Where am I?Using Vanishing Points

Where was I (how high)?

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

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

Ground plane

Where am I w.r.t. Ground Plane?

3D world

Camera

Xu

Yv

ZK r r r tC

1 2 3 W

11

How to compute?

3D world

r1

r2

r3

What can a Vanishing Point tell us about?Camera

3D world

r1

r2

r3

v

Ground plane

Ground plane

What can a Vanishing Point tell us about?Camera

3D world

r1

r2

r3

v

Z

0

0

1

0

Ground plane

What can a Vanishing Point tell us about?Camera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

C

1 2 3 W

0

0

1

0

v K r r r t

Ground plane

Single Vanishing PointCamera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

C

1 2 3 W

0

0

1

0

v K r r r t

-1

3

K v r

3

v Kr

Ground plane

Single Vanishing PointCamera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

C

1 2 3 W

0

0

1

0

v K r r r t

-1

3

K v r

3

v Kr

K v

rK v

-1

3 -1because r3 is a unit vector.

Z vanishing point tells us about the

surface normal of the ground plane

Ground plane

Single Vanishing PointCamera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

C

1 2 3 W

0

0

1

0

v K r r r t

-1

3

K v r

3

v Kr

K v

rK v

-1

3 -1because r3 is a unit vector.

Z vanishing point tells us about the

surface normal of the ground plane

Ground plane

Rotation ambiguity

Ground plane

Single Vanishing PointCamera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-1

3

K v r

K v

rK v

-1

3 -1

Roll and pitch angle can be computed by the ground plane.

Ground plane

Single Vanishing PointCamera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-1

3

K v r

K v

rK v

-1

3 -1

Roll and pitch angle can be computed by the ground plane.

Yaw: rotation about z axis:yaw

cos -sin 0

sin cos 0

0 0 1

R

Ground plane

Single Vanishing PointCamera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-1

3

K v r

K v

rK v

-1

3 -1

Roll and pitch angle can be computed by the ground plane.

Yaw: rotation about z axis:yaw

cos -sin 0

sin cos 0

0 0 1

R

Pitch: rotation about y axis:pitch

cos 0 sin

0 1 0

-sin 0 cos

R

Ground plane

Single Vanishing PointCamera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-1

3

K v r

K v

rK v

-1

3 -1

Roll and pitch angle can be computed by the ground plane.

Yaw: rotation about z axis:yaw

cos -sin 0

sin cos 0

0 0 1

R

Pitch: rotation about y axis:pitch

cos 0 sin

0 1 0

-sin 0 cos

R

Roll: rotation about x axis:roll

1 0 0

0 cos -sin

0 sin cos

R

Ground plane

Single Vanishing PointCamera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-1

3

K v r

K v

rK v

-1

3 -1

Roll and pitch angle can be computed by the ground plane.

Yaw: rotation about z axis:yaw

cos -sin 0

sin cos 0

0 0 1

R

Pitch: rotation about y axis:pitch

cos 0 sin

0 1 0

-sin 0 cos

R

Roll: rotation about x axis:roll

1 0 0

0 cos -sin

0 sin cos

R

T

1 2 3 yaw pitch roll

-sin

cos sin

cos cos

r r r R R R ?

Ground plane

Single Vanishing PointCamera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-1

3

K v r

K v

rK v

-1

3 -1

Roll and pitch angle can be computed by the ground plane.

Yaw: rotation about z axis:yaw

cos -sin 0

sin cos 0

0 0 1

R

Pitch: rotation about y axis:pitch

cos 0 sin

0 1 0

-sin 0 cos

R

Roll: rotation about x axis:roll

1 0 0

0 cos -sin

0 sin cos

R

T

1 2 3 yaw pitch roll

-sin

cos sin

cos cos

r r r R R R

Ground plane

Single Vanishing PointCamera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-1

3

K v r

K v

rK v

-1

3 -1

Roll and pitch angle can be computed by the ground plane.

-1 31

2 2

32 33

-1 32

33

tan

tan

r

r r

r

r

T

1 2 3 yaw pitch roll

-sin

cos sin

cos cos

r r r R R R

Pitch:

Roll:

Ground plane

Single Vanishing PointCamera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-1

3

K v r

K v

rK v

-1

3 -1

Roll and pitch angle can be computed by the ground plane.

-1 31

2 2

32 33

-1 32

33

tan

tan

r

r r

r

r

T

1 2 3 yaw pitch roll

-sin

cos sin

cos cos

r r r R R R

Pitch:

Roll:

Single Vanishing Point (Exercise)

f = 1224;

K = [f 0 size(im,2)/2;

0 f size(im,1)/2;

0 0 1];

m1 = [2563;25;1];

m2 = [2439;545;1];

m3 = [571;25;1];

m4 = [723;498;1];

l1 = GetLineFromTwoPoints(m1,m2);

l2 = GetLineFromTwoPoints(m3,m4);

v1 = GetPointFromTwoLines(l1,l2);

v1 = v1/v1(3);

r3 = inv(K)*v1/norm(inv(K)*v1)

pitch = atan(-r3(1)/norm(r3(2:3)))

roll = atan(r3(2)/r3(3))

ComputeCameraUsingVanishingPoint.m

r3 =

-0.07360.89590.4381

pitch =

0.0736

roll =

1.1160

v

Single Vanishing Point (Exercise)

f = 1224;

K = [f 0 size(im,2)/2;

0 f size(im,1)/2;

0 0 1];

m1 = [2563;25;1];

m2 = [2439;545;1];

m3 = [571;25;1];

m4 = [723;498;1];

l1 = GetLineFromTwoPoints(m1,m2);

l2 = GetLineFromTwoPoints(m3,m4);

v1 = GetPointFromTwoLines(l1,l2);

v1 = v1/v1(3);

r3 = inv(K)*v1/norm(inv(K)*v1)

pitch = atan(-r3(1)/norm(r3(2:3)))

roll = atan(r3(2)/r3(3))

ComputeCameraUsingVanishingPoint.m

r3 =

-0.07360.89590.4381

pitch =

0.0736

roll =

1.1160

Ground plane

Two Vanishing Points

3D world

Camera

r1

r2

r3

Yv

v

X

3D world

Ground plane

Two Vanishing PointsCamera

0

1

0

0

Y

X

1

0

0

0

3D world

r1

r2

r3

Yv

v

X

3D world

Ground plane

Two Vanishing PointsCamera

X

1

0

0

0

v K r r r t X

Kr

C

X 1 2 3 W

1

0

1

0

0

Y

3D world

r1

r2

r3

Yv

v

X

3D world

Ground plane

Two Vanishing PointsCamera

X

1

0

0

0

C

Y 1 2 3 W

2

v K r r r t Y

Kr

v K r r r t X

Kr

C

X 1 2 3 W

1

0

1

0

0

Y

3D world

r1

r2

r3

Yv

v

X

3D world

Ground plane

Two Vanishing PointsCamera

X

1

0

0

0

-1 -1

X Y1 2-1 -1

X Y

,

K v K v

r rK v K v

3 1 2 r r r : Orthogonality constraint

C

Y 1 2 3 W

2

v K r r r t Y

Kr0

1

0

0

Y

3D world

r1

r2

r3

v K r r r t X

Kr

C

X 1 2 3 W

1

Yv

v

X

3D world

Ground plane

Geometric Interpretation

3D world

r1

r2

r3

X

1

0

0

0

-1

X

K v

Ground plane

Geometric Interpretation

X

1

0

0

0

-1

X

K v

0

1

0

0

Y-1

Y

K v

3D world

r1

r2

r3

Ground plane

Geometric Interpretation

X

1

0

0

0

-1

X

K v

3D world

r1

r2

r3

-1

Y

K v

0

1

0

0

Y

Ground plane

Geometric Interpretation

X

1

0

0

0

-1

X

K v

-1 -1

X Y K v K v

3D world

r1

r2

r3

-1

Y

K v

0

1

0

0

Y

Ground plane

Geometric Interpretation (Translation Ambiguity)

Ground plane

Geometric Interpretation (Translation Ambiguity)

Ground plane

Geometric Interpretation (Translation Ambiguity)

Ground plane

Two Vanishing Points

3D world

Camera

r1

r2

r3

f = 4000;K = [f 0 size(im,2)/2;

0 f size(im,1)/2;0 0 1];

l11 = GetLineFromTwoPoints(m11,m12);l12 = GetLineFromTwoPoints(m13,m14);

l21 = GetLineFromTwoPoints(m21,m22);l22 = GetLineFromTwoPoints(m23,m24);

v1 = GetPointFromTwoLines(l11,l12);v2 = GetPointFromTwoLines(l21,l22);

r1 = inv(K)*v1/norm(inv(K)*v1);r2 = inv(K)*v2/norm(inv(K)*v2);

r3 = Vec2Skew(r1)*r2;

R =0.2448 -0.5178 0.0424

-0.1737 -0.1960 -0.69780.9539 0.8327 -0.1379

det(R) = 0.5077

R’*R =0.3299 0.0294 -0.20360.0294 0.5555 -0.2327

-0.2036 -0.2327 1.6224

Not orthogonal matrix!

ComputeCameraUsingTwoVanishingPoints.m

Yv

v

X

3D world

Two Vanishing Pointsf = 1224;K = [f 0 size(im,2)/2;

0 f size(im,1)/2;0 0 1];

l11 = GetLineFromTwoPoints(m11,m12);l12 = GetLineFromTwoPoints(m13,m14);

l21 = GetLineFromTwoPoints(m21,m22);l22 = GetLineFromTwoPoints(m23,m24);

v1 = GetPointFromTwoLines(l11,l12);v2 = GetPointFromTwoLines(l21,l22);

r1 = inv(K)*v1/norm(inv(K)*v1);r2 = inv(K)*v2/norm(inv(K)*v2);

r3 = Vec2Skew(r1)*r2;

ComputeCameraUsingTwoVanishingPoints.m

R =0.5846 -0.8496 0.0508

-0.4149 -0.3216 -0.83670.6972 0.4180 -0.5405

det(R) =0.9948

R’*R =1.0662 -0.0118 0.0250

-0.0118 0.9757 0.02850.0250 0.0285 0.9530

Orthogonal matrix!

Change focal length