Where am I? Using Vanishing Pointshspark/CSci5980/Lec4_Localization... · JJ JJ ªº «» «»...

Where am I? Using Vanishing Points

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


3D world

r1

r2

r3

v

Z

0

0

1

0

Ground plane


3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

C1 2 3 W

0

0

1

0

v K r r r t

Ground plane

Single Vanishing Point Camera

3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

C1 2 3 W

0

0

1

0

v K r r r t

-13

K v r

3v Kr

Ground plane


3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

C1 2 3 W

0

0

1

0

v K r r r t

-13

K v r

3v Kr

K v

rK v

-1

3 -1 because r3 is a unit vector.

Z vanishing point tells us about the surface normal of the ground plane

Ground plane


3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

C1 2 3 W

0

0

1

0

v K r r r t

-13

K v r

3v Kr

K v

rK v

-1

3 -1 because r3 is a unit vector.

Z vanishing point tells us about the surface normal of the ground plane

Ground plane

Rotation ambiguity

Ground plane


3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-13

K v r

K v

rK v

-1

3 -1

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

Ground plane


3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-13

K v r

K v

rK v

-1

3 -1


3D spatial rotation: a sequence of Euler angle rotation (roll/pitch/yaw).

Ground plane


3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-13

K v r

K v

rK v

-1

3 -1


Yaw: rotation about z axis: yaw

cos -sin 0

sin cos 0

0 0 1

R

Ground plane


3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-13

K v r

K v

rK v

-1

3 -1



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


3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-13

K v r

K v

rK v

-1

3 -1



cos -sin 0

sin cos 0

0 0 1

R


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


3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-13

K v r

K v

rK v

-1

3 -1



cos -sin 0

sin cos 0

0 0 1

R


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


3D world

r1

r2

r3

v

0 0

0 0

1

1 0

Z

-13

K v r

K v

rK v

-1

3 -1


-1 31

2 232 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 = -0.0736 0.8959 0.4381 pitch = 0.0736 roll = 1.1160

Ground plane

Two Vanishing Points

3D world

Camera

r1

r2

r3

Yv

vX

3D world

Ground plane

Two Vanishing Points Camera

0

1

0

0

Y

X

1

0

0

0

3D world

r1

r2

r3

Yv

vX

3D world

Ground plane


X

1

0

0

0

v K r r r t X

Kr

CX 1 2 3 W

1

0

1

0

0

Y

3D world

r1

r2

r3

Yv

vX

3D world

Ground plane


X

1

0

0

0

CY 1 2 3 W

2

v K r r r t Y

Kr

v K r r r t X

Kr

CX 1 2 3 W

1

0

1

0

0

Y

3D world

r1

r2

r3

Yv

vX

3D world

Ground plane


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

CY 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

vX

3D world

Ground plane

Geometric Interpretation

3D world

r1

r2

r3

X

1

0

0

0

-1X

K v

Ground plane


X

1

0

0

0

-1X

K v

0

1

0

0

Y-1Y

K v

3D world

r1

r2

r3

Ground plane


X

1

0

0

0

-1X

K v

3D world

r1

r2

r3

-1Y

K v

0

1

0

0

Y

Ground plane


X

1

0

0

0

-1X

K v

-1 -1X Y K v K v

3D world

r1

r2

r3

-1Y

K v

0

1

0

0

Y

Ground plane

Geometric Interpretation (Translation Ambiguity)

Ground plane

Two Vanishing Points

3D world

Camera

r1r2

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 -0.6978 0.9539 0.8327 -0.1379 det(R) = 0.5077 R’*R = 0.3299 0.0294 -0.2036 0.0294 0.5555 -0.2327 -0.2036 -0.2327 1.6224

Not orthogonal matrix!

ComputeCameraUsingTwoVanishingPoints.m

Yv

vX

3D world

Two Vanishing Points f = 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 -0.8367 0.6972 0.4180 -0.5405 det(R) = 0.9948 R’*R = 1.0662 -0.0118 0.0250 -0.0118 0.9757 0.0285 0.0250 0.0285 0.9530

Orthogonal matrix!

Change focal length

Where am I? Using Vanishing Pointshspark/CSci5980/Lec4_Localization... · JJ JJ ªº «» «»...

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