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Calibration. 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, cos 4 , etc.)

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camera calibration
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.)
geometric calibration issues
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)
geometric calibration issues1
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?
caveat 2d coordinate systems
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

camera calibration example 1
Camera Calibration – Example 1
  • Given:
    • 3D  2D correspondences
    • General perspective camera model (no distortion)
  • Don’t care about “z” after transformation
  • Homogeneous scale ambiguity  11 free parameters
camera calibration example 12
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)
camera calibration example 13
Camera Calibration – Example 1
  • Standard linear least squares methods forAx=0 will give the solution x=0
  • Instead, look for a solution with |x|= 1
  • That is, minimize |Ax|2 subject to |x|2=1
camera calibration example 14
Camera Calibration – Example 1
  • Minimize |Ax|2 subject to |x|2=1
  • |Ax|2 = (Ax)T(Ax) = (xTAT)(Ax) = xT(ATA)x
  • Expand x in terms of eigenvectors of ATA: x = m1e1+ m2e2+… xT(ATA)x = l1m12+l2m22+… |x|2 = m12+m22+…
camera calibration example 15
Camera Calibration – Example 1
  • To minimizel1m12+l2m22+…subject tom12+m22+… = 1set mmin= 1 and all other mi=0
  • Thus, least squares solution is eigenvector corresponding to minimum (non-zero) eigenvalue of ATA
camera calibration example 2
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
camera calibration example 21
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
radial distortion
Radial Distortion
  • Radial distortion can not be representedby matrix
  • (cu, cv) is image center,u*img= uimg– cu, v*img= vimg– cv,k is first-order radial distortion coefficient
camera calibration example 3
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
calibration targets
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
calibration targets1
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
coded circles
Coded Circles

[Marschner et al.]

concentric coded circles
Concentric Coded Circles

[Gortler et al.]

calibrating projector
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
multi camera geometry
Multi-Camera Geometry
  • Epipolar geometry – relationship between observed positions of points in multiple cameras
  • Assume:
    • 2 cameras
    • Known intrinsics and extrinsics
epipolar geometry2
Epipolar Geometry

P

Epipolar line

l2

p1

p2

C1

C2

Epipoles

epipolar geometry3
Epipolar Geometry
  • Goal: derive equation for l2
  • Observation: P, C1, C2 determine a plane

P

l2

p1

p2

C1

C2

epipolar geometry4
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

epipolar geometry5
Epipolar Geometry
  • p1 is perpendicular to this normal: p1 (T  Rp2) = 0

P

l2

p1

p2

C1

C2

epipolar geometry6
Epipolar Geometry
  • Write cross product as matrix multiplication

P

l2

p1

p2

C1

C2

epipolar geometry7
Epipolar Geometry
  • p1 T* R p2 = 0  p1TEp2 = 0
  • E is the essential matrix

P

l2

p1

p2

C1

C2

essential matrix
Essential Matrix
  • E depends only on camera geometry
  • Given E, can derive equation for line l2

P

l2

p1

p2

C1

C2

fundamental matrix
Fundamental Matrix
  • Can define fundamental matrix F analogously, operating on pixel coordinates instead of camera coordinates u1TFu2 = 0
  • Advantage: can sometimes estimate F without knowing camera calibration
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