tracking overview and mathematics. christoph krautz 2 motivation technologies – advantages and...

TrackingOverview and Mathematics

Christoph Krautz 2

Motivation Technologies – Advantages and Disadvantages

– Common Problems and Errors– Acoustic Tracking– Mechanical Tracking– Inertial Tracking– Magnetic Tracking– Optical Tracking– Inside-out versus Outside-in

Mathematics– Transformations in the 2D-space– Transformations in the 3D-space

Discussion

Motivation TechnologiesTracking

Mathematics

Content

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What is tracking?

The repeated localization of the position and orientation (pose) of one or several real physical objects

Why is tracking needed in AR?

Integration of virtual objects into real world (images)


Mathematics

Motivation

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Discussion


Mathematics

Content

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Mathematics

Common Problems and Errors

High update rate required (usually in real-time systems)

Dynamic tracker error, e.g. sensor‘s motion Distortion due to environmental influences,

e.g. noise Long-term variations

– Cause readings to change from one day to the next day

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Mathematics

Acoustic Tracking

The Geometry– The intersection of two spheres is a circle.– The intersection of three spheres is two points.

• One of the two points can easily be eliminated.

Ultrasonic– 40 [kHz] typical (Slide taken from SIGGRAPH 2001 Course

11 – Slides by Allen, Bishop, Welch)

From [1]

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Mathematics

Acoustic Tracking - Methods

Time of Flight– Measures the time required for a sonic pulse to

travel from a transmitter to a receiver.– d [m] = v [m/s] * t [s], v = speed of sound– Absolute range measurement

Phase Coherence– Measures phase difference between transmitted and

received sound waves– Relative to previous measurement

• still absolute!!

(Slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch)

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Mathematics

Acoustic Tracking – Discussion

Advantages– Small and lightweight (miniaturization of transmitters

and receivers)– Only sensitive to influences by noise in the ultrasonic

range

Disadvantages– Speed of Sound (~331 [m/s] in air at 0°C)

• Varies with temperature, pressure and humidity Slow Low update rate

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Mathematics

Mechanical Tracking

Ground-based or Body-based Used primarily for motion capture Provide angle and range measurements

– Gears– Bend sensors

Elegant addition of force feedback


From [1]From [1]

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Mathematics

Mechanical Tracking – Discussion

Advantages– Good accuracy– High update rate– No suffering from environmental linked errors

Disadvantages– Small working volume due to mechanical linkage

with the reference

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Mathematics

Inertial Tracking

Inertia– Rigidity in space

Newton’s Second Law of Motion– F = ma (linear)– M = I (rotational)

Accelerometers and Gyroscopes– Provide derivative measurements

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Mathematics

Inertial Tracking - Accelerometers

Measure force exerted on a mass since we cannot measure acceleration directly.

Proof-mass and damped spring– Displacement proportional to acceleration

Potentiometric and Piezoelectric Transducers(Slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch)

From [1]

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Mathematics

Inertial Tracking - Gyroscopes

Conservation of angular momentum Precession

– If torque is exerted on a spinning mass, its axis of rotation will precess at right angles to both itself and the axis of the exerted torque

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Mathematics


From [1]

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Mathematics


From [1]

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Mathematics


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Mathematics


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Mathematics

Inertial Tracking – Discussion

Advantages– Lightweight– No physical limits on the working volume

Disadvantages– Error accumulation due to integration (numerical)

• Periodic recalibration– Hybrid systems typical

– Drift in the axis of rotation of a gyroscope due to the remaining friction between the axis of the wheel and the bearings

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Mathematics

Magnetic Tracking

Three mutually-orthogonal coils– Each transmitter coil activated serially

• Induced current in the receiver coils is measured– Varies with

» the distance (cubically) from the transmitter and» their orientation relative to the transmitter (cosine of

the angle between the axis and the local magnetic field direction)

• Three measurements apiece (three receiver coils)• Nine-element measurement for 6D pose

AC at low frequency DC-pulses

(Parts of the slide taken from SIGGRAPH 2001 Course 11 – Slides by Allen, Bishop, Welch)

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Mathematics

Magnetic Tracking – Discussion

Advantages– Small– Good update rate

Disadvantages– Small working volume– Ferromagnetic interference– Eddy currents induced in conducting materials

Distortions Inaccurate pose estimates

– Use of DC transmitters overcomes that problem– Sensitive to electromagnetic noise

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Mathematics

Optical Tracking

Provides angle measurements– One 2D point

defines a ray– Two 2D points

define a pointfor 3D position

– Additional pointsrequired fororientation

Speed of Light– 2.998 * 108 [m/s]


From [1]

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Mathematics

Optical Tracking – Active Targets

Typical detectors– Lateral Effect PhotoDiodes (LEPDs)– Quad Cells

Active targets– LEDs

From [1]

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Mathematics

Optical Tracking – Passive Targets

Typical detectors– Video and CCD cameras

• Computer vision techniques

Passive targets– Reflective materials, high contrast patterns

From [1]

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Mathematics

Optical Tracking – Passive Targets

From [A.R.T. GmbH]

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Mathematics

Optical Tracking – Discussion

Advantages– Good update rate (due to the speed of light)

• Well suited for real-time systems

Disadvantages– Accuracy tends to worsen with increased distance– Sensitive to optical noise and spurious light

• Can be minimized by using infrared light

– Ambiguity of surface and occlusion

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Mathematics

Inside-out versus Outside-in

Inside-out

From [3]

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Mathematics

Inside-out versus Outside-in

Outside-in

From [3]

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Discussion


Mathematics

Content

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Representation– x, y, z (position) and , , (orientation)– with respect to a given reference coordinate system


Mathematics

Position and Orientation (Pose)

From [1]

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Translation


Mathematics

Transformations in the 2D-space

),('),( byaxPyxP

1 2 3

1

2

X

Y

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Scale


Mathematics


),('),( 21 ysxsPyxP

2

1

0

0

'

s

sS

SPP

1 2 3

1

2

X

Y

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Rotation


Mathematics


cossin'

sincos'

)','('),(

yxy

and

yxx

where

yxPyxP

cossin

sincos

'

R

RPP

1 2 3

1

2

X

Y

XY

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Scale and Rotation can be combined by multiplication of their matrices

Translation cannot be combined with them by multiplication

Introduction of Homogeneous Coordinates


Mathematics


)1,,(),( yxyx

From [1]

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Mathematics


100

10

01

'

b

a

T

TPP

100

00

00

'

2

1

s

s

S

SPP

100

0cossin

0sincos

'

R

RPP

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Translation


Mathematics


1000

100

010

001

'

c

b

a

T

TPP

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Scale


Mathematics


1000

000

000

000

'

3

2

1

s

s

s

S

SPP

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Rotation


Mathematics


1000

0

0

0

'

333231

232221

131211

rrr

rrr

rrr

R

RPP

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e.g. Rotation through about the z axis


Mathematics


1000

0100

00cossin

00sincos

'

R

RPP

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Rotation-Sequences– Concatenation of several rotations– Can be performed by using

• Rotation matrices (matrix multiplication)• Euler-angles• Quaternions


Mathematics


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Euler-angles– Three angles , and

• Each represents a rotation about one of the coordinate axes (X, Y and Z).

– Gimbal Lock– Ambiguities

• R(, 0, 0) = R(0, , )


Mathematics


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Quaternions


Mathematics


),,(),,(: zyxvvskzjyixsq

Unit Quaternions

12222 zyxs

A unit quaternionrepresents a rotation about the axisthrough the angle

),( vs

v

sarccos2

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Multiplication-operator for quaternions:


Mathematics


),( vvvrvrvvrr rqqrrqrqrqqrp

The result is a rotation p composed by the rotations q and r.

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Mathematics


Advantages of quaternions:– No gimbal lock– Unique representation of a rotation– Interpolation can be properly carried out

(spherical interpolation on the 4-sphere; Shoemake, 1985)

– Rotation-sequences can be easily performed

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Mathematics

Conclusion

Each tracking technology has advantages and disadvantages

Multi-Sensor-Fusion for minimizing the measurement errors

Transformations in the 3D-space have to be handled with care

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Mathematics

Thank you for your attention!

Any questions?

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Mathematics

References:

[1] G. Bishop, G. Welch and B. D. Allen, „Tracking: Beyond 15 Minutes of Thought”,SIGGRAPH 2001 Course Notes, University of North Carolina at Chapel Hill

[2] G. Bishop, G. Welch and B. D. Allen, „Tracking: Beyond 15 Minutes of Thought”,SIGGRAPH 2001 Course Slides, University of North Carolina at Chapel Hill

[3] Ribo, Miguel, “State of the Art Report on Optical Tracking”, 2001

tracking overview and mathematics. christoph krautz 2 motivation technologies – advantages and...

Documents

tracking overview

mathematics slide

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welch slide

mathematics transformations

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