Camera models and calibration
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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). 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

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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)


2D Ideal points

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)


Conics and quadrics

l

Conics

x

C

l=Cx

Quadrics


Theorem:

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 conics

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)


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


Affine properties from images

projection

rectification


Affine rectification

l∞

v1

v2

l1

l3

l2

l4


The circular points

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∞


Projective:

(orthogonal)

Angles

Euclidean:


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:


Camera model

Relation between pixels and rays in space

?


Pinhole camera

Gemma Frisius, 1544


Distant objects appear smaller


Parallel lines meet

  • vanishing point


Vanishing points

VPL

H

VPR

VP2

VP1

To different directions

correspond different vanishing points

VP3


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!


Pinhole camera model


Principal point offset

principal point


Principal point offset

calibration matrix


Camera rotation and translation

~


CCD camera


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


Camera model

Relation between pixels and rays in space

?


Projector model

Relation between pixels and rays in space

(dual of camera)

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

?


Meydenbauer camera

vertical lens shift

to allow direct

ortho-photographs


Affine cameras


Action of projective camera on points and lines

projection of point

forward projection of line

back-projection of line


Action of projective camera on conics and quadrics

back-projection to cone

projection of quadric


Resectioning


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


Line correspondences

Extend DLT to lines

(back-project line)

(2 independent eq.)


Geometric error


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)


Errors in the image

(standard case)

Errors in the world

Errors in the image and in the world


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


Image of absolute conic


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/


from Szeliski’s book


Next week:Image features


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