Camera parameters

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Camera parameters
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• Extrinisic parameters define location and
  orientation of camera reference frame with
  respect to world frame
• Intrinsic parameters define pixel coordinates of
  image point with respect to coordinates in camera
  reference frame

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Homogenous coordinates
Add an extra coordinate and define
equivalence Relationship
(x,y) -gt (kx, ky, k)
(X,Y,Z) -gt (wX, wY, wZ, w)
Makes it possible to write the Perspective
projection as a linear Transformation
(matrix) (from projective space to projective
plane)
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Central projection
HC
NHC
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Scaled orthographic projection
HC
NHC
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In a simpler notation

• T describes the position of the origin of camera
  frame with respect to world frame
• R describes the rotation which aligns the camera
  frame with the world frame
• Pc R(Pw T)
• (here RT BOA)

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Translation and Rotation
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Intrinsic parameters
y
ypix
x
xpix
Scaling
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Intrinsic parameters
y
ypix
x
xpix
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Intrinsic parameters
y
ypix
x
xpix

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The internal calibration parameters
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with
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Properties of matrix M

• M has 11 degrees of freedom (5 internal 3
  rotation, 3 translation parameters) , 3x4 matrix
  defined up to scale
• The 3x3 submatrix MMintR is non-singular (Mint
  is upper triangular, R is orthogonal -gt essential
  QR decomposition)

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Radial distortion
from lens distortion (pin cushioning effect)
(significant error for cheap optics and short
focal length)
Straight lines are not imaged straight
x and xd measured from image center
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Radial calibration
Using lines to be straight (x,y) is radial
projection of (xd, yd) on straight line
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Calibration Procedure

• Calibration target 2 planes at right angle with
  checkerboard (Tsai grid)
• We know positions of corners of grid with respect
  to a coordinate system of the target
• Obtain from images the corners
• Using the equations (relating pixel coordinates
  to world coordinates) we obtain the camera
  parameters (the internal parameters and the
  external (pose) as a side effect)

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Image Processing

• Canny edge detection
• Straight line fitting to detect long edges
• Intersection of lines to detect image corners
• Matching image corners and 3D checkerboard corners

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Estimation procedure

• First estimate M from corresponding image points
  and scene points (solving homogeneous equation)
• Second decompose M into internal and external
  parameters
• Use estimated parameters as starting point to
  solve calibration parameters non-linearly.

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(homogeneous equation)
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Solving A m 0
Linear homogeneous system
Have at least 5 times as many equations as
unknowns (28 points)
Minimize Am2 with the constraint m21
M is the unit singular value of A
corresponding to the smallest singular value (the
last column of V, where A UDVT is the SVD of
A), or the eigenvector (corresponding to smallest
eigenvalue ) of ATA
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Finding camera translation
(position of camera center)
Let be the homogeneous representation of T
is the null vector of M
Null vector is found using SVD ( is the unit
singular vector corresponding to the smallest
singular value of M)
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Finding camera orientation and internal parameters

• Left 3x3 submatrix of M is of the form M
• Mint R
• Mint upper triangular
• R orthogonal
• Any nonsingular matrix can be decomposed
• into the product of an upper triangular and
• an orthogonal matrix (RQ factorizationhere
• R refers to upper triangular and Q to
  orthogonal)
• (Similar to QR factorization)

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RQ factorization of M

• Givens rotations

To set M32 to zero, solve equation Thus
Multiply M by Rx ( such that term (3,2) is 0),
then by, Ry (choosing c, s such that term
(3,1) is zero), then by Rz (with c, s such
that term (2,1) is zero)
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Improving solution with nonlinear optimization
Find m using the linear constraint Use as
initialization for nonlinear optimization
(Levenberg-Marquardt iterative minimization)
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Algorithm described in Multiple View Geometry in
Computer Vision (Hartley, Zisserman)