Numerical methods for simultaneous diagonalization
SIAM Journal on Matrix Analysis and Applications
Jacobi Angles for Simultaneous Diagonalization
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Efficient Linear Solution of Exterior Orientation
IEEE Transactions on Pattern Analysis and Machine Intelligence
A multidimensional scaling framework for mobile location using time-of-arrival measurements
IEEE Transactions on Signal Processing
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We consider the problem of tracking multiple targets in the presence of imperfect and incomplete ranging information using an MDS-based tracking algorithm. An advantage of this technique is that tracking accuracy is independent on target dynamics. The main feature of the aforementioned algorithm, which we proposed in an earlier work, is that tracking is performed over the eigenspace of a Nyström-Gram kernel matrix constructed with no a-priori knowledge of the statistics of target trajectories. Consequently tracking becomes a problem of updating the eigenspace given new input data, which is achieved with an iterative Jacobian eigen-decomposition technique. In this paper it is first shown how to improve the aforementioned eigendecomposition to fully exploit the structure of the reconstructed Gram kernel matrix, then how to use the similarity existing between subsequent Gram matrices to efficiently track the relative sub-spaces. The performance and computational complexity of two techniques, namely, the Multidimensional Scaling (MDS)- based tracking algorithm and SMACOF are investigated. As a result, the MDS-based tracking algorithm with Jacobian eigenspace updating is shown to achieve the same performance as the SMACOF algorithm, but at a significantly lower complexity.