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Camera models and calibration. Read tutorial chapter 2 and 3.1 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

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








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



2D projective transformations


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


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



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

The line at infinity l∞



Ratios of lengths, angles.

The circular points I,J





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



Affine rectification








The circular points

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


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






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



Intersection and tangency

Parallellism of planes,

Volume ratios, centroids,

The plane at infinity π∞





Angles, ratios of length

The absolute conic Ω∞




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






To different directions

correspond different vanishing points


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


General projective camera

11 dof (5+3+3)

intrinsic camera parameters

extrinsic camera parameters

Radial distortion

  • Due to spherical lenses (cheap)

  • Model:



straight lines are not straight anymore

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


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


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

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.

from Szeliski’s book

Next week:Image features

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