Tracking - Overview and Mathematics

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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. sensors 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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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 0C)
• Varies with temperature, pressure and humidity
• ? Slow ? Low update rate

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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
• Newtons Second Law of Motion
• F ma (linear)
• M I? (rotational)
• Accelerometers and Gyroscopes
• Provide derivative measurements

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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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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 - 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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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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Tracking
Motivation
Technologies
Mathematics
Optical Tracking Active Targets

• Typical detectors
• Lateral Effect PhotoDiodes (LEPDs)
• Quad Cells
• Active targets
• LEDs

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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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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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Tracking
Motivation
Technologies
Mathematics
Inside-out versus Outside-in

• Inside-out

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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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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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Tracking
Motivation
Technologies
Mathematics
Transformations in the 2D-space

• Translation

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Tracking
Motivation
Technologies
Mathematics
Transformations in the 2D-space

• Scale

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Tracking
Motivation
Technologies
Mathematics
Transformations in the 2D-space

• Rotation

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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