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Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH
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Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Jan 03, 2016

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Page 1: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Camera Model &

Camera Calibration

Slides are from Marc Pollefeys @ETH

Page 2: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

TT ZfYZfXZYX )/,/(),,(

101

0

0

1

Z

Y

X

f

f

Z

fY

fX

Z

Y

X

Pinhole camera model

Page 3: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Pinhole camera model

101

0

0

Z

Y

X

f

f

Z

fY

fX

101

01

01

1Z

Y

X

f

f

Z

fY

fX

PXx

0|I)1,,(diagP ff

Page 4: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Principal point offset

Tyx

T pZfYpZfXZYX )/,/(),,(

principal pointT

yx pp ),(

101

0

0

1

Z

Y

X

pf

pf

Z

ZpfY

ZpfX

Z

Y

X

y

x

x

x

Page 5: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Principal point offset

101

0

0

Z

Y

X

pf

pf

Z

ZpfY

ZpfX

y

x

x

x

camX0|IKx

1y

x

pf

pf

K calibration matrix

Page 6: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Camera rotation and translation

C~

-X~

RX~

cam

X10

RCR

1

10

C~

RRXcam

Z

Y

X

camX0|IKx XC~

|IKRx

t|RKP C~

Rt PXx

Page 7: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

CCD camera

1yx

xx

p

p

K

11y

x

x

x

pf

pf

m

m

K

Page 8: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Finite projective camera

1yx

xx

p

ps

K

1yx

xx

p

p

K

C~

|IKRP

non-singular

11 dof (5+3+3)

decompose P in K,R,C?

4p|MP 41pMC

~ MRK, RQ

{finite cameras}={P4x3 | det M≠0}

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

Page 9: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Camera matrix decomposition

Finding the camera center

0PC (use SVD to find null-space)

432 p,p,pdetX 431 p,p,pdetY

421 p,p,pdetZ 321 p,p,pdetTFinding the camera orientation and internal parameters

KRM (use RQ decomposition ~QR)

Q R=( )-1= -1 -1QR

(if only QR, invert)

Page 10: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Cameras at infinity

00

dP

Camera center at infinity

0Mdet

Affine and non-affine cameras

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

Page 11: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Affine cameras

Page 12: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Camera calibration

Page 13: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

ii xX ? P

Resectioning

Page 14: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Basic equations

ii PXx

ii PXx

0Ap

Page 15: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Basic equations

0Ap

minimal solution

Over-determined solution

5½ correspondences needed (say 6)

P has 11 dof, 2 independent eq./points

n 6 points

Ap

1p

1p̂3 3p̂P

minimize subject to constraint

Page 16: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Degenerate configurations

More complicate than 2D case

(i) Camera and points on a twisted cubic

(ii) Points lie on plane or single line passing through projection center

Page 17: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Less obvious

(i) Simple, as before

(ii) Anisotropic scaling

Data normalization

32

Page 18: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Geometric error

Page 19: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Gold Standard algorithmObjective

Given n≥6 2D to 2D point correspondences {Xi↔xi’}, determine the Maximum Likelyhood 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

~ ~~

Page 20: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Calibration example

(i) Canny edge detection(ii) Straight line fitting to the detected edges(iii) Intersecting the lines to obtain the images corners

typically precision <1/10 (HZ rule of thumb: 5n constraints for n unknowns

Page 21: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Errors in the world

Errors in the image and in the world

ii XPx

iX

Page 22: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

short and long focal length

Radial distortion

Page 23: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.
Page 24: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.
Page 25: Camera Model & Camera Calibration Slides are from Marc Pollefeys @ETH.

Correction of distortion

Choice of the distortion function and center

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