Projective Line

Parallel lines in 3D converge to a point in the image.

MLPS-St. Paul International Airport

Indoor point at infinity

3D Parallel Line Projection

Ground plane

Camera plane

3D Parallel Line Projection

Ground plane

Camera plane

3D Parallel Line Projection

Ground plane

Camera plane

3D Parallel Line Projection

Ground plane

Camera plane

3D Parallel Line Projection

Ground plane

Camera plane

3D Parallel Line Projection

Ground plane

Camera plane

3D Parallel Line Projection

Vanishing point

Ground plane

Camera plane

3D Parallel Line Projection

Vanishing point

Ground plane

Camera plane

3D Parallel Line Projection

Vanishing point

Ground plane

1. Parallel lines in 3D meet at the same vanishing point in image.

Camera plane

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

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

Vanishing point

Vanishing point

Vanishing point

Multiple vanishing point

Vanishing point

Vanishing point

Vanishing line: Horizon

Vanishing line for horizon

Vanishing point

Vanishing point

What can vanishing line tell us about me?

Vanishing line for horizon

Keller Hall

Vanishing point

Vanishing point

What can vanishing line tell us about me? • Horizon

Vanishing line for horizon

Keller Hall

Vanishing point

Vanishing point

What can vanishing line tell us about me? • Horizon • Camera pitch angle (looking down)

Vanishing line for horizon

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

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.

Parallel 3D planes share the vanishing line.

Different plane produces different vanishing line.

Different plane produces different vanishing line.

Different plane produces different vanishing line.

How to compute a vanishing point?

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 )

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

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

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

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:

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

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

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

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

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

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

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)

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)

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)

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

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:

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

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

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

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

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

?

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

Geometric Interpretation (Point)

CO u

u

Geometric Interpretation (Point)

CO u

u

-1K u

Geometric Interpretation (Point)

CO

u

-1 u K uu

Normalized coordinate:

-1u K u

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

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

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

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

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

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

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

CO

l

Geometric Interpretation (Line-Line) 1π

CO

Geometric Interpretation (Line-Line)

l1l2

1π

2π

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

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

Vanishing Point

11l

1l 2

Ground plane

1u

2u3u

4u

Parallel lines:

11 4 3 l u u 12 1 2 l u u

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

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

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

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

Geometric Interpretation of Vanishing Line

Ground plane

Camera plane

Geometric Interpretation of Vanishing Line

Ground plane

Camera plane

Geometric Interpretation of Vanishing Line

Ground plane

Camera plane

Geometric Interpretation of Vanishing Line

Ground plane

Camera plane

Side view

Ground plane

Plane of vanishing line

Plane of vanishing line

Geometric Interpretation of Vanishing Line

Side view

Ground plane

Plane of vanishing line

Height

Where was I (how high)?

Where was I (how high)?

Taken from my hotel room (6th floor) Taken from beach

Where was I (how high)?

Taken from my hotel room (6th floor) Taken from beach

First person video Cylindrical projection

Height of the camera wearer