Camera Model - University of Minnesotahspark/CSci5980/Lec1...3D Point Projection (Metric Space) 3D point fm ( , , )X Y Z: Focal length in meter Pinhole -fm T X Z tan ccd X uf ccd m

Camera Model

3D object

Lens

CCD sensor

3D Point Projection (Metric Space) 3D point

mf

( , , )X Y Z

: Focal length in meter

Pinhole -fm


mf

( , , )X Y Z


Pinhole -fm

X

Ztan


mf

( , , )X Y Z

ccd ccdu v( , )


Pinhole -fm

X

Ztan

ccd X

u f fZ

m mtan


mf

( , , )X Y Z


Pinhole -fm

X

Ztan

ccd X

u fZ

m

ccd Y

v fZ

m

ccd ccdu v( , )


mf

( , , )X Y Z


Pinhole -fm

X

Ztan

ccd X

u fZ

m

ccd Y

v fZ

m

ccd ccd X Y

u v f fZ Z

m m( , ) ( , )

2D projection onto CCD plane

( , , ) X Y Z

ccd ccdu v( , )


mf

( , , )X Y Z


Pinhole -fm

ccd X

u fZ

m

ccd Y

v fZ

m

ccd ccd X Y

u v f fZ Z

m m( , ) ( , )


( , , ) X Y Z

Projection plane

( , )u vccd ccd

fm

ccd ccdu v( , )


mf

( , , )X Y Z


Pinhole -fm

ccd X

u fZ

m

ccd Y

v fZ

m

ccd ccd X Y

u v f fZ Z

m m( , ) ( , )


( , , ) X Y Z

Projection plane

( , )u vccd ccd

fm

ccd ccdu v( , )


mf

( , , )X Y Z

: Focal length in meter ccd

Xu f

Zm

ccd Y

v fZ

m

ccd ccd X Y

u v f fZ Z

m m( , ) ( , )


( , , ) X Y Z

Projection plane

( , )u vccd ccd

fm


mf

( , , )X Y Z

: Focal length in meter ccd

Xu f

Zm

ccd Y

v fZ

m

ccd ccd X Y

u v f fZ Z

m m( , ) ( , )


( , , ) X Y Z

Projection plane

( , )u vccd ccd

fm

( , )u vimg img

img img ccd ccdu v u v( , ) ( , ) ?

Metric Pixel

3D Point Projection (Pixel Space)

CCD sensor (mm) Image (pixel)



u vccd ccd( , )

u vimg img( , )



u vccd ccd( , )

ccdw

ccdh

u vimg img( , )



u vccd ccd( , )

ccdw

ccdh

u vimg img( , )



u vccd ccd( , )

ccdw

ccdh

(0,0)

Projection of pinhole

u vimg img( , )



u vccd ccd( , )

ccdw

ccdh

(0,0)


imgw

imgh

u vimg img( , )



u vccd ccd( , )

ccdw

ccdh

(0,0)


(0,0) imgw

imgh

x yp p( , ) : Image principal point

u vimg img( , )



u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)


(0,0) imgw

imgh


xu pu

w w

imgccd

ccd img



u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)


(0,0) imgw

imgh


xu pu

w w

imgccd

ccd img

yv pv

h h

imgccd

ccd img



u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)


(0,0) imgw

imgh


xu pu

w w

imgccd

ccd img

yv pv

h h

imgccd

ccd img

x

wu u p

w

img

img ccd

ccd

y

hv v p

h

img

img ccd

ccd


O

( , )u vccd ccd

mf

Projection plane ( , , )X Y Z

Z

ccd ccd m m( , ) ( , )X Y

u v f fZ Z


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

: Metric projection


O

( , )u vccd ccd

mf


Z


u v f fZ Z


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

: Metric projection

x x

w w Xu u p f p

w w Z

img img

img ccd m

ccd ccd

y y

h h Yv v p f p

h h Z

img img

img ccd m

ccd ccd

Pixel projection


O

( , )u vccd ccd

mf


Z


u v f fZ Z


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

: Metric projection

x x

w w Xu u p f p

w w Z

img img

img ccd m

ccd ccd

y y

h h Yv v p f p

h h Z

img img

img ccd m

ccd ccd

Pixel projection

Exercise

324 m

1280 pix

960 pix

fm ?

What f to make the height of Eifel tower appear 960 pixel distance?

21.8 mm

ccd size

1500 m

Exercise

324 m

1280 pix

960 pix

fm ?


img

img

ccd

m

h Yy

hf

Z

21.8 mm

ccd size

1500 m

Exercise

324 m

1280 pix

960 pix

fm ?


21.8 mm

ccd size

1280 324960 0.0757m

0.0218 1500 m mf f

1500 m

img

img

ccd

m

h Yy

hf

Z

Focal Length

Exercise

324 m

Z ?

1280 pix

960 pix

What Z to make the height of Eifel tower appear 960 pixel distance?

img

img

ccd

m

h Yy f

h Z

21.8 mm

ccd size

fm = 50 mm

Do I need to move backward or forward?

Exercise

324 m

Z ?

1280 pix

960 pix

What Z to make the height of Eifel tower appear 960 pixel distance?

21.8 mm

ccd size

1280 324960 0.05 990.826m

0.0218 Z

Z

fm = 50 mm

Do I need to move backward or forward?

Forward

img

img

ccd

m

h Yy f

h Z

Exercise

324 m

1500

What Zp to make the height of Eifel tower appear twice of the person?

fm = 50 mm

Z p

1.7 m

Where Was I?

Circa 1984

Where Was I?

Circa 1984

250 pix

670 pix

Where Was I?

Circa 1984

324 m

fm = 50 mm

0.9 m

250 pix

670 pix

Where Was I?

Circa 1984

324 m

Z 1

0.9 m

250 pix

670 pix

fm = 50 mm

img img1 11

ccd cc

1

11 d

1280 0.90.05 8.03m

0.0218 250 m m

h hY Yy f

h yZ

Zf

h

Where Was I?

Circa 1984

324 m

fm = 50 mm

Z 1

0.9 m

250 pix

670 pix

img img1 11

ccd cc

1

11 d

1280 0.90.05 8.03m

0.0218 250 m m

h hY Yy f

h yZ

Zf

h

Where Was I?

Circa 1984

324 m

Z 1

0.9 m

Z 2

250 pix

670 pix

img img2 22

ccd cc

2

22 d

1280 3240.05 1079m

0.0218 670 m m

h hY Yy f

h yZ

Zf

h

fm = 50 mm

img img1 11

ccd cc

1

11 d

1280 0.90.05 8.03m

0.0218 250 m m

h hY Yy f

h yZ

Zf

h

Where Was I?

Circa 1984

250 pix

670 pix

m m

h hY Y Yy f f Z f

Z h Z h y

img img2 2 22 2

2 ccd 2 ccd 2

1280 3240.05 1079m

0.0218 670

200m

400m

600m

800m

1000m

800 m

Where Was I?


O

( , )u vccd ccd

mf


Z


u v f fZ Z


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

: Metric projection

x x

w w Xu u p f p

w w Z

img img

img ccd m

ccd ccd

y y

h h Yv v p f p

h h Z

img img

img ccd m

ccd ccd

Pixel projection


O


Z


u v f fZ Z

x x

w w Xu u p f p

w w Z

img img

img ccd m

ccd ccd

y y

h h Yv v p f p

h h Z

img img

img ccd m

ccd ccd


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

: Metric projection

Focal length in pixel

Pixel projection

img imgu v( , )

f


O


Z


u v f fZ Z

x x

w w Xu u p f p

w w Z

img img

img ccd m

ccd ccd

y y

h h Yv v p f p

h h Z

img img

img ccd m

ccd ccd


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

: Metric projection


xf

yf

img

m

ccd

x

wf f

w

img

m

ccd

y

hf f

hwhere

Pixel projection

img imgu v( , )

f


O


Z


u v f fZ Z

x x

w w Xu u p f p

w w Z

img img

img ccd m

ccd ccd

y y

h h Yv v p f p

h h Z

img img

img ccd m

ccd ccd


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

: Metric projection


f

f

img img

m m

ccd ccd

w h

f f fw h

where

Pixel projection

img img

ccd ccd

w h

w hif

img imgu v( , )

f


O


Z

img x

Xu f p

Z

img y

Yv f p

Z


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

img imgu v( , )

f


O


Z

img x

Xu f p

Z

img y

Yv f p

Z


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

img imgu v( , )

f

img xZu fX p Z

img yZv fY p Z


O


Z

img x

Xu f p

Z

img y

Yv f p

Z


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

img imgu v( , )

f

img xZu fX p Z

img yZv fY p Z

img

img

11

x

y

u f p X

Z v f p Y

Z


O


Z

img x

Xu f p

Z

img y

Yv f p

Z


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

img imgu v( , )

f

img xZu fX p Z

img yZv fY p Z

img

img

11

x

y

u f p X

v f p Y

Z

Computer Graphics

img x

Xu f p

Z

img y

Yv f p

Z

img xZu fX p Z

img yZv fY p Z

Graphics

img

img

11

x

y

u f p X

v f p Y

Z

Computer Vision = inv(Computer Graphics)

img x

Xu f p

Z

img y

Yv f p

Z

img xZu fX p Z

img yZv fY p Z

Graphics

Vision

img

img

11

x

y

u f p X

v f p Y

Z


O


Z

img x

Xu f p

Z

img y

Yv f p

Z


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

img imgu v( , )

f

img xZu fX p Z

img yZv fY p Z

img

img

11

x

y

u f p X

v f p Y

Z

Metric space Pixel space


O


Z

img x

Xu f p

Z

img y

Yv f p

Z


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

img imgu v( , )

f

img xZu fX p Z

img yZv fY p Z

img

img

11

x

y

u f p X

v f p Y

Z


+

Camera intrinsic parameter : metric space to pixel space


O


Z

img x

Xu f p

Z

img y

Yv f p

Z


u vccd ccd( , )

u vimg img( , )

ccdw

ccdh

(0,0)

(0,0) imgw

imgh

x yp p( , )

img imgu v( , )

f

img xZu fX p Z

img yZv fY p Z

img

img

11

x

y

u f p X

v f p Y

Z

K

+

Camera intrinsic parameter : metric space to pixel space


Camera Intrinsic Parameter

O

Projection plane

Z

img

img

11

x

y

u f p X

v f p Y

Z

K

Metric space

( , , )X Y Z


O

Projection plane

Z

img

img

11

x

y

u f p X

v f p Y

Z

K

Metric space

( , , )X Y Z

f


O

Projection plane

Z

img

img

11

x

y

u f p X

v f p Y

Z

K

Metric space

( , , )X Y Z

f

img imgu v( , )

Pixel space

2D Inverse Projection

O

Projection plane

img

img

11

x

y

u f p X

v f p Y

Z

K

Metric space

f

img imgu v( , )

Pixel space

u X

v Y

Z

K

img

-1img

1


O

Projection plane

img

img

11

x

y

u f p X

v f p Y

Z

K

Metric space

f

img imgu v( , )

Pixel space

u X

v Y

Z

K

img

-1img

1

u

vK

img

-1img

1

2D point == 3D ray

3D ray


O

Projection plane

img

img

11

x

y

u f p X

v f p Y

Z

K

Metric space

f

img imgu v( , )

Pixel space

u X

v Y

Z

K

img

-1img

1

u

vK

img

-1img

1

( , , )X Y Z

2D point == 3D ray

3D ray


O

Projection plane

img

img

11

x

y

u f p X

v f p Y

Z

K

Metric space

f

img imgu v( , )

Pixel space

u X

v Y

Z

K

img

-1img

1

u

vK

img

-1img

1

( , , )X Y Z

2D point == 3D ray

3D ray

The 3D point must lie in the 3D ray passing through the origin and 2D image point.

Camera Model - University of Minnesotahspark/CSci5980/Lec1...3D Point Projection (Metric Space) 3D point fm ( , , )X Y Z: Focal length in meter Pinhole -fm T X Z tan ccd X uf ccd m

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