Old summary of camera modelling

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Old summary of camera modelling

• 3 coordinate frame
• projection matrix
• decomposition
• intrinsic/extrinsic param

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World coordinate frame extrinsic parameters
Finally, we should count properly ...
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new way of looking at old modeling
abstract camera projection from P3 to P2
Math central proj. Physics pin-hole
As lines are preserved so that it is a linear
transformation and can be represented by a 34
matrix
This is the most general camera model without
considering optical distortion
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Properties of the 34 matrix P

• 11 d.o.f.
• Rank(P) ?
• ker(P)c
• row vectors, planes
• column vectors, directions
• principal plane w0
• calibration, 6 pts
• decomposition by QR,
• K intrinsic (5). R, t, extrinsic (6)
• geometric interpretation of K, R, t (backward
  from u/xv/yf/z to P)
• internal parameters and absolute conic

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What is the calibration matrix K?
It is the image of the absolute conic, prove it
first!
Point conic
The dual conic
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(No Transcript)
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Dont forget when the world is planar
A general plane homography!
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Camera calibration
Given
from image processing or by hand ?

• Estimate C
• decompose C into intrinsic/extrinsic

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Calibration set-up
3D calibration object
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The remaining pb is how to solve this trivial
system of equations!
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Review of some basic numerical algorithms

• linear algebra how to solve Axb?
• (non-linear optimisation)
• (statistics)

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Linear algebra review

• Gaussian elimination
• LU decomposition
• orthogonal decomposition
• QR (Gram-Schmidt)
• SVD (the high(est)light of linear algebra!)

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Solving (full rank) square matrix linear sys Ax
b elimination LU factorization

1. factor A into LU
2. solve Lc b (lower triangular, forward
   substitution)
3. solve Uxc (upper tri., backward substitution)

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Solving for Least squares solution for Axb,
minAx-b pseudo-inverse x
(ATA)-1(AT A)b (theoretically, but not
numerically)
Orthogonal bases and Gram-Schmidt A QR
Numerically, QR does it well as ATA RTR,
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Solving for homogeneous system Axo subject to
x1, It is equivalent to minAx, i.e. xT
AT A x, the solution is the eigenvector of
ATA associated with the smallest eigenvalue
Triangular systems not bad, but diagonal system
is better!
Diagonalization eigen vectors gt doable for
symmtric matrices
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• row space first Vs
• null space last Vs
• col space first Us
• null space of the trans last Us

SVD gives orthogonal bases for all subspaces
You get everything with svd
A x b, pseudo-inverse, x A b for both
square system and least squares sol. Even better
with homogeneous sys A x 0, x v_n !
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Linear methods of computing P

• p341
• p1
• p31

Geometric interpretation of these constraints
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Decomposition

• analytical by equating K(R,t)P
• (QR (more exactly it is RQ))

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1. Renormalise by c3
2. tz c34
3. r3 c3
4. u0 c1T c3
5. v0 c2T c3
6. alpha u
7. alpha v

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Linear, but non-optimal,but we want optima, but
non-linear, methods of computing P
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How to solve this non-linear system of equations?
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(Non-linear iterative optimisation)

• J d r from vector F(xd)F(x)J d
• minimize the square of y-F(xd)y-F(x)-J d r
  J d
• normal equation is JT J d JT r
  (Gauss-Newton)
• (Hlambda I) d JT r (LM)

Note F is a vector of functions, i.e. min
f(y-F)T(y-F)
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Using a planar pattern
Why? it is more convenient to have a planar
calibration pattern than a 3D calibration
object, so its very popular now for amateurs.
Cf. the paper by Zhengyou Zhang (ICCV99), Sturm
and Maybank (CVPR99)
(Homework read these papers.)
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• first estimate the plane homogrphies Hi from u
  and x,
• 1. How to estimate H?
• 2. Why one may not be sufficient?
• extract parameters from the plane homographies

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How to extract intrinsic parameters?
Relationship between H and parameters
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(How to extract intrinsic parameters?)
The absolute conic in image
The (transformed) absolute conic in the plane
The circular points of the Euclidean plane
(i,1,0) and (-i,1,0) go thru this conic two
equations on K!
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What does the calibration give us?
It turns the camera into an spherical one, or
angular/direction sensor!
Normalised coordinates
Direction vector
Angle between two rays ...
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Summary of calibration

1. Get image-space points
2. Solve the linear system
3. Optimal sol. by non-linear method
4. Decomposition by RQ