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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.)

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 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

- 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

- 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 1

- Write equations:

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 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 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 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

- 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 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 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

- 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

- 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 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

Planar Checkerboard Target

[Bouguet]

Coded Circles

[Marschner et al.]

Concentric Coded Circles

[Gortler et al.]

Color Coded Circles

[Culbertson]

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

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

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

Essential Matrix

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

P

l2

p1

p2

C1

C2

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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