1 pb. camera model calibration separation (int/ext) pose don’t get lost! what are we doing?...
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pb. camera model calibration separation (int/ext) pose
Don’t get lost! What are we doing?
Projective geometry Numerical tools
Uncalibrated cameras Calibrated cameras
Projective geometry Euclidean geometry
Classical (Euclidean) tools
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First application: camera pose estimation
• 3-point algebraic method
• 4 coplanar points linear method
Two most popular methods:
In vision, robotics, virtual reality, …
Pose estimation = extrinsic calibration = navigation by reference =where is the camera? = where am I in the scene?
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Pose estimation = calibration of only extrinsic parameters
33ii , KXu• Given
• Estimate R and t
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3-point algebraic method
• 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
222 cos2 ijjiijji dxxxx
3 reference points == 3 beacons
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X
x
O
X’
x’
Fundamental euclidean geometric constraint:
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Solving the algebraic system by elimination:
• (using a symbolic computation software (Maple or Mathematica))• using … our hands
a polynomial of degree m and a polynomial of degree n leads to a polynomial of degree m*n
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given 3 corresponding 3D points:
3D transformation estimation
• 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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nvv 2 aroundtorotatedis1
v
v’
n
nvvvvvv 21212 ,1
)()( 1 212 vvnvv k
Geometry of 3D rotation about an axis with angle theta
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k
matrixsymmetricantitheiswhere
/
,[*]
,][ 1
nx
vvxvv 212
2 equations for 3 unknowns, so two vectors are needed!
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Rotation axis is obtained, but not yet the angle …
vv
vvv parallel
vv
vv
2tan
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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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O
A
B
CD
x_a
x_d
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DC,B,A,,,,),( dcbT
aaa vu uuuu
,..., ba xx
,..., bbaa dOBdOA xx
ACABAD
)()()(
OAOCOAOBOAOD
0
....
....
....
d
c
b
a
d
d
d
d
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||||cos222 ABdddd baabba
Now we get only the ratios of the unknown distances, to fix the ratio,
bada 2
b
a
d
d