Robot Vision SS 2010 Matthias Rüther 1

ROBOT VISION Lesson 4: Camera Models and Calibration

Matthias Rüther

Slides partial courtesy of Marc Pollefeys Department of Computer Science

University of North Carolina, Chapel Hill

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Content

Camera Models– Pinhole Camera– CCD Camera– Finite Projective Camera– Affine Camera– Pushbroom Camera

Calibration– Inner Orientation– Nonlinear Distortion– Calibration using Planar Targets– Calibration using a 3D Target

The Cyclops, 1914 by Odilon Redon

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Basic Pinhole Camera Model

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TT ZfYZfXZYX )/,/(),,(

10100

1ZYX

ff

ZfYfX

ZYX

Basic Pinhole Camera Model

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10100

ZYX

ff

ZfYfX

1010101

1ZYX

ff

ZfYfX

PXx

0|I)1,,(diagP ff

Basic Pinhole Camera Model

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Tyx

T pZfYpZfXZYX )/,/(),,( principal point

Tyx pp ),(

00

1 01 1

x x

y y

X XfX Zp f p

Y YfY Zp f p

Z ZZ

Principal Point Offset

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10100

ZYX

pfpf

ZZpfYZpfX

y

x

x

x

camX0|IKx

1y

x

pfpf

K calibration matrix

Principal Point Offset

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C~-X~RX~ cam

X10RCR

110

C~RRXcam

ZYX

camX0|IKx XC~|IKRx

t|RKP C~Rt PXx

Camera Rotation and Translation

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1yy

xx

pp

K

11y

x

y

x

pfpf

mm

K

CCD Camera

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1yy

xx

pps

K

1yy

xx

pp

K

C~|IKRP

non-singular

11 dof (5+3+3)

decompose P in K,R,C? 4p|MP 4

1pMC~ MRK, RQ

{finite cameras}={P3x4 | det M≠0}

If rank P=3, but rank M<3, then cam at infinity

Finite Projective Camera

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PXx

MdDp|MPDx 4

Forward projection (3D -> 2D)

Back-projection (2D -> 3D)

xPX 1PPPP

TT IPP

(pseudo-inverse)

0PC

λCxPλX

1p-μxM

1pM-

0xM

μλX 4-1

4-1-1

xMd -1

CD

Action of Projective Cameras on Points

D…direction

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1yx

xx

pps

K

1

arctan(1/s)

for CCD/CMOS, always s=0

Image from image, s≠0 possible(non coinciding principal axis)

HPresulting camera:

When is skew non-zero?

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00d

P

Camera center at infinity

0Mdet

Affine and non-affine cameras

Definition: affine camera has P3T=(0,0,0,1)

Moving the Camera Center to Infinity

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Affine Cameras

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100000100001

P canonical representation

100K

K 22affine calibration matrix

principal point is not defined

Parallel Projection: Summary

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Orthographic projection

Scaled orthographic projection

100000100001

P

10tR

H

1T1

2T2

rP r

0 1

tt

1T1

2T2

rP r

0 1/

tt

k

(5dof)

(6dof)

A Hierarchy of Affine Cameras

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Weak perspective projection

ktt

y

x

/10rr

1α

αP 2

1T1

1T

(7dof)

A Hierarchy of Affine Cameras

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1. Affine camera=camera with principal plane coinciding with ∞

2. Affine camera maps parallel lines to parallel lines

3. No center of projection, but direction of projection PAD=0(point on ∞)

Affine camera

ktts

y

x

A/10

rr

1α

αP 2

1T1

1T(8dof)

1000P 2232221

1131211tmmmtmmm

A

affine 44100000100001

affine 33P

A

A Hierarchy of Affine Cameras

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T)(X X,Y,X,T T),,(PX wyx T)/,( wyx

Straight lines are not mapped to straight lines!(otherwise it would be a projective camera)

(11dof)

Pushbroom Cameras

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ZYX

pppppp

yx

232221

131211 (5dof)

Null-space PC=0 yields camera center

Also decomposition c~|IRKP 22222232

Line Cameras

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Camera calibration

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ii xX ? P

Problem Statement

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ii PXx

ii PXx

0Ap

Basic Equations

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0Ap

minimal solution

Over-determined solution

5½ correspondences needed (say 6) P has 11 dof, 2 independent eq./points

n 6 points

Apminimize subject to constraint

1p

1p̂3 3p̂

P

Basic Equations

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Geometric Error

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Gold Standard algorithm

ObjectiveGiven n≥6 2D to 2D point correspondences {Xi↔xi’}, determine the Maximum Likelihood Estimation of P

Algorithm(i) Linear solution:

(a) Normalization: (b) DLT:

(ii) Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error:

(iii) Denormalization:

ii UXX~ ii Txx~

UP~TP -1

~ ~~

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Calibrated camera, position and orientation unkown

Pose estimation

6 dof 3 points minimal (4 solutions in general)

Exterior Orientation

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short and long focal length

Nonlinear Distortion

Radial Component

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Nonlinear Distortion Radial Component

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Correction of radial distortion

Computing the parameters of the distortion function(i) Minimize with additional unknowns(ii) Straighten lines(iii) …

Correction of radial Distortion

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Nonlinear Distortion Tangential Component

Distortion function:

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Distortion Workflow (KU wp1)

Project world points: x = P*X

Normalize projected points: xn = K-1 * x

Apply distortion to xn : xd = fdistort(xn)

Denormalize: xp = K * xd