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### General Imaging Model

Michael Grossberg and Shree Nayar

CAVE Lab, Columbia University

ICCV Conference

Vancouver, July 2001

Partially funded by NSF ITR Award, DARPA/ONR MURI

General Imaging Model

- Essential components:
- Photosensitive elements
- optics

i

Pi

- Maps incoming pixels to rays

symbol

Index

Geometry

Radiometry

Position

Direction

Point Spread

Fall-off

Response

Raxel = Ray + Pixel- Small perspective camera
- Simple lens
- One pixel photo-detector

- Most general model is a list of raxels

(raxels)

- (qq, qf)

(pX,pY,pZ)

physical detectors

(pixels)

ray surface

imaging optics

Ray SurfacesPosition: (pX,pY,pZ)

Direction: (qq, qf)

Rays in 2D

perspective

non-perspective

- Singularity of rays called a caustic

position-direction

space

q

Y

X

position

space

Caustic Ray Surface

- Caustic is a singularity or envelope of incoming rays
- Caustic represents loci of view-points

imaging optics

raxels

Caustic curve

- Linear fall-off of optical elements

Normalized

Fall-off

Raxel index

g(e)

Normalized

Response

Normalized Exposure (e)

Raxel Radiometry- Non-linear response of photosensitive element

sb

Image plane

sa

Point Spread- Elliptical gaussian model of point spread.
- Major and minor deviation lengths, sa (d), sb (d)
- Angle of axis y(when sa (d), sb (d) are different)

Chief ray

d, Scene depth

Impulse at Scene point

Finding the Parameters

- Known optical components: Compute

- Unknown optical components: Calibration Environment

Calibration Apparatus

- Structured light at two planes
- Geometry from binary patterns
- Radiometry from uniform patterns

pf

i

pn

qf

z

Finding the parameters: Perspective System

video camera with perspective lens

laptop LCD

sample image

translating stage

Computed Raxel Model: Geometry

180

160

140

120

X in mm

100

80

60

180

160

Y in mm

140

360

120

340

320

100

300

80

280

Z in mm

260

Computed Raxel Model: Radiometry

- Pointwise fall-offh(x,y)

- Radiometric response g(e)

1

.

0

0

.

9

1

1

.

.

0

0

0

.

8

0

0

.

.

8

8

0

.

7

normalized

response

normalized

fall-off

0

.

6

0

0

.

.

6

6

0

.

5

0

0

.

.

4

4

0

.

4

0

.

3

0

0

.

.

2

2

0

.

2

0

0

.

.

0

0

0

.

1

0

.

0

0

0

.

.

0

0

0

0

.

.

1

1

0

0

.

.

2

2

0

0

.

.

3

3

0

0

.

.

4

4

0

0

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.

5

5

0

0

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.

6

6

0

0

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.

7

7

0

0

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8

8

0

0

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.

9

9

1

1

.

.

0

0

0

50

100

150

200

250

300

normalized exposure

radius in pixels

Finding the parameters: Non-single Viewpoint System

video camera with perspective lens

laptop LCD

sample image

parabolic Mirror

translating stage

5

0

-5

60

40

-10

20

0

-15

-20

-40

-20

-60

-25

-30

-35

-60

-40

-20

0

20

40

60

Computed Raxel Model: Geometry- Rotationally symmetric

mm from caustic max

mm from axis of symmetry

mm from axis of symmetry

Computed Raxel Model: Radiometry

- Fall-off toward edge as resolution increases:
- less light collected

normalized

fall-off

radius in pixels

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