December 11, 2002 MVA2002 1
Determining Shapes of Transparent Determining Shapes of Transparent Objects from Two Polarization ImagesObjects from Two Polarization Images
Daisuke Miyazaki
Masataka Kagesawa
Katsushi Ikeuchi
The University of Tokyo, Japan
December 11, 2002 MVA2002 2
Modeling transparent objectsModeling transparent objects
Polarization-based vision system
Unambiguous determination of surface normal using geometrical invariant
Transparentobject
VR
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Related worksRelated works
Koshikawa 1979Koshikawa et al. 1987
Wolff 1990Wolff et al. 1991
Saito et al. 1999
Miyazaki et al. 2002
Our method
Rahmann et al. 2001
Need many light sourcesNot search corresponding points
Not solve ambiguity problem
Not needcamera calibration
Not needinfrared camera
Sphericaldiffuser
DOP
ThermalradiationBinocular
stereo
Optimizationmethod Searching
correspondingpoints
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OutlineOutline
Rotatethe object
Targetobject
DOP(Degree Of
Polarization)images
Regionsegmentation
Searchcorresponding
points
3D model
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PolarizationPolarization
PolarizerObjectAir
Incident light Reflected light
Surface normal
Transmitted light
Incidentangle
Reflectionangle
DOP(Degree Of Polarization): the ratio of how much the light polarized
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
Lightsource
Unpolarized light(DOP 0)
Pefectly polarized light(DOP 1)
Partially polarized light(DOP 0~1)
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Object
ObservationObservation
Surface normal
Lightsource
Polarizer
Camera
Lightsource
Surfa
ce n
orm
al
P Q
P
Incident angle P
Reflection angle
Q Incident angleQ
Ref
lect
ion
angl
e
PPhase angle
Q Phase angle
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Ambiguity of phase angle Ambiguity of phase angle
Determination of phase angle Propagate the determination from occluding boundary to the inner area (Assume C2 surface)
Inten
sity
255
0
IminP
3601P 2
P
Phase angleAzimuth angle
-ambiguity
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Ambiguity of reflection angle Ambiguity of reflection angle
Reflection angleZenith angle
DO
P(D
eg
ree O
f Pola
rizatio
n)
1
P
0 1P 2
P
Brewsterangle
B 90
-ambiguity
Determination of reflection angle Explain in the following slides
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Object rotationObject rotation
Rotate the object at a small angle Solve the ambiguity from two DOP images
taken from two directions
Rotate
Camera
Object
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Region segmentationRegion segmentation
Measure DOP of the
object
DOP image
DOP 1: whiteDOP 0: black
Result ofregion segmentation
Divided into3 regions
Re
gio
n
segm
en
tation
Divide DOP image with curves of 1 DOP (Brewster angle)
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Gauss’ mapGauss’ mapN
B-E region B-B region B-N region
N NNN
or
EF
B
B: Brewster curve N: North pole E: Equator F: Folding curve
B B B
E E E
F
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B-E region & B-N regionB-E region & B-N region
B-E region (B<<90o)Definition: A region enclosed by occlu
ding boundariesDetermine the occluding boundary fro
m background subtraction
B-N region (0o<<B)Definition: A region where a point of 0o
is included is 0o or 90o when DOP is 0Assume there is no self-occlusion, so
is 0o when DOP is 0
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B-B regionB-B region
B-B region (0o<<B or B<<90o)Definition: A region which is not the previous twoApply the following disambiguation method to this
region
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Folding curveFolding 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)
Folding curve
Brewster curve
Equator
North pole
Gaussian sphere
G=Gaussian sphere
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Parabolic curveParabolic curve
Theorem: Folding curve is parabolic curveParabolic curve = a curve where Gaussian
curvature is 0
Folding curve = geometrical invariantNorth pole
Folding curve
Equator
Gaussian sphereObject surface
Folding curve
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Corresponding pointCorresponding point
Corresponding point folding curve great circle arg min DOP, s. t. surface normal // rotation plane
Correspondingpoint
Rotate the object
North side
Correspondingpoint
Rotate the object
South side
point
[= rotation direction]
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Difference of DOPDifference of DOP
De
rivative
of
DO
PD
OP
+
–
1
0
B 90
0
sgn)(sgn)()(sgn
DO
P b
efo
re
rotation
DO
P a
fter
rotation
Ro
tation
ang
le
De
rivative
of
DO
P
Compare two DOPsat the pair of corresponding points
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Acquisition systemAcquisition system
Object
Camera
Polarizer
Light
Light
Light
Optica
ldiffuse
r
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PrecisionPrecision
Result of region segmentation
Estimated shape
Graph of DOP
Reflectionangle
DO
P
Error (Average absolute difference)
Error
DOP 0.17
Reflection angle 8.5
Height 2.6mm
Plastic transparent hemisphere [diameter 3cm]
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Target objectTarget object
Photo [Acrylic bell-shaped object]
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DOP imagesDOP images
DOP image when the object is not rotated
DOP image when the object is rotated at a small angle
Ro
tation d
irectio
n
We rotate the object about 8°
DOP0:whiteDOP1:black
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Region segmentation resultRegion 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
Ro
tation d
irectio
n
We rotate the object about 8°
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Disambiguation of B-B regionDisambiguation of B-B region
De
rivative
of
DO
P
Positive
NegativeB 90
0
sgn)(sgn)()(sgn
Negative
Po
sitive
Su
rface n
orm
al w
as
Ro
tation d
irectio
n wa
s
0.0890.084
Negative
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Rendered imageRendered image
Shading image
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Rendered imageRendered image
Photo Raytracing image
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ErrorError
Comparison of true value and estimated value
The diameter(width) of the object is 24mmError is 0.4mm (Average of the difference of the height)
True
Estimated
True value is made by hand
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ConclusionsConclusions
A method to measure the surface shape of transparent object based on the analysis of polarization and geometrical characteristicsDetermined the surface normal with no ambiguityDetected a pair of corresponding points of
transparent surfaceDetermined the surface normal of the entire
surface at onceMeasured a transparent object which is not a
hemisphere
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Future worksFuture works
Higher precision (dealing with interreflections) Estimation of refractive index More elegant method for determining phase a
ngle
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(c) Daisuke Miyazaki 2002(c) Daisuke Miyazaki 2002All rights reserved.All rights reserved.
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