Differential Geometry

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Marr ??????
3D ??
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2-1/2D ??
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3D ???? ?????
2D ??
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wi
tj
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????

• 3??????????????
• Detectability
• Reliability
• Uniqueness

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Detectability

• 1????????????????

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?
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Reliability

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b2
a2
b2
a2
a1
b1
??
b1
a1
????
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Uniqueness

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a1
a2
a1
b1
unique
a2
b1
????
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• 15????????????

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?? 090 11 135 6 270 150
30 285 39 345 7 165 7 270
56 155 300 7 315 19 330 7
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????(????)
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3?????

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???????????

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???? (3)

• 3. ???????????Template matching
• ??? 86

Classified as Real Class Real Class Real Class Real Class
Classified as Sedan Minivan Hatchback Wagon
Sedan 19 0 0 1
Minivan 0 10 0 0
Hatchback 0 3 13 0
Wagon 2 2 0 15
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• ???85??

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• 2001/9/5
• 11001600
• ????
• ??246??
• ????? (3.5km)
• ??????? (1.0km)

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?????????
Face Label
F1 F2 F3 F4 F5 F6 F7 F8 1 1 1 0 0 0
0 0
Visible
Invisible
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?????????2
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• ??CAD???????

??????? ??? --- ????????? ???? ? --- ???? ??
--- ????
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???????
PCW
NCCW
PVert
Nface
Pface
NVert
NCW
PCCW
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Surface based representation on
sphere(Well-tessellated sphere rep)

• represent directions as points on the unit sphere
  (usually geodesic dome)
• moving the points on the sphere out by their
  associated magnitude (usually the distance
  between a surface point and the mass center)

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• 1. curve -- s arc length
• a(s) ( x(s), y(s) )
• 2. tangent of a curve
• a(s) ( x(s), y(s) )
• 3. curvature of a curve
• a(s) ( x(s), y(s) )
• a(s) -- curvature

s
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?(?)

• 1. Arc length, s
• 2. coordinates
• 3. tangent
• 4. curvature

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• The normal direction (n) toward the empty side.

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s0
a
b
d
c
a
b
s0
d
c
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a

b
e
a

d
c
b
f
h
k

d
a
c
i
e
g
j
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F. Klein???
F.Klein ??? ??????????????? F.Klein??? ????????
?????????? F.Kline??? ????????????????
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????(lines of curvature)
???( Principal direction) ?????????????
???? ??????????
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??????
1.????? 2.??????(EGI) 3.????????????????
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?????
gauss map
1D
gauss map
2D
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?????? Extended Gaussian Image (EGI)
??????????????????????
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????????
side view
top view
Cylinder
Ellipsoid
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?????????
1. EGI??????????2??????????? 2.
?????????????????????????????? 3.
?????EGI???????????? 4. EGI???????? 5.
??????EGI??????
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EGI?????????
object
Gaussian sphere
small
large
small
(K small)
small
large
large
(K large)
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EGI???

• ???????
• ?????????????
• ????????????
• ????????

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EGI????
10
20
0
viewing direction
0
8
5
EGI table
0
8
5
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??EGI
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EGI????????????????
Photometric stereo segmentation Region
selection Photometric stereo EGI
generation EGI matching Grasp planning
Needle map isolated regions target
region precise needle map EGI object attitude
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???????

• 1. ????
• 3. ?????
• 4. ??????
• 5. ????

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????

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????
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input scene
Segmentation result
grasp execution
grasp plan
Superquadric representation of a rock
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???(?????????)
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Symmetry seeking ??
Snakes?3???
x
s
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??????(???)
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????
2D ??
l
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O
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2?????????
rotation
similar shapes?
contours
circular representations
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3??????
Regularization Each Mech has roughly equal
area Each mesh has the same topology
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????????Spherical Attribute Image (SAI)
Gaussian Curvature
Deformed Surface
SAI
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?????
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3?????
??
S
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???????????
Warped
Skewed
Layered
GC
axis
Transformation rule
Cross sections
Cross section planes
Bilinear
Straight
Torridal
Right
Circular
Open
Polygonal
Homogeneous
Linear
Uniform
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Straight Homogeneous Generalized Cylinders
1. a(s) is a linear func 2. n-b planes are
parallel 3. All cross sections have the same
shape but may vary in size.
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• Initial light-strip
  image
• initial axis
  estimation
• preliminary center and
  axis estimation
• Corn with
  smoothed radius function
• completed
  analysis

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ACRONYM
USERS
Data Structure
LIBRARY
HI-LEVEL MOOELER
CONTEXT GRAPH
modelary module (user ? GC)
OBJECT GRAPH
GRAPHICS MODULE
prediction module
PREDICTOR PLANNER
OBSERVABILITY GRAPH
MATCHER
interpretation module
INTERPRETATION GRAPH
description module (image ? GC)
SURFACE MAPPER
PICTURE GRAPH
PIX
PIX
EDGE MAPPER
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L?f?d?????????????????
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???
Edge image (PIX)
GC description
Interpretation graph
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7 lt WING-WIDTH lt 12 1000 lt M lt 12000
7 lt WING-WIDTH lt 10.5677531 2199 lt M lt 3322
MATCH -- L1011!!!
2356 lt M lt 2489
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z
x
y
1. ????? 2. ??????
r
z
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• ?????????????????
• ????
• 1?????????????????????

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2?????????
Quad-tree
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3????????
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Oct-tree
0
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Constructive solid geometry(CSG)??

• ?????????????????????????
• CAD????????
• ??CSG????????????????

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CSG???
ltCSG repgt ltprimitive solidgt
move ltCSG Repgt by ltMotion paramsgt
ltCSG RepgtltoperationgtltCSG Repgt leaf
node either primitive or motion params other
node Boolean operations (UNION, DIF, INT)
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• Symmetry Seeking
• ????
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• constructive solid geometry
• ??????