General imaging model
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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. Imaging. What is a general imaging model ? How do we Compute its Parameters ?. Scene. Imaging System. Images.

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General Imaging Model

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General imaging model

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


Imaging

Imaging

  • What is a general imaging model ?

  • How do we Compute its Parameters ?

Scene

Imaging System

Images


Perspective imaging model

rays become

image points

rays selected

Perspective Imaging Model

Camera Obscura


Systems that are not perspective

Systems that are not perspective

compound eyes

catadioptric

system

multiple camera

system

fisheye lens


General imaging model1

General Imaging Model

  • Essential components:

    • Photosensitive elements

    • optics

i

Pi

  • Maps incoming pixels to rays


Raxel ray pixel

Raxel

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


Ray surfaces

virtual detectors

(raxels)

  • (qq, qf)

(pX,pY,pZ)

physical detectors

(pixels)

ray surface

imaging optics

Ray Surfaces

Position: (pX,pY,pZ)

Direction: (qq, qf)


Rays in 2d

caustic

Rays in 2D

perspective

non-perspective

  • Singularity of rays called a caustic

position-direction

space

q

Y

X

position

space


Computing caustics

  • Solve for d

Computing Caustics

  • Change coordinates

    • (x,y,d) (X,Y,Z)


Caustic ray surface

Caustic Ray Surface

  • Caustic is a singularity or envelope of incoming rays

  • Caustic represents loci of view-points

imaging optics

raxels

Caustic curve


Simple examples

Simple Examples

perspective

single viewpoint

multi-viewpoint


Raxel radiometry

h(x)

  • 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


Point spread

y

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

Finding the Parameters

  • Known optical components: Compute

  • Unknown optical components: Calibration Environment


Calibration apparatus

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

Finding the parameters: Perspective System

video camera with perspective lens

laptop LCD

sample image

translating stage


Computed raxel model geometry

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

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

.

.

5

5

0

0

.

.

6

6

0

0

.

.

7

7

0

0

.

.

8

8

0

0

.

.

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

Finding the parameters: Non-single Viewpoint System

video camera with perspective lens

laptop LCD

sample image

parabolic Mirror

translating stage


Computed raxel model geometry1

10

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 radiometry1

Computed Raxel Model: Radiometry

  • Fall-off toward edge as resolution increases:

    • less light collected

normalized

fall-off

radius in pixels


Summary

Index

Geometry

Radiometry

Position

Direction

Point Spread

Fall-off

Response

x, y

pX, pY, pZ

qq, qf

sa, sb, y

h

g(e)

Summary

  • Most general model simply list of raxels

  • Caustics summarize geometry

  • Simple procedure for obtaining parameters from a black box system


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