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Old summary of camera modelling

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... is how to solve this trivial' system of equations! 11 ... Linear algebra review. Gaussian elimination. LU decomposition ... light of linear algebra!) 13 ... – PowerPoint PPT presentation

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Title: Old summary of camera modelling


1
Old summary of camera modelling
  • 3 coordinate frame
  • projection matrix
  • decomposition
  • intrinsic/extrinsic param

2
World coordinate frame extrinsic parameters
Finally, we should count properly ...
3
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
4
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

5
What is the calibration matrix K?
It is the image of the absolute conic, prove it
first!
Point conic
The dual conic
6
(No Transcript)
7
Dont forget when the world is planar
A general plane homography!
8
Camera calibration
Given
from image processing or by hand ?
  • Estimate C
  • decompose C into intrinsic/extrinsic

9
Calibration set-up
3D calibration object
10
The remaining pb is how to solve this trivial
system of equations!
11
Review of some basic numerical algorithms
  • linear algebra how to solve Axb?
  • (non-linear optimisation)
  • (statistics)

12
Linear algebra review
  • Gaussian elimination
  • LU decomposition
  • orthogonal decomposition
  • QR (Gram-Schmidt)
  • SVD (the high(est)light of linear algebra!)

13
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)

14
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,
15
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
16
  • 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 !
17
Linear methods of computing P
  • p341
  • p1
  • p31

Geometric interpretation of these constraints
18
Decomposition
  • analytical by equating K(R,t)P
  • (QR (more exactly it is RQ))

19
  1. Renormalise by c3
  2. tz c34
  3. r3 c3
  4. u0 c1T c3
  5. v0 c2T c3
  6. alpha u
  7. alpha v

20
Linear, but non-optimal,but we want optima, but
non-linear, methods of computing P
21
How to solve this non-linear system of equations?
22
(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)
23
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.)
24
  • 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

25
How to extract intrinsic parameters?
Relationship between H and parameters
26
(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!
27
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 ...
28
Summary of calibration
  1. Get image-space points
  2. Solve the linear system
  3. Optimal sol. by non-linear method
  4. Decomposition by RQ
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