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# General Imaging Model PowerPoint PPT Presentation

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.

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

rays become

image points

rays selected

Camera Obscura

compound eyes

system

multiple camera

system

fisheye lens

### General Imaging Model

• Essential components:

• Photosensitive elements

• optics

i

Pi

• Maps incoming pixels to rays

Raxel

symbol

Index

Geometry

Position

Direction

Fall-off

Response

### Raxel = Ray + Pixel

• Small perspective camera

• Simple lens

• One pixel photo-detector

• Most general model is a list of raxels

virtual detectors

(raxels)

• (qq, qf)

(pX,pY,pZ)

physical detectors

(pixels)

ray surface

imaging optics

### Ray Surfaces

Position: (pX,pY,pZ)

Direction: (qq, qf)

caustic

### Rays in 2D

perspective

non-perspective

• Singularity of rays called a caustic

position-direction

space

q

Y

X

position

space

• Solve for d

### Computing Caustics

• Change coordinates

• (x,y,d) (X,Y,Z)

### 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

perspective

single viewpoint

multi-viewpoint

h(x)

• Linear fall-off of optical elements

Normalized

Fall-off

Raxel index

g(e)

Normalized

Response

Normalized Exposure (e)

• Non-linear response of photosensitive element

y

sb

Image plane

sa

• 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

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

• Pointwise fall-offh(x,y)

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

### Finding the parameters: Non-single Viewpoint System

video camera with perspective lens

laptop LCD

sample image

parabolic Mirror

translating stage

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

• Fall-off toward edge as resolution increases:

• less light collected

normalized

fall-off

Index

Geometry

Position

Direction

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