the ‘plane space’ a ‘dual’ of the euclidean space jean-yves bouguet 9/30/97
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The ‘plane space’
a ‘dual’ of the Euclidean space
Jean-Yves Bouguet9/30/97
Historic: 3D Photography on your desk
Object
Desk plane
Edge ofthe shadow
d
Light sourceS
s
Shadow plane
P
s
Goal: Estimate the 3D location of P
Stick
Principle
xc
Oc
Xc
Yc
Zc
s
Imageplane
Optical ray(Oc,xc)
P
d
s
s
S
Camera
Intersecting s with the optical ray (Oc,xc):
scc xOP ),(
What is s?
xc
Oc
Xc
Yc
Zc
s
Imageplane
Optical ray(Oc,xc)
P
d
s
s
S
Camera
The shadow plane s contains S and s:
),( ss S
What is s?
xc
Oc
Xc
Yc
Zc
s
Imageplane
Optical ray(Oc,xc)
P
d
s
s
S
Camera
The line es is the projection of the edge s, ors is the intersection of the planes (Oc,s) and d:
dscs O ),(
Conclusion
xc
Oc
Xc
Yc
Zc
s
Imageplane
Optical ray(Oc,xc)
P
d
s
s
S
Camera
)),(,(),( dsccc OSxOP
How do we write the math?
• Preliminary observations:– The key element is the shadow plane s
– Neither s nor d cross the origin Oc
Shadowplane ??
d
s
s
Central objects: Planes
Definition of a plane that does not cross the origin
dXn ,
n
d
1nwith and 0d
Oc
XY
Z
cO
normalvector
distance tothe origin
X
P
P
.,.Note: dot product
Central objects: Planes
or: 1, X n
d
d
nwith:
Oc
XY
Z
T321 Plane vector:
)0(
The ‘plane space’
Oc
XY
Z
O
Euclidean space Plane space
()
Observation
b
a
Consider two planes a and b
that intersect along the line
Oc
Xc
Yc
Zc
Imageplane
Let be the projection of on the image plane
Observation (cont’d)
b
a
Oc
Xc
Yc
Zc
Imageplane
Projected line:
aa
bb Parameterization:
such that:
0, xx
Tyxx 1with:
Observation (cont’d)
b
a
Oc
Xc
Yc
Zc
Imageplane
Proof:
)( a
ba Proposition:)( b
Let P be a point on
Z
Y
X
X:P
1
1y
x
XZ
x
3D space Image plane
P
x
Observation (cont’d)
b
a
Oc
Xc
Yc
Zc
Imageplane
Proof (cont’d):
)( a
ba Proposition:)( b
1, XP aa
P
x
1, XP bb
0,
0,1
0,
x
XZ
X
ba
ba
ba
ba .,.Note: dot product
Observation (cont’d)
b
a
Oc
Xc
Yc
Zc
Imageplane
)( a
)( b
O
a
b
Plane space
)(()
The dual of a line
Oc
XY
Z
O
Euclidean space Plane space
Set of planes that contain the line
Perspective projection of onto the image plane
()
The dual of a point
Oc
XY
Z
O
Euclidean space Plane space
P
Set of planes that contain point P
xX P
X
1, X
()
What about the shadow plane?
O
Set of candidate shadow planes
Perspective projection of s onto the image plane
s
Where is the Shadow plane s?
d
s
s
)( d
)( s
d
s
Need of an additional constraint!
?
s()
Where is the shadow plane? (cont’d)
O
s
d
s
s
)( d
)( s
d
sS
Extra constraint: sS S Dual of S
s
Sssˆˆ Shadow plane s:
()
SS ssˆ
Where is the shadow plane? (cont’d)
s
d
s
s
)( d
)( s
d
sUse of an extra plane r
s
rss ˆˆShadow plane s:
r
)( rr
r
r
r
Alternative method:
Projection of r onto the image plane
Note: Least squares estimate in case of noise
()
O
Properties (1)
1 )( 1
2 )( 2
1
2
O
()
Intersecting planes
Properties (2)
O
1
()
1 )( 1
2 )( 22 1 )( 1
2 )( 2
O
1
()
2
0, 21
Parallel planes
Orthogonal planes
Dual of the horizon line
HHorizon
lineH
Properties (3)
O
()
)(1
2
12P
P
)(
1
Coplanar intersecting lines
Parallel lines
2
O
()
12
V
VO ˆ
V
vanishing point
Horizon line
H
H
Properties (4)
O
)(1
2
12P
)ˆ,ˆ(ˆ21 P
Coplanar orthogonal lines
(not shown)
()
Properties (4)
O
)(1
2
12P
)ˆ,ˆ(ˆ21 P
Coplanar orthogonal lines
(not shown)
Set of orthogonal planes to
Properties (4)
O
)(1
2
12P
)ˆ,ˆ(ˆ21 P
Coplanar orthogonal lines
(not shown)
Set of orthogonal planes to
Properties (4)
O
)(1
2
12P
)ˆ,ˆ(ˆ21 P
Coplanar orthogonal lines
(not shown)
Set of orthogonal planes to
3
3
Example 1
O
() )(
Horizon line H
Image plane
H
H
Set of candidate ground planes
d
O
()
H
H
1/d
Ground plane
Example 1 (cont’d)
)(
H
Image plane
V
12
()
O
H
H
1
2
V
Ground plane
a
b
W
Vanishing point
road lines
Width of the road:ba
W11
12
VHHV ˆˆ
Example 2: Calibration
()O
1
2
V
Desk plane?
Vanishing points
Gridedges
Image plane
12
V
W
12
H
U
Horizon: VUHVUH ˆˆˆ),(
L
U
3
4
Example 2: Calibration
()
O
1
2
Gridedges
Image plane
12
V
W
12
H
UL
Set of orthogonal planes to the desk
4
3
Gridedges
baW
11
a
b
cd
dcL
11
3
4
3D Photography on your desk
s
d
s
s
)( d
)( s
d
sSS
s
sds . 1, SS ss S
S
s
d
,
,1
Shadow plane:
xZX .
3D coordinates of P: is the plane of direction thatcontainsP x
sx
Z s ,1
3D Photography on your desk
OX
Y
Z
s
Imageplane
P
d
s
S
Camera
x
ss d
sS
s
x
P
Note: 0, ss xx
Psˆˆ
sP
)( s
)( d
Interesting features
• Simple formalism
• Convenient for plane estimation
• Natural link with the perspective projection operator
• Vanishing points and Horizon lines are natural objects in that space
Looking at pictures differently
Future: What about curves?
Final remark
There exists a strong similarity between this formalism and the way the Reciprocal Lattice is defined in Crystallography.
Reference: “Solid State Physics” by Neil W. Ashcroft and N. David Merminl, Saunders College Publishing international Edition, Chapter 5, pp 85-94