Robot Vision Systems
Camera Model
Calibration
1
1
“perspectograph”
Alberti’s Grid
1
2
Pinhole Camera
scene
image plane
1
iris,
optical center
3
Pinhole Camera
“image coordinates”
1
4
Pinhole Camera
“Alberti’s Grid”
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5
Pinhole Camera
• Classical pin-hole
r’
1
x
f
z
6
Classical Pinhole Camera
• Similar Triangles
c’
1
x
f
z
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Classical Pinhole Camera
• Similar Triangles
c’
Image
coordinates
Projective
coordinates
1
x
f
Camera
matrix
(projection)
z
Point expressed
in the camera
frame
8
Classical Pinhole Camera
• Similar Triangles
Convert
pixels to
mm:
1
c’
x
f
z
world
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Camera Calibration
• Used to determine the elements in
the camera matrix
• Use pairs of known world points and
their corresponding image points
e.g. use calibration grid
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10
Camera Model – Perspective Projection
• Need to determine the parameters!
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11
Camera Calibration
• camera matrix
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12
Camera Calibration
• Projective Equivalence
Two equations in 12 unknowns
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13
Camera Calibration
• Have 6 point pairs (c,r) and (x,y,z)
– Correspondence known
– World coordinates known accurately
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14
Camera Calibration
1
•
Method 1: assume a34 = 0 and solve for aij
•
Solve using pseudo-inverse
15
Camera Calibration
1
•
Method 2: make no assumption about a34 and
solve for aij
•
The vector a = [a11, …, a34]T is in the nullspace of
the design matrix
16
Camera Calibration
•
Ba = 0,
•
Use SVD,
then a is the column of V
corresponding to the null singular value of B
– One property of the SVD is that the columns of V
corresponding to the zero singular value span the
null space of B
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17
Camera Calibration
• Given the camera matrix solved w.r.t. the
robot base frame
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18
Using the Camera Matrix
• Projection (
1
is 3 x 4)– cannot invert!
19
Measurements from Images
• Must have relationship
between the image
“pixels” and the world
• 2D imaging
– the image plane and the
“world” plane are in 1-1
correspondence
1
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Using the Camera Matrix in 2D
• If all world points are on a plane
• Then z is a linear function of x & y
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Using the Camera Matrix in 2D
• Now the projection equations
• Can be written
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Using the Camera Matrix in 2D
• Now the projection equations
• Can be written
1
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Using the Camera Matrix in 2D
• Now the projection equations
• Can be written
1
2 equations, 9 unknowns
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Using the Camera Matrix in 2D
• Four known points, i=1,..4
8 equations, 9 unknowns up
to a scale (8 unknowns)
• Linear equations -- can be solved
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25
Using the Camera Matrix in 2D
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Using the Camera Matrix in 2D
• Now the projection equations are simpler
… and we can “invert” (map image points
back to the world plane)
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Measurements from Images
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28
Robot to Plane
• Homogeneous transformation from base (or
end-effector) frame to the work-plane of the
imaging system
– Use “known” corresponding points to solve for
the elements of T (6 unknowns)
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