General Imaging Model
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Transcript of General Imaging Model
General Imaging ModelMichael Grossberg and Shree Nayar
CAVE Lab, Columbia University
ICCV ConferenceVancouver, 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
?
Perspective Imaging Model
Camera Obscura
rays selectedrays become image points
Systems that are not perspective
multiple camera system
catadioptric system
fisheye lens
compound eyes
General Imaging Model
• Essential components:– Photosensitive elements– optics
i
Pi
• Maps incoming pixels to rays
Raxel = Ray + Pixel
• Small perspective camera– Simple lens– One pixel photo-detector
Raxel symbol
Index Geometry Radiometry
Position Direction Point Spread Fall-off Response
• Most general model is a list of raxels
Ray Surfaces
(pX, pY, pZ) (q, q)
imaging optics
virtual detectors(raxels)
physical detectors
(pixels)
ray surface
Position: (pX, pY, pZ)Direction: (q, q)
perspective
Rays in 2D
• Singularity of rays called a caustic
position-directionspace
positionspace
XY
non-perspective
caustic
Computing Caustics
• Change coordinates– (x,y,d) (X,Y,Z)
ddx
y(X,Y,Z)
• Solve for d
ZZZZZ
YYYYY
XXXXX
qyq
dyp
xq
dxp
qyq
dyp
xq
dxp
qyq
dyp
xq
dxp
J
)det(
Caustic Ray Surface
• Caustic is a singularity or envelope of incoming rays• Caustic represents loci of view-points
raxels
Caustic curve
imaging optics
Simple Examples
perspective single viewpoint multi-viewpoint
Raxel Radiometry
• Non-linear response of photosensitive element
• Linear fall-off of optical elements
Raxel index
Normalized Fall-off
h(x)
Normalized Exposure (e)
Normalized Response
g(e)
Point Spread
• Elliptical gaussian model of point spread.
– Major and minor deviation lengths, a (d), b (d)
– Angle of axis (when a (d), b (d) are different)
Impulse at Scene point
d, Scene depth
Chief ray
a
b
Image plane
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
z
pfpnqf
i
Finding the parameters: Perspective System
laptop LCD
video camera with perspective lens
translating stage sample image
Computed Raxel Model: Geometry
180
160
360
140
120
100
80
60
180160
140120
10080
340320
300280
260
X in mm
Y in mm
Z in mm
Computed Raxel Model: Radiometry
• Radiometric response g(e)
normalized exposure
normalizedresponse
• Pointwise fall-off h(x,y)
radius in pixels
normalizedfall-off
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
1.00.90.80.70.60.50.40.30.20.10.0 1.00.90.80.70.60.50.40.30.20.10.0 0 50 100 150 200 250 300
0.1
0.0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Finding the parameters: Non-single Viewpoint System
laptop LCDvideo camera with perspective lens
translating stageparabolic Mirror sample image
Computed Raxel Model: Geometry
• Rotationally symmetric
10
5
-35
0
-5
-10
-15
-20
-25
-30
-60-40
-200
6040
20
-60-40
-200
6040
20
mm from caustic max
mm from axis of symmetrymm from axis of symmetry
Computed Raxel Model: Radiometry
• Fall-off toward edge as resolution increases:– less light collected
radius in pixels
normalizedfall-off
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
27025023021019017015013011090 290
Summary
• Most general model simply list of raxels
• Caustics summarize geometry• Simple procedure for obtaining
parameters from a black box system
Index Geometry Radiometry
Position Direction Point Spread Fall-off Response
x, y pX, pY, pZ q, q a, b, h g(e)