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Camera models and calibration - PowerPoint PPT Presentation

Camera models and calibration. Read tutorial chapter 2 and 3.1 http://www.cs.unc.edu/~marc/tutorial/ Szeliski ’ s book pp.29-73. Schedule (tentative). 2D Ideal points. 3D Ideal points. 2D line at infinity. 3D plane at infinity. Brief geometry reminder.

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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

Schedule (tentative)

3D Ideal points

2D line at infinity

3D plane at infinity

Brief geometry reminder

2D 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)

l

Conics

x

C

l=Cx

A mapping h:P2P2is 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 transformations

Definition:

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 dual conics

Transformation of 2D points, lines and conics

For a point transformation

Transformation for lines

(eigenvectors H-T =fixed lines)

Fixed points and lines

(eigenvectors H =fixed points)

(1=2 pointwise fixed line)

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 l is a fixed line under a projective transformation H if and only if H is an affinity

Note: not fixed pointwise

projection

rectification

l∞

v1

v2

l1

l3

l2

l4

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

l

Algebraically, encodes orthogonal directions

The circular points

“circular points”

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 points

l∞

(orthogonal)

Angles

Euclidean:

For a point transformation

(cfr. 2D equivalent)

Transformation for lines

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 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 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 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:

Relation between pixels and rays in space

?

Gemma Frisius, 1544

• vanishing point

VPL

H

VPR

VP2

VP1

To different directions

correspond different vanishing points

VP3

• 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

linear projection in homogeneous coordinates!

principal point

calibration matrix

General projective camera

11 dof (5+3+3)

intrinsic camera parameters

extrinsic camera parameters

• Due to spherical lenses (cheap)

• Model:

R

R

straight lines are not straight anymore

Relation between pixels and rays in space

?

Relation between pixels and rays in space

(dual of camera)

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

?

vertical lens shift

to allow direct

ortho-photographs

projection of point

forward projection of line

back-projection of line

back-projection to cone

rank-2 matrix

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

• Points lie on plane or single line passing through projection center

• Camera and points on a twisted cubic

• Scale data to values of order 1

• move center of mass to origin

• scale to yield order 1 values

Extend DLT to lines

(back-project line)

(2 independent eq.)

• 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:

~

~

~

• 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)

(standard case)

Errors in the world

Errors in the image and in the world

• 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)||

• 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

• 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)

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/

from Szeliski’s book

Next week:Image features