Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection...
Transcript of Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection...
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Projective Line
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Parallel lines in 3D converge to a point in the image.
MLPS-St. Paul International Airport
Indoor point at infinity
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3D Parallel Line Projection
Ground plane
Camera plane
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3D Parallel Line Projection
Ground plane
Camera plane
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3D Parallel Line Projection
Ground plane
Camera plane
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3D Parallel Line Projection
Ground plane
Camera plane
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3D Parallel Line Projection
Ground plane
Camera plane
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3D Parallel Line Projection
Ground plane
Camera plane
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3D Parallel Line Projection
Vanishing point
Ground plane
Camera plane
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3D Parallel Line Projection
Vanishing point
Ground plane
Camera plane
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3D Parallel Line Projection
Vanishing point
Ground plane
1. Parallel lines in 3D meet at the same vanishing point in image.
Camera plane
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3D Parallel Line Projection
Vanishing point
Ground plane
1. Parallel lines in 3D meet at the same vanishing point in image. 2. The 3D ray passing camera center and the vanishing point is parallel to the lines.
Camera plane
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Vanishing Point
Vanishing point
1. Parallel lines in 3D meet at the same vanishing point in image. 2. The 3D ray passing camera center and the vanishing point is parallel to the lines. 3. Multiple vanishing points exist.
Ground plane
Camera plane
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Vanishing point
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Vanishing point
Vanishing point
Multiple vanishing point
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Vanishing point
Vanishing point
Vanishing line: Horizon
Vanishing line for horizon
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Vanishing point
Vanishing point
What can vanishing line tell us about me?
Vanishing line for horizon
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Keller Hall
Vanishing point
Vanishing point
What can vanishing line tell us about me? • Horizon
Vanishing line for horizon
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Keller Hall
Vanishing point
Vanishing point
What can vanishing line tell us about me? • Horizon • Camera pitch angle (looking down)
Vanishing line for horizon
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Keller Hall
Vanishing point
Vanishing point
What can vanishing line tell us about me? • Horizon • Camera pitch angle (looking down) • Camera roll angle (tilted toward right)
Vanishing line for horizon
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Celestial Navigation Far far away: point at infinity
Horizon
Earth
At least two stars are needed.
Two points at infinity (vanishing points) tells us about where I am.
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Parallel 3D planes share the vanishing line.
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Different plane produces different vanishing line.
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Different plane produces different vanishing line.
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Different plane produces different vanishing line.
How to compute a vanishing point?
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Point-Line in Image
A 2D line passing through 2D point (u,v ):
au bv c 0 au bv c 0
( , )u v
Line parameter: (a,b,c )
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Point-Line in Image
A 2D line passing through 2D point (u,v ):
au bv c 0 au bv c 0
( , )u v
Line parameter: (a,b,c )
u
a b c v l xT 0
1
au bv c 0
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Point-Line in Image
A 2D line passing through 2D point (u,v ):
au bv c 0 au bv c 0
( , )u v
Line parameter: (a,b,c )
u
a b c v l xT 0
1
where
u
vx
1
a
b
c
land
au bv c 0
Line parameter 2D point
=
a
b
c
l2D line:
1
u
vx
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Point-Point in Image A 2D line passing through two 2D points:
1 1 0 au bv c
1 1( , )u v
1xx2
2 2( , )u v
2 2 0 au bv c
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Point-Point in Image
1 1 0 au bv c
1 1( , )u v
1xx2
2 2( , )u v
2 2 0 au bv c
x lT1 0 x lT
2 0
1
1 1
1
x
u
v2
2 2
1
x
u
v
a
b
c
lwhere
A 2D line passing through two 2D points:
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Point-Point in Image
1 1 0 au bv c
1 1( , )u v
1xx2
2 2( , )u v
2 2 0 au bv c
x lT1 0 x lT
2 0
1
1 1
1
x
u
v2
2 2
1
x
u
v
a
b
c
lwhere
A 2D line passing through two 2D points:
xl 0
x
T1
T2
2x3
0 0 A
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Point-Point in Image
1 1 0 au bv c
1 1( , )u v
1xx2
2 2( , )u v
2 2 0 au bv c
x lT1 0 x lT
2 0
1
1 1
1
x
u
v2
2 2
1
x
u
v
a
b
c
lwhere
A 2D line passing through two 2D points:
xl 0
x
T1
T2
0 0 A
null A l x x1 2=or
2x3
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Point-Point in Image
(1804,934) (1052,1323)
x1 = [1804;934;1]; x2 = [1052;1323;1];
l
1 1( , )u v
1xx2
2 2( , )u v
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Point-Point in Image
(1804,934) (1052,1323)
x1 = [1804;934;1]; x2 = [1052;1323;1]; l = Vec2Skew(x1)*x2;
Cross product
l
1 1( , )u v
1xx2
2 2( , )u v
GetLineFromTwoPoints.m
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Point-Point in Image
(1804,934) (1052,1323)
x1 = [1804;934;1]; x2 = [1052;1323;1]; l = Vec2Skew(x1)*x2; l = -389 -752 1404124
Cross product
l
1 1( , )u v
1xx2
2 2( , )u v
GetLineFromTwoPoints.m
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Point-Point in Image
(1804,934) (1052,1323)
x1 = [1804;934;1]; x2 = [1052;1323;1]; l = Vec2Skew(x1)*x2; l = -389 -752 1404124
Cross product
a b a b
a b a b
a b a b
a a b
a -a b
-a a b
a b
a b
2 3 3 2
3 1 1 3
1 2 2 1
3 2 1
3 1 2
2 1 3
-
= -
-
0 -
0
0
Cross product with skew-symmetric matrix representation:
function skew = Vec2Skew(v) skew = [0 -v(3) v(2); v(3) 0 -v(1); -v(2) v(1) 0];
l
1 1( , )u v
1xx2
2 2( , )u v
Vec2Skew.m
GetLineFromTwoPoints.m
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Two 2D lines in an image intersect at a 2D point:
Line-Line in Image
1
1 1
1
l
a
b
c
( , )u vx
2
2 2
2
l
a
b
c
1 1 1 0 a u b v c2 2 2 0 a u b v c
(1804,934)
![Page 40: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/40.jpg)
Two 2D lines in an image intersect at a 2D point:
Line-Line in Image
l xT1 0 l xT
2 0
1
1 1
1
l
a
b
c
( , )u vx
2
2 2
2
l
a
b
c
1 1 1 0 a u b v c2 2 2 0 a u b v c
1
x
u
v1
1 1
1
l
a
b
c
where
2
2 2
2
l
a
b
c(1804,934)
![Page 41: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/41.jpg)
Two 2D lines in an image intersect at a 2D point:
Line-Line in Image
l xT1 0 l xT
2 0
lx 0
l
T1
T2
1 2= x l l
1
1 1
1
l
a
b
c
( , )u vx
2
2 2
2
l
a
b
c
1 1 1 0 a u b v c2 2 2 0 a u b v c
1
x
u
v1
1 1
1
l
a
b
c
where
2
2 2
2
l
a
b
c
2x3
0 0 A
null A
or
(1804,934)
![Page 42: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/42.jpg)
Line-Line in Image
1
1 1
1
l
a
b
c
( , )u vx
2
2 2
2
l
a
b
c
l1 = [-398;-752;1404124]; l2 = [310;-924;303790]; x = Vec2Skew(l1)*l2; x = x/x(3) x = 1779.0 925.6 1
similar to (1804,934)
(1804,934)
GetPointFromTwoLines.m
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2D Point and Line Duality
l x x1 2
The intersection between two lines:
x l l1 2
Given any formula, we can switch the meaning of point and line to get another formula.
The 2D line joining two points:
![Page 44: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/44.jpg)
2D Point and Line Duality
l x x1 2
The intersection between two lines:
x l l1 2
Given any formula, we can switch the meaning of point and line to get another formula.
1x
2x
The 2D line joining two points:
T: Transformation -T2 1 2 1 x Tx l T l
![Page 45: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/45.jpg)
2D Point and Line Duality
l x x1 2
The intersection between two lines:
x l l1 2
Given any formula, we can switch the meaning of point and line to get another formula.
1x
2x
The 2D line joining two points:
T
T T -1 -T T1 1 1 1 1 1 2 2 l x l T Tx T l Tx l x
T: Transformation -T2 1 2 1 x Tx l T l
0
![Page 46: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/46.jpg)
2D Point and Line Duality
l x x1 2
The intersection between two lines:
x l l1 2
Given any formula, we can switch the meaning of point and line to get another formula.
1x
2x
The 2D line joining two points:
T
T T -1 -T T1 1 1 1 1 1 2 2 l x l T Tx T l Tx l x
T: Transformation -T2 1 2 1 x Tx l T l
0
![Page 47: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/47.jpg)
2D Point and Line Duality
l x x1 2
The intersection between two lines:
x l l1 2
Given any formula, we can switch the meaning of point and line to get another formula.
1x
2x
The 2D line joining two points:
T
T T -1 -T T1 1 1 1 1 1 2 2 l x l T Tx T l Tx l x
T: Transformation -T2 1 2 1 x Tx l T l
![Page 48: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/48.jpg)
2D Point and Line Duality
l x x1 2
The intersection between two lines:
x l l1 2
Given any formula, we can switch the meaning of point and line to get another formula.
2 1
cos sin 0
-sin cos 0
0 0 1
x x
1x
2x
The 2D line joining two points:
T
T T -1 -T T1 1 1 1 1 1 2 2 l x l T Tx T l Tx l x
T: Transformation
T
2 1 1
cos sin 0 cos sin 0
-sin cos 0 -sin cos 0
0 0 1 0 0 1
l l l
-T2 1 2 1 x Tx l T l
?
![Page 49: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/49.jpg)
2D Point and Line Duality
l x x1 2
The intersection between two lines:
x l l1 2
Given any formula, we can switch the meaning of point and line to get another formula.
2 1
cos sin 0
-sin cos 0
0 0 1
x x
1x
2x
The 2D line joining two points:
T
T T -1 -T T1 1 1 1 1 1 2 2 l x l T Tx T l Tx l x
T: Transformation
T
2 1 1
cos sin 0 cos sin 0
-sin cos 0 -sin cos 0
0 0 1 0 0 1
l l l
-T2 1 2 1 x Tx l T l
![Page 50: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/50.jpg)
Geometric Interpretation (Point)
CO u
u
![Page 51: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/51.jpg)
Geometric Interpretation (Point)
CO u
u
-1K u
![Page 52: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/52.jpg)
Geometric Interpretation (Point)
CO
u
-1 u K uu
Normalized coordinate:
-1u K u
![Page 53: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/53.jpg)
Geometric Interpretation (Point)
CO
1u
-11 1 u K u
1u
2u
-12 2 u K u
2u
-12 2u K u
Normalized coordinate:
-11 1u K u
![Page 54: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/54.jpg)
Geometric Interpretation (Line)
CO
l
l
Normalized coordinate:
-T
-1 T l K l K l due to duality
1l = u 2u
where ?
1u
2u
-12 2 u K u
-11 1 u K u
1u
2u
-11 1u K u -1
2 2u K u
![Page 55: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/55.jpg)
Geometric Interpretation (Line)
CO
l
l
Normalized coordinate:
due to duality
1l = u 2u1u
2u
-12 2 u K u
-11 1 u K u
1u
2u
-11 1u K u -1
2 2u K u
-T
-1 T l K l K l due to duality where
![Page 56: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/56.jpg)
CO
l
l
Geometric Interpretation (Line) 3D plane π Normalized coordinate:
due to duality where
A 2D line in an image defines to a 3D plane passing the camera center:
l π
1u-1
1 1 u K u
2u
-12 2 u K u
1u
2u
-11 1u K u -1
2 2u K u
1l = u 2u
-T
-1 T l K l K l
![Page 57: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/57.jpg)
CO
l
l
Geometric Interpretation (Line) 3D plane π Normalized coordinate:
due to duality where
A 2D line in an image defines to a 3D plane passing the camera center:
l π
Plane normal: 1 1 x 2 2 x l
l
?
1u-1
1 1 u K u
2u
-12 2 u K u
1u
2u
-11 1u K u -1
2 2u K u
1l = u 2u
-T
-1 T l K l K l
![Page 58: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/58.jpg)
CO
l
l
Geometric Interpretation (Line) 3D plane π Normalized coordinate:
due to duality where
A 2D line in an image defines to a 3D plane passing the camera center:
l π
Plane normal: 1 1 u 2 2 u l
l
1u-1
1 1 u K u
2u
-12 2 u K u
1u
2u
-11 1u K u -1
2 2u K u
1l = u 2u
-T
-1 T l K l K l
![Page 59: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/59.jpg)
CO
l
l
Geometric Interpretation (Line) 3D plane π Normalized coordinate:
due to duality where
A 2D line in an image defines to a 3D plane passing the camera center:
l π
Plane normal: 1 1 u 2 2 u l
l
1u-1
1 1 u K u
2u
-12 2 u K u
1u
2u
-11 1u K u -1
2 2u K u
1l = u 2u
-T
-1 T l K l K l
0
l
0
lπ
![Page 60: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/60.jpg)
CO
l
Geometric Interpretation (Line-Line) 1π
![Page 61: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/61.jpg)
CO
Geometric Interpretation (Line-Line)
l1l2
1π
2π
![Page 62: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/62.jpg)
CO
Geometric Interpretation (Line-Line)
l1l2
1π
2π
x
u
2D lines in an image intersect a 2D point corresponding to a 3D ray:
u1= l 2l
![Page 63: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/63.jpg)
CO
Geometric Interpretation (Line-Line)
l1l2
1π
2π
u
2D lines in an image intersect a 2D point corresponding to a 3D ray:
1= l 2l1l
2lu
: the 3D ray is perpendicular to two plane normals.
uu
![Page 64: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/64.jpg)
Vanishing Point
11l
1l 2
Ground plane
1u
2u3u
4u
Parallel lines:
11 4 3 l u u 12 1 2 l u u
![Page 65: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/65.jpg)
Vanishing Point
11l
Ground plane
4u
Parallel lines:
11 4 3 l u u 12 1 2 l u u
21 4 1 l u u 22 3 4 l u u
2l 1
l221l 2
1u
2u3u
![Page 66: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/66.jpg)
Vanishing Point
11l
x1
Ground plane
4u
Parallel lines:
11 4 3 l u u 12 1 2 l u u
21 4 1 l u u 22 3 4 l u u
2l 1
l221l 2
1u
2u3u
x l l1 11 12
Vanishing points:
x l l2 21 22
x2
![Page 67: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/67.jpg)
Vanishing Point
11l
1v
Ground plane
4u
Parallel lines:
11 4 3 l u u 12 1 2 l u u
21 4 1 l u u 22 3 4 l u u
2l 1
l221l 2
1u
2u3u
1 11 12 v l l
Vanishing points:
2 21 22 v l l
2v Vanishing line:
1 2 l v vl
![Page 68: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/68.jpg)
Vanishing Point
11l
Ground plane
2l 1
l221l 2
l
l11 = GetLineFromTwoPoints(m11,m12); l12 = GetLineFromTwoPoints(m13,m14); l21 = GetLineFromTwoPoints(m21,m22); l22 = GetLineFromTwoPoints(m23,m24); v1 = GetPointFromTwoLines(l11,l12); v2 = GetPointFromTwoLines(l21,l22); vanishing_line = GetLineFromTwoPoints(x1, x2);
m11
m12
m13
m14
ComputeVanishingLine.m
2v
1v
![Page 69: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/69.jpg)
Geometric Interpretation of Vanishing Line
Ground plane
Camera plane
![Page 70: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/70.jpg)
Geometric Interpretation of Vanishing Line
Ground plane
Camera plane
![Page 71: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/71.jpg)
Geometric Interpretation of Vanishing Line
Ground plane
Camera plane
![Page 72: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/72.jpg)
Geometric Interpretation of Vanishing Line
Ground plane
Camera plane
Side view
Ground plane
Plane of vanishing line
Plane of vanishing line
![Page 73: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/73.jpg)
Geometric Interpretation of Vanishing Line
Side view
Ground plane
Plane of vanishing line
Height
![Page 74: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/74.jpg)
Where was I (how high)?
![Page 75: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/75.jpg)
Where was I (how high)?
Taken from my hotel room (6th floor) Taken from beach
![Page 76: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/76.jpg)
Where was I (how high)?
Taken from my hotel room (6th floor) Taken from beach
![Page 77: Projective Line - University of Minnesotahspark/CSci5980/Lec3... · 3D Parallel Line Projection Vanishing point Ground plane 1. Parallel lines in 3D meet at the same vanishing point](https://reader033.fdocuments.in/reader033/viewer/2022042923/5f706f0738b88934f347ee9e/html5/thumbnails/77.jpg)
First person video Cylindrical projection
Height of the camera wearer