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Tracking. Overview and Mathematics. Tracking. Motivation. Technologies. Mathematics. Content. Motivation Technologies – Advantages and Disadvantages Common Problems and Errors Acoustic Tracking Mechanical Tracking Inertial Tracking Magnetic Tracking Optical Tracking

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Tracking

Overview and Mathematics


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Tracking

Motivation

Technologies

Mathematics

Content

  • 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


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Tracking

Motivation

Technologies

Mathematics

Motivation

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)


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Tracking

Motivation

Technologies

Mathematics

Content

  • 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


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Tracking

Motivation

Technologies

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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From [1]

Tracking

Motivation

Technologies

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)


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Tracking

Motivation

Technologies

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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Tracking

Motivation

Technologies

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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From [1]

From [1]

Tracking

Motivation

Technologies

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

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


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Tracking

Motivation

Technologies

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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Tracking

Motivation

Technologies

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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From [1]

Tracking

Motivation

Technologies

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)


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Tracking

Motivation

Technologies

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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From [1]

Tracking

Motivation

Technologies

Mathematics

Inertial Tracking - Gyroscopes


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From [1]

Tracking

Motivation

Technologies

Mathematics

Inertial Tracking - Gyroscopes


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Tracking

Motivation

Technologies

Mathematics

Inertial Tracking - Gyroscopes


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Tracking

Motivation

Technologies

Mathematics

Inertial Tracking - Gyroscopes


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Tracking

Motivation

Technologies

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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Tracking

Motivation

Technologies

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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Tracking

Motivation

Technologies

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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From [1]

Tracking

Motivation

Technologies

Mathematics

Optical Tracking

  • Provides angle measurements

    • One 2D pointdefines a ray

    • Two 2D pointsdefine a pointfor 3D position

    • Additional pointsrequired fororientation

  • Speed of Light

    • 2.998 * 108 [m/s]

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


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From [1]

Tracking

Motivation

Technologies

Mathematics

Optical Tracking – Active Targets

  • Typical detectors

    • Lateral Effect PhotoDiodes (LEPDs)

    • Quad Cells

  • Active targets

    • LEDs


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From [1]

Tracking

Motivation

Technologies

Mathematics

Optical Tracking – Passive Targets

  • Typical detectors

    • Video and CCD cameras

      • Computer vision techniques

  • Passive targets

    • Reflective materials, high contrast patterns


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From [A.R.T. GmbH]

Tracking

Motivation

Technologies

Mathematics

Optical Tracking – Passive Targets


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Tracking

Motivation

Technologies

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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From [3]

Tracking

Motivation

Technologies

Mathematics

Inside-out versus Outside-in

  • Inside-out


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From [3]

Tracking

Motivation

Technologies

Mathematics

Inside-out versus Outside-in

  • Outside-in


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Tracking

Motivation

Technologies

Mathematics

Content

  • 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


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From [1]

Tracking

Motivation

Technologies

Mathematics

Position and Orientation (Pose)

  • Representation

    • x, y, z (position) and , ,  (orientation)

    • with respect to a given reference coordinate system


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Y

2

1

X

2

3

1

Tracking

Motivation

Technologies

Mathematics

Transformations in the 2D-space

  • Translation


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Y

2

1

X

2

3

1

Tracking

Motivation

Technologies

Mathematics

Transformations in the 2D-space

  • Scale


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Y

Y

2

1

X

X

2

3

1

Tracking

Motivation

Technologies

Mathematics

Transformations in the 2D-space

  • Rotation


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From [1]

Tracking

Motivation

Technologies

Mathematics

Transformations in the 2D-space

  • Scale and Rotation can be combined by multiplication of their matrices

  • Translation cannot be combined with them by multiplication

  • Introduction of Homogeneous Coordinates


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Tracking

Motivation

Technologies

Mathematics

Transformations in the 2D-space


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Tracking

Motivation

Technologies

Mathematics

Transformations in the 3D-space

  • Translation


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Tracking

Motivation

Technologies

Mathematics

Transformations in the 3D-space

  • Scale


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Tracking

Motivation

Technologies

Mathematics

Transformations in the 3D-space

  • Rotation


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Tracking

Motivation

Technologies

Mathematics

Transformations in the 3D-space

  • e.g. Rotation through  about the z axis


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Tracking

Motivation

Technologies

Mathematics

Transformations in the 3D-space

  • Rotation-Sequences

    • Concatenation of several rotations

    • Can be performed by using

      • Rotation matrices (matrix multiplication)

      • Euler-angles

      • Quaternions


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Tracking

Motivation

Technologies

Mathematics

Transformations in the 3D-space

  • 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, , )


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Tracking

Motivation

Technologies

Mathematics

Transformations in the 3D-space

  • Quaternions

  • Unit Quaternions

  • A unit quaternionrepresents a rotation about the axisthrough the angle


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Tracking

Motivation

Technologies

Mathematics

Transformations in the 3D-space

  • Multiplication-operator for quaternions:

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


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Tracking

Motivation

Technologies

Mathematics

Transformations in the 3D-space

  • 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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Tracking

Motivation

Technologies

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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Tracking

Motivation

Technologies

Mathematics

Thank you for your attention!

Any questions?


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Tracking

Motivation

Technologies

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


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