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### Camera models and calibration

Read tutorial chapter 2 and 3.1

http://www.cs.unc.edu/~marc/tutorial/

Szeliski’s book pp.29-73

2D Ideal points

3D Ideal points

2D line at infinity

3D plane at infinity

Brief geometry reminder2D line-point coincidence relation:

Point from lines:

Line from points:

3D plane-point coincidence relation:

Point from planes: Plane from points:

3D line representation:

(as two planes or two points)

Theorem:

A mapping h:P2P2is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P2 reprented by a vector x it is true that h(x)=Hx

Definition: Projective transformation

or

8DOF

2D projective transformationsDefinition:

A projectivity is an invertible mapping h from P2 to itself such that three points x1,x2,x3lie on the same line if and only if h(x1),h(x2),h(x3) do.

projectivity=collineation=projective transformation=homography

Transformation for conics

Transformation for dual conics

Transformation of 2D points, lines and conicsFor a point transformation

Transformation for lines

(eigenvectors H-T =fixed lines)Fixed points and lines

(eigenvectors H =fixed points)

(1=2 pointwise fixed line)

Hierarchy of 2D transformations

transformed squares

invariants

Concurrency, collinearity, order of contact (intersection, tangency, inflection, etc.), cross ratio

Projective

8dof

Parallellism, ratio of areas, ratio of lengths on parallel lines (e.g midpoints), linear combinations of vectors (centroids).

The line at infinity l∞

Affine

6dof

Ratios of lengths, angles.

The circular points I,J

Similarity

4dof

Euclidean

3dof

lengths, areas.

The line at infinity

The line at infinity l is a fixed line under a projective transformation H if and only if H is an affinity

Note: not fixed pointwise

The circular points

The circular points I, J are fixed points under the projective transformation H iff H is a similarity

The dual conic is fixed conic under the

projective transformation H iff H is a similarity

Note: has 4DOF

l∞ is the nullvector

Conic dual to the circular pointsl∞

Transformation of 3D points, planes and quadrics

For a point transformation

(cfr. 2D equivalent)

Transformation for lines

Transformation for quadrics

Transformation for dual quadrics

Hierarchy of 3D transformations

Projective

15dof

Intersection and tangency

Parallellism of planes,

Volume ratios, centroids,

The plane at infinity π∞

Affine

12dof

Similarity

7dof

Angles, ratios of length

The absolute conic Ω∞

Euclidean

6dof

Volume

The plane at infinity

The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity

- canonical position
- contains directions
- two planes are parallel line of intersection in π∞
- line // line (or plane) point of intersection in π∞

The absolute conic

The absolute conic Ω∞ is a (point) conic on π.

In a metric frame:

or conic for directions:

(with no real points)

The absolute conic Ω∞ is a fixed conic under the projective transformation H iff H is a similarity

- Ω∞is only fixed as a set
- Circle intersect Ω∞ in two circular points
- Spheres intersect π∞ in Ω∞

The absolute dual quadric

The absolute dual quadric Ω*∞ is a fixed conic under the projective transformation H iff H is a similarity

- 8 dof
- plane at infinity π∞ is the nullvector of Ω∞
- Angles:

Pinhole camera

Gemma Frisius, 1544

Parallel lines meet

- vanishing point

Geometric properties of projection

- Points go to points
- Lines go to lines
- Planes go to whole image

or half-plane

- Polygons go to polygons
- Degenerate cases:
- line through focal point yields point
- plane through focal point yields line

Pinhole camera model

linear projection in homogeneous coordinates!

Principal point offset

principal point

Principal point offset

calibration matrix

non-singular

General projective camera

11 dof (5+3+3)

intrinsic camera parameters

extrinsic camera parameters

Radial distortion

- Due to spherical lenses (cheap)
- Model:

R

R

straight lines are not straight anymore

http://foto.hut.fi/opetus/260/luennot/11/atkinson_6-11_radial_distortion_zoom_lenses.jpg

Projector model

Relation between pixels and rays in space

(dual of camera)

(main geometric difference is vertical principal point offset to reduce keystone effect)

?

Action of projective camera on points and lines

projection of point

forward projection of line

back-projection of line

Direct Linear Transform (DLT)

rank-2 matrix

minimize subject to constraint

Direct Linear Transform (DLT)

Minimal solution

P has 11 dof, 2 independent eq./points

- 5½ correspondences needed (say 6)

Over-determined solution

n 6 points

use SVD

Degenerate configurations

- Points lie on plane or single line passing through projection center
- Camera and points on a twisted cubic

Data normalization

- Scale data to values of order 1
- move center of mass to origin
- scale to yield order 1 values

Gold Standard algorithm

- Objective
- Given n≥6 2D to 3D point correspondences {Xi↔xi’}, determine the Maximum Likelyhood Estimation of P
- Algorithm
- Linear solution:
- Normalization:
- DLT
- Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error:
- Denormalization:

~

~

~

Calibration example

- Canny edge detection
- Straight line fitting to the detected edges
- Intersecting the lines to obtain the images corners
- typically precision <1/10
- (H&Z rule of thumb: 5n constraints for n unknowns)

Restricted camera estimation

- Find best fit that satisfies
- skew s is zero
- pixels are square
- principal point is known
- complete camera matrix K is known

- Minimize geometric error
- impose constraint through parametrization

- Minimize algebraic error
- assume map from param q P=K[R|-RC], i.e. p=g(q)
- minimize ||Ag(q)||

Restricted camera estimation

- Initialization
- Use general DLT
- Clamp values to desired values, e.g. s=0, x= y
- Note: can sometimes cause big jump in error
- Alternative initialization
- Use general DLT
- Impose soft constraints
- gradually increase weights

A simple calibration device

- compute H for each square
- (corners (0,0),(1,0),(0,1),(1,1))
- compute the imaged circular points H(1,±i,0)T
- fit a conic to 6 circular points
- compute K from w through cholesky factorization

(≈ Zhang’s calibration method)

Some typical calibration algorithms

Tsai calibration

Zhangs calibration

http://research.microsoft.com/~zhang/calib/

Z. Zhang. A flexible new technique for camera calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330-1334, 2000.

Z. Zhang. Flexible Camera Calibration By Viewing a Plane From Unknown Orientations. International Conference on Computer Vision (ICCV'99), Corfu, Greece, pages 666-673, September 1999.

http://www.vision.caltech.edu/bouguetj/calib_doc/

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