Calibration

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Calibration

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Camera Calibration

• Geometric
• Intrinsics Focal length, principal point,
  distortion
• Extrinsics Position, orientation
• Radiometric
• Mapping between pixel value and scene radiance
• Can be nonlinear at a pixel (gamma, etc.)
• Can vary between pixels (vignetting, cos4, etc.)
• Dynamic range (calibrate shutter speed, etc.)

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Geometric Calibration Issues

• Camera Model
• Orthogonal axes?
• Square pixels?
• Distortion?
• Calibration Target
• Known 3D points, noncoplanar
• Known 3D points, coplanar
• Unknown 3D points (structure from motion)
• Other features (e.g., known straight lines)

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Geometric Calibration Issues

• Optimization method
• Depends on camera model, available data
• Linear vs. nonlinear model
• Closed form vs. iterative
• Intrinsics only vs. extrinsics only vs. both
• Need initial guess?

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Caveat - 2D Coordinate Systems

• y axis up vs. y axis down
• Origin at center vs. corner
• Will often write (u, v) for image coordinates

u
v
v
u
v
u
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Camera Calibration Example 1

• Given
• 3D ? 2D correspondences
• General perspective camera model (no distortion)
• Dont care about z after transformation
• Homogeneous scale ambiguity ? 11 free parameters

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Camera Calibration Example 1

• Write equations

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Camera Calibration Example 1

• Linear equation
• Overconstrained (more equations than unknowns)
• Underconstrained (rank deficient matrix any
  multiple of a solution, including 0, is also a
  solution)

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Camera Calibration Example 1

• Standard linear least squares methods forAx0
  will give the solution x0
• Instead, look for a solution with x 1
• That is, minimize Ax2 subject to x21

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Camera Calibration Example 1

• Minimize Ax2 subject to x21
• Ax2 (Ax)T(Ax) (xTAT)(Ax) xT(ATA)x
• Expand x in terms of eigenvectors of ATA x
  m1e1 m2e2 xT(ATA)x l1m12l2m22 x2
  m12m22

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Camera Calibration Example 1

• To minimize l1m12l2m22subject to
  m12m22 1set mmin 1 and all other mi0
• Thus, least squares solution is eigenvector
  corresponding to minimum (non-zero) eigenvalue of
  ATA

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Camera Calibration Example 2

• Incorporating additional constraints intocamera
  model
• No shear (u, v axes orthogonal)
• Square pixels
• etc.
• Doing minimization in image space
• All of these impose nonlinear constraints
  oncamera parameters

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Camera Calibration Example 2

• Option 1 nonlinear least squares
• Usually gradient descent techniques
• e.g. Levenberg-Marquardt
• Option 2 solve for general perspective model,
  find closest solution that satisfies constraints
• Use closed-form solution as initial guess
  foriterative minimization

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Radial Distortion

• Radial distortion can not be representedby
  matrix
• (cu, cv) is image center,uimg uimg cu, vimg
  vimg cv,k is first-order radial distortion
  coefficient

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Camera Calibration Example 3

• Incorporating radial distortion
• Option 1
• Find distortion first (e.g., straight lines
  incalibration target)
• Warp image to eliminate distortion
• Run (simpler) perspective calibration
• Option 2 nonlinear least squares

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Calibration Targets

• Full 3D (nonplanar)
• Can calibrate with one image
• Difficult to construct
• 2D (planar)
• Can be made more accuracte
• Need multiple views
• Better constrained than full SFM problem

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Calibration Targets

• Identification of features
• Manual
• Regular array, manually seeded
• Regular array, automatically seeded
• Color coding, patterns, etc.
• Subpixel estimation of locations
• Circle centers
• Checkerboard corners

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Calibration Target w. Circles
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3D Target w. Circles
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Planar Checkerboard Target


Bouguet
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Coded Circles
Marschner et al.
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Concentric Coded Circles
Gortler et al.
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Color Coded Circles
Culbertson
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Calibrating Projector

• Calibrate camera
• Project pattern onto a known object(usually
  plane)
• Can use time-coded structured light
• Form (uproj, vproj, x, y, z) tuples
• Use regular camera calibration code
• Typically lots of keystoning relative to cameras

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Multi-Camera Geometry

• Epipolar geometry relationship between observed
  positions of points in multiple cameras
• Assume
• 2 cameras
• Known intrinsics and extrinsics

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Epipolar Geometry
P
p1
p2
C1
C2
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Epipolar Geometry
P
l2
p1
p2
C1
C2
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Epipolar Geometry
P
Epipolar line
l2
p1
p2
C1
C2
Epipoles
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Epipolar Geometry

• Goal derive equation for l2
• Observation P, C1, C2 determine a plane

P
l2
p1
p2
C1
C2
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Epipolar Geometry

• Work in coordinate frame of C1
• Normal of plane is T ? Rp2, where T is relative
  translation, R is relative rotation

P
l2
p1
p2
C1
C2
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Epipolar Geometry

• p1 is perpendicular to this normal p1 ?
  (T ? Rp2) 0

P
l2
p1
p2
C1
C2
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Epipolar Geometry

• Write cross product as matrix multiplication

P
l2
p1
p2
C1
C2
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Epipolar Geometry

• p1 ? T R p2 0 ? p1T E p2 0
• E is the essential matrix

P
l2
p1
p2
C1
C2
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Essential Matrix

• E depends only on camera geometry
• Given E, can derive equation for line l2

P
l2
p1
p2
C1
C2
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Fundamental Matrix

• Can define fundamental matrix F analogously,
  operating on pixel coordinates instead of camera
  coordinates u1T F u2 0
• Advantage can sometimes estimate F without
  knowing camera calibration