1. Uncalibrated Epipolar - Calibration
Jana Kosecka
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2. Uncalibrated Camera calibrated coordinates Linear transformation pixel
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3. Overview
Calibration with a rig
Uncalibrated epipolar geometry
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4. Uncalibrated Camera Calibrated camera Uncalibrated camera
Image plane coordinates
Camera extrinsic parameters
Perspective projection
Calibrated camera
Pixel coordinates
Projection matrix
Uncalibrated camera
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5. Taxonomy on Uncalibrated Reconstruction
is known, back to calibrated case
is unknown
Calibration with complete scene knowledge (a rig) – estimate
Uncalibrated reconstruction despite the lack of knowledge of
Autocalibration (recover from uncalibrated images)
Use partial knowledge
Parallel lines, vanishing points, planar motion, constant intrinsic
Ambiguities, stratification (multiple views)
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6. Calibration with a Rig
Use the fact that both 3-D and 2-D coordinates of feature
points on a pre-fabricated object (e.g., a cube) are known.
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7. Calibration with a Rig Given 3-D coordinates on known object
Eliminate unknown scales
Recover projection matrix
Factor the into and using QR decomposition
Solve for translation
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8. More details
Direct calibration by recovering and decomposing the projection matrix
2 constraints per point
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9. More details Recover projection matrix
Collect the constraints from all N points into matrix M (2N x 12)
Solution eigenvector associated with the smallest eigenvalue
Unstack the solution and decompose into rotation and translation
Factor the into and using QR decomposition
Solve for translation
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10. Calibration with a planar pattern
To eliminate unknown depth, multiply both sides by
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11. Calibration with a planar pattern
Because are orthogonal and unit norm vectors of rotation matrix
We get the following two constraints
We want to recover S
Unknowns in K (S)
Skew is often close 0 -> 4 unknowns
S is symmetric matrix (6 unknowns) in general we need at least 3 views
To recover S (2 constraints per view) - S can be recovered linearly
Get K by Cholesky decomposition of directly from entries of S
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12. Alternative camera models/projections
Orthographic projection
Scaled orthographic projection
Affine camera model
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13. Barrel and Pincushion Distortion
wideangle
tele
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14. Models of Radial Distortion
distance from center
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15. Tangential Distortion
cheap CMOS chip
cheap lens
image
cheap
glue
cheap camera
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16. Barrel distortion
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17. Distorted Camera Calibration
Set k1=k2=0, solve for undistorted case
Find optimal k1,k2 via nonlinear least squares
Iterate
Tends to generate good calibrations
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18. Calibration Software: Matlab
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19. Calibration Software: OpenCV
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20. Calibration by nonlinear Least Squares
Least Mean Square
Gradient descent:
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21. The Calibration Problem Quiz
Given
Calibration pattern with N corners
K views of this calibration pattern
How large would N and K have to be?
Can we recover all intrinsic parameters? NO N 1 3 4 6 K
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22. Hint: may not be co-linear
Constraints
N points
K images  NK constraints
4 intrinsics (distortion: +2)
6K extrinsics
 need 2NK ≥ 6K+4
 (N-3)K ≥ 2
Hint: may not be co-linear
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23. The Calibration Problem Quiz1 3 4 6 K No
Yes
need (N-3)K ≥ 2
Hint: may not be co-linear
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24. Problem with Least Squares
Many parameters (=slow)
Many local minima! (=slower)
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