Multi-View Geometry (Cont.)

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Multi-View Geometry (Cont.)
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Stereo Constraints (Review)
p
?
p
Given p in left image, where can the
corresponding point pin right image be?
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Stereo Constraints (Review)
M
Image plane
Y1
p
O1
Z1
X1
Focal plane
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Epipolar Constraint (Review)
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From Geometry to Algebra (Review)
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From Geometry to Algebra (Review)
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Linear ConstraintShould be able to express as
matrix multiplication.
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The Essential Matrix (Review)
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The Essential Matrix

• Based on the Relative Geometry of the Cameras
• Assumes Cameras are calibrated (i.e., intrinsic
  parameters are known)
• Relates image of point in one camera to a second
  camera (points in camera coordinate system).
• Is defined up to scale
• 5 independent parameters

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The Essential Matrix
Similarly p is the epipolar line
corresponding to p in theright camera
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The Essential Matrix
Similarly,
Essential Matrix is singular with rank 2
e
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Small Motions and Epipolar Constraint
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Motion Models (Review)
3D Rigid Motion
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Small Motions
Velocity Vector
Translational Component of Velocity
Angular Velocity
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Translating Camera
Focus of expansion (FOE) Under pure translation,
the motionfield at every point in the image
points toward the focus ofexpansion
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FOE for Translating Camera
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FOE from Basic Equations of Motion
q
p
v
O
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What if Camera Calibration is not known
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Review Intrinsic Camera Parameters
Y
M
Image plane
C
Z
v
X
Focal plane
m
u
P
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Fundamental Matrix
If u and u are corresponding image coordinates
then we have
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Fundamental Matrix
Fundamental Matrix is singular with rank 2
In principal F has 7 parameters up to scale and
can be estimatedfrom 7 point correspondences
Direct Simpler Method requires 8 correspondences
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Estimating Fundamental Matrix
The 8-point algorithm
Each point correspondence can be expressed as a
linear equation
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The 8-point Algorithm
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Shape from Stereo
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Pinhole Camera Model
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Basic Stereo Derivations
Derive expression for Z as a function of x1, x2,
f and B
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Basic Stereo Derivations
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Basic Stereo Derivations
Define the disparity