uncalibrated epipolar - calibration jana kosecka cs223b
Post on 21-Dec-2015
230 Views
Preview:
TRANSCRIPT
CS223b4
Uncalibrated Camera
• Pixel coordinates
• Projection matrix
Uncalibrated camera
• Image plane coordinates • Camera extrinsic parameters
• Perspective projection
Calibrated camera
CS223b5
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)
CS223b6
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.
CS223b7
Calibration with a Rig
• Eliminate unknown scales
• Factor the into and using QR decomposition
• Solve for translation
• Recover projection matrix
• Given 3-D coordinates on known object
CS223b8
More details• Direct calibration by recovering and decomposing the projection matrix
2 constraints per point
CS223b9
More details
• Factor the into and using QR decomposition
• Solve for translation
• 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
CS223b11
Calibration with a planar pattern
Because are orthogonal and unit norm vectors of rotation matrixWe get the following two constraints
• Unknowns in K (S)
Skew is often close 0 -> 4 unknowns
• We want to recover S
• 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
CS223b12
Alternative camera models/projections
Orthographic projection
Scaled orthographic projection
Affine camera model
CS223b14
Models of Radial Distortion
)1( 42
21 rkrk
y
x
y
x
d
d ++⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
distance from center
CS223b17
Distorted Camera Calibration
Set k1k2, solve for undistorted case
Find optimal k1k2 via nonlinear least squares
Iterate
Tends to generate good calibrations
CS223b20
Calibration by nonlinear Least Squares
Least Mean Square
Gradient descent:
J
},,,,]},[{]},[{]},[{]},[{{ yxyx oossfkTkkkX ψϕφ=
0X
0X
J
∂∂
)(1.0 11 −− ∂∂
⋅−← kkk XXJ
XX
CS223b21
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?
N 1 3 1 3 4 4 6
K 1 1 3 3 3 4 6
CS223b22
Constraints
N points K images 2NK constraints
4 intrinsics (distortion: +2) 6K extrinsics need 2NK ≥ 6K+4
(N-3)K ≥ 2
Hint: may not be co-linear
CS223b23
The Calibration Problem Quiz
N 1 3 1 3 4 4 6
K 1 1 3 3 3 4 6
No No No No Yes Yes Yes
need (N-3)K ≥ 2
Hint: may not be co-linear
top related