Uncalibrated Epipolar Calibration

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Uncalibrated Epipolar - Calibration
Jana Kosecka CS223b
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Uncalibrated Camera
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Overview

• Calibration with a rig
• Uncalibrated epipolar geometry

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Uncalibrated Camera
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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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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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Calibration with a Rig

• Recover projection matrix

• Factor the into and
  using QR decomposition

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More details

• Direct calibration by recovering and decomposing
  the projection matrix

2 constraints per point
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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

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Calibration with a planar pattern
To eliminate unknown depth, multiply both sides
by
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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 -gt 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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Alternative camera models/projections
Orthographic projection
Scaled orthographic projection
Affine camera model
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Barrel and Pincushion Distortion
tele
wideangle
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Models of Radial Distortion
distance from center
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Tangential Distortion
cheap CMOS chip
cheap lens
image
cheap glue
cheap camera
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Barrel distortion
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Distorted Camera Calibration

• Set k1k20, solve for undistorted case
• Find optimal k1,k2 via nonlinear least squares
• Iterate
• ?Tends to generate good calibrations

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Calibration Software Matlab
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Calibration Software OpenCV
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Calibration by nonlinear Least Squares

• Least Mean Square
• Gradient descent

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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
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Constraints

• N points
• K images ? 2NK constraints
• 4 intrinsics (distortion 2)
• 6K extrinsics
• ? need 2NK 6K4
• ? (N-3)K 2

Hint may not be co-linear
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The Calibration Problem Quiz
need (N-3)K 2
Hint may not be co-linear
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Problem with Least Squares

• Many parameters (slow)
• Many local minima! (slower)