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Dont get lost! What are we doing?
Classical (Euclidean) tools
pb. ? camera model ? calibration ? separation
(int/ext) ? pose
Numerical tools
Projective geometry
Calibrated cameras
Uncalibrated cameras
Euclidean geometry
Projective geometry
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First application camera pose estimation
Pose estimation extrinsic calibration
navigation by reference where is the camera?
where am I in the scene?
In vision, robotics, virtual reality,
Two most popular methods

• 3-point algebraic method
• 4 coplanar points linear method

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Pose estimation calibration of only extrinsic
parameters

• Given
• Estimate R and t

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3-point algebraic method
3 reference points 3 beacons

• First convert pixels u into normalized points x
  by knowing the intrinsic parameters
• Write down the fundamental equation
• Solve this algebraic system to get the point
  distances first
• Compute a 3D transformation

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Fundamental euclidean geometric constraint
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Solving the algebraic system by elimination
a polynomial of degree m and a polynomial of
degree n leads to a polynomial of degree mn

• (using a symbolic computation software (Maple or
  Mathematica))
• using our hands ?

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3D transformation estimation
given 3 corresponding 3D points

• Compute the centroids as the origin
• Compute the scale
• (compute the rotation by quaternion)
• Compute the rotation axis
• Compute the rotation angle

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Geometry of 3D rotation about an axis with angle
theta
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2 equations for 3 unknowns, so two vectors are
needed!
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Rotation axis is obtained, but not yet the angle
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Linear pose estimation from 4 coplanar points

• (Projective method based on a homography)
• (Similar to plane-based calibration)
• Vector based (or affine geometry) method

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Now we get only the ratios of the unknown
distances, to fix the ratio,