Calibration
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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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Calibration

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Calibration

Calibration


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 11

Camera Calibration – Example 1

  • Write equations:


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


Calibration target w circles

Calibration Target w. Circles


3d target w circles

3D Target w. Circles


Planar checkerboard target

Planar Checkerboard Target

[Bouguet]


Coded circles

Coded Circles

[Marschner et al.]


Concentric coded circles

Concentric Coded Circles

[Gortler et al.]


Color coded circles

Color Coded Circles

[Culbertson]


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 geometry

Epipolar Geometry

P

p1

p2

C1

C2


Epipolar geometry1

Epipolar Geometry

P

l2

p1

p2

C1

C2


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 coordinatesu1TFu2 = 0

  • Advantage: can sometimes estimate F without knowing camera calibration


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