system) and we have no way to ascertain the scale
of the scene.
We have only the essential matrix or fundamental
matrix to work with.
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pr, pl are the left and right image points in
camera coordinates
Intrinsic parameters allow to go from pixel
coordinates to camera coordinates.
We get E from a few correspondences. But E is
only determined up to a scale!
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We have no T, no information on scale.
From E Et E St S
Find a set of (T, R).
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We have two images, and thats it!
The reconstruction is only up to a global
projective transformation.
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The ambiguity is easy to see.
Only F and pr, pl are known and F is known only
up to a scale. (xl, xr are 4-by-1 vectors in
homogeneous coordinates).
H a nonsigular 4x4 matrix
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Projective Transform
Given a 3D point, x(x1, x2, x3). In homogenous
coordinates, it is x (x1, x2, x3, 1). If Hx
(y1, y2, y3, y4), then the image of the 3D point
x under the projective transform H is (y1/y4,
y2/y4, y3/y4).
It is a 15-dimensional (non-linear)
transformation group.
It is important that we know there is ambiguity
in reconstruction, but it is only up to a
15-dimensional transformation group. Ambiguity
is global not local.
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You have a weird camera.
A better camera perhaps.
Impossible result
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Normalize Prt to I3 0 . Find a Plt that
satisfy the equations above. Pl S F e
for some skew-symmetric matrix S and e the left
epipole will do
Let S e x
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• More about the class
• We will cover two (and half) more topics Shape
  from shading (differential geometry), optical
  flows (motions) and perhaps recognition (Chapters
  10-13 in Horns book )
• Office hours Normally 2-4 on Friday. But come
  by anytime you need to discuss issues/problems
  with me. (Send email to see if I am in office.)
• Assignments To be discussed.
• Solutions Will be available starting today. TA
  has a busy semester so far.
• Problem 4 will be available shortly ( couldnt
  make the Monday deadline).

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• More sophisticated method using other constraints
  will reduce the projective ambiguity down to a
  global unknown similarity transform. Assume
• both cameras have the same intrinsic parameters
• Sufficiently many orthogonal lines have been
  identified.

Covered in advanced vision class.
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• (Projective) Reconstruction from a possibly large
  set of images.
• Problem
• Set of 3D points, Xj
• Set of cameras Pi
• For each camera, image points xji (the input
  data)
• Find Pi, Xj, such that Pi Xj xji

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N views and M points Total number of
parameters 11N3M. Number of Equations
NM With enough points and views, we have number
of equations gt total number of parameters. The
problem is over-constrained. (What about N2?)
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m is the number views and n is the number of
points.
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Input A video sequence
Output Camera Matrices and 3D locations of the
points (up to a global similarity transform).
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1. Intensity Correlation
2. Edge Matching

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Distorted Subwindows if disparity is not constant
(complicates correlation)
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