Mathematical Background Information for Ubiquitous Tracking

This page collects some papers on the math background needed for ubiquitous tracking.

Representing Rigid Motion in 3D

An overview over the possibilities to represent rigid body motion in 3D (including the exponential map) is given in:

• F. Sebastian Grassia
  Practical parameterization of rotations using the exponential map
  Journal of Graphics Tools, 3(3):29-48, 1998

One application of the exponential map is given here:

• C. Bregler, J. Malik
  Tracking People with Twists and Exponential Maps
  Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 8- (1998)

Uncertainty Models

(from Tom Drummond)

A paper which describes most of the early work including tracking of articulated structures (ie imposing constraints) is the PAMI paper:

• T. Drummond, R. Cipolla
  Real-time Visual Tracking of Complex Structures
  IEEE Transactions on Pattern Analysis and Machine Intelligence 24(7): 932-946 (2002)

The ECCV paper talks more about uncertainties and presents a linear time algorithm for optimising over long kinematic chains (this could be applied to resolving ubitrack graphs):

• Tom Drummond, Roberto Cipolla
  Real-Time Tracking of Multiple Articulated Structures in Multiple Views
  ECCV (2) 2000: 20-36

Finally the recent ICCV paper describes the uncertainty models much more carefully and gives more detail on propagaing uncertainties along a chain (through time in this instance):

• Paul Smith, Tom Drummond
  Computing MAP trajectories by representing, propagating and combining PDFs over groups
  IEEE International Conference on Computer Vision, Volume 2, October 13 - 16, 2003

And two more:

• Tom Drummond, Roberto Cipolla
  Application of Lie Algebras to Visual Servoing
  International Journal of Computer Vision, 37(1):21--41, June 2000
• Tom Drummond, Roberto Cipolla
  Visual tracking and control using Lie algebras
  Proc. of IEEE Conf. on Computer Vision and Pattern Recognition (CVPR'1999), volume 2, pages 652--657

Interpolating between Rotations

Smooth interpolation between rotations is usually done using quaternion representations:

• Alan H. Barr, Bena Currin, Steven Gabriel, John F. Hughe
  Smooth interpolation of orientations with angular velocity constraints using quaternions
  19th Annual Conference on Computer Graphics and Interactive Techniques 313-320 (1992)  
• Ken Shoemake
  Animating rotation with quaternion curves
  19th Annual Conference on Computer Graphics and Interactive Techniques 245-254 (1985)  
• Myoung-Jun Kim, Myung-Soo Kim, Sung Yong Shin
  A general construction scheme for unit quaternion curves with simple high order derivatives
  22th Annual Conference on Computer Graphics and Interactive Techniques 369-376  (1995)  
• Erik B. Dam, Martin Koch and Martin Lillholm
  Quaternions, Interpolation and Animation
  University of Copenhagen, Institute of Computer Science (DIKU), technical report 98/5