Determining Shapes of Transparent
Objects from Two Polarization Images
Daisuke Miyazaki
Masataka Kagesawa
Katsushi Ikeuchi
The University of Tokyo, Japan
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Modeling transparent objects
 Polarization-based vision system
 Unambiguous determination of
surface normal using geometrical
invariant
Transparent
object
VR
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Related works
Wolff 1990
Wolff et al. 1991
Koshikawa 1979
Koshikawa et al. 1987
Not search corresponding points
Need many light sources
DOP
Optimization
method
Spherical
diffuser
Saito et al. 1999
Searching
corresponding
points
Rahmann et al. 2001
Not need
camera calibration
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Not solve ambiguity problem
Thermal
radiation
Binocular
stereo
Our method
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Miyazaki et al. 2002
Not need
infrared camera
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Outline
3D model
Target
object
Rotate
the object
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DOP
Region
(Degree Of segmentation
Polarization)
images
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Search
corresponding
points
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Polarization
DOP
Origin
Unpolarized light
0 Sunlight / incandescent light
Perfectly polarized light
1 The light transmitted the polarizer
Partially polarized light
0~1 The light hit the object surface
DOP(Degree Of Polarization): the ratio of how much the light polarized
Surface normal
Incident Reflection
angle angle
Light
source
Unpolarized light
(DOP 0)
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Air
Object
q q
Polarizer
Pefectly polarized light
(DOP 1)
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Partially polarized light
(DOP 0~1)
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Observation
Camera
Polarizer
qP
qP
qQ
Light
source
Phase angle
qQ
P
P
Q
Object
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Light
source
Q Phase angle
6
Ambiguity of phase angle 
255
Intensity
IminP
0
1P
2P
-ambiguity
360
Phase angle
Azimuth angle 
Determination of phase angle 
Propagate the determination from occluding boundary
to the inner area (Assume C2 surface)
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Ambiguity of reflection angle q
DOP
(Degree Of Polarization) 
1
P
0
q1P
qB
Brewster
angle
q2P
90
Reflection angle
Zenith angle q
q-ambiguity
Determination of reflection angle q
Explain in the following slides
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Object rotation
 Rotate the object at a small angle
 Solve the ambiguity from two DOP images
taken from two directions
Camera
Rotate
Object
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Region segmentation
Divide DOP image with curves of 1 DOP (Brewster angle)
Region segmentation
Measure
DOP of the
object
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DOP image
DOP 1: white
DOP 0: black
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Result of
region segmentation
Divided into
3 regions
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Gauss’ map
N
N
N
N
N
F
B
B
E
E
B
F
E
B
E
or
B-E region
B: Brewster curve
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B-B region
N: North pole
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E: Equator
B-N region
F: Folding curve
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B-E region & B-N region
o
 B-E region (qB<q<90 )
Definition: A region enclosed by
occluding boundaries
Determine the occluding boundary
from background subtraction
o
 B-N region (0 <q<qB)
Definition: A region where a point of 0
is included
o
o
q is 0 or 90 when DOP is 0
Assume there is no self-occlusion, so
o
q is 0 when DOP is 0
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o
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B-B region
 B-B region (0 <q<qB or qB<q<90 )
o
o
Definition: A region which is not the previous two
Apply the following disambiguation method to this
region
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Folding curve
 A curve (on G) that is a part of the boundary
of the region (on G) and is not a Brewster
curve (on G) is called a folding curve (on G)
G=Gaussian sphere
North pole
Folding curve
Brewster curve
Equator
Gaussian sphere
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Parabolic curve
 Theorem: Folding curve is parabolic curve
Parabolic curve = a curve where Gaussian
curvature is 0
 Folding curve = geometrical invariant
North pole
Folding curve
Equator
Folding curve
Object surface
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Gaussian sphere
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Corresponding point
 Corresponding point
 folding curve  great circle [= rotation direction]
 arg min DOP, s. t.
point
surface normal // rotation plane
Corresponding
point
Corresponding
point
Rotate the object
Rotate the object
North side
South side
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Difference of DOP
sgn  (q  q )   (q )   sgn  (q ) sgn q 
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Derivative of DOP
Rotation angle
Derivative of DOP
DOP before rotation
DOP after rotation
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DOP
Compare two DOPs
at the pair of corresponding points
0
+
0
–
qB
90
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Acquisition system
Camera
Light
Light
Polarizer
Light
Optical
diffuser
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Object
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Precision
Plastic transparent hemisphere [diameter 3cm]
Estimated shape
DOP 
Result of region segmentation
Error
DOP
0.17
Reflection angle
8.5
Height
Reflection
angle q
2.6mm
Error (Average absolute difference)
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Graph of DOP
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Target object
 Photo [Acrylic bell-shaped object]
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DOP images
 DOP image when the
object is not rotated
 DOP image when the
object is rotated at a
small angle
DOP0:white
DOP1:black
Rotation direction
We rotate the object about 8°
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Region segmentation result
 Result of region
segmentation when the
object is not rotated
 Result of region
segmentation when the
object is rotated at a
small angle
Rotation direction
We rotate the object about 8°
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Disambiguation of B-B region
sgn  (q  q )   (q )   sgn  (q ) sgn q 
0.089
Negative
Positive
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Surface normal was 
Rotation direction was 
Negative
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Derivative of DOP
0.084
Positive
0
Negative
qB
90
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Rendered image
 Shading image
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Rendered image
 Photo
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 Raytracing image
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Error
 Comparison of true value and estimated
value
True
Estimated
The diameter(width) of the object is 24mm
Error is 0.4mm (Average of the difference of the height)
True value is made by hand
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Conclusions
 A method to measure the surface shape of
transparent object based on the analysis of
polarization and geometrical characteristics
Determined the surface normal with no ambiguity
Detected a pair of corresponding points of
transparent surface
Determined the surface normal of the entire
surface at once
Measured a transparent object which is not a
hemisphere
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Future works
 Higher precision (dealing with interreflections)
 Estimation of refractive index
 More elegant method for determining phase
angle 
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Daisuke Miyazaki 2002
Creative Commons Attribution 4.0
International License.
http://www.cvl.iis.u-tokyo.ac.jp/
D. Miyazaki, M. Kagesawa, K. Ikeuchi, "Determining
Shapes of Transparent Objects from Two Polarization
Images," in Proceedings of IAPR Workshop on
Machine Vision Applications, pp.26-31, Nara, Japan,
2002.12
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