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Hilbert-Schmidt Lower Bounds for Estimators on Matrix Lie Groups for ATR
August 1998 (vol. 20 no. 8)
pp. 790-802

Abstract—Deformable template representations of observed imagery, model the variability of target pose via the actions of the matrix Lie groups on rigid templates. In this paper, we study the construction of minimum mean squared error estimators on the special orthogonal group, SO(n), for pose estimation. Due to the nonflat geometry of SO(n), the standard Bayesian formulation, of optimal estimators and their characteristics, requires modifications. By utilizing Hilbert-Schmidt metric defined on GL(n), a larger group containing SO(n), a mean squared criterion is defined on SO(n). The Hilbert-Schmidt estimate (HSE) is defined to be a minimum mean squared error estimator, restricted to SO(n). The expected error associated with the HSE is shown to be a lower bound, called the Hilbert-Schmidt bound (HSB), on the error incurred by any other estimator. Analysis and algorithms are presented for evaluating the HSE and the HSB in case of both ground-based and airborne targets.

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Index Terms:
Pose estimation, ATR, Hilbert-Schmidt bounds, Bayesian approach, performance analysis, orthogonal groups.
Ulf Grenander, Michael I. Miller, Anuj Srivastava, "Hilbert-Schmidt Lower Bounds for Estimators on Matrix Lie Groups for ATR," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20, no. 8, pp. 790-802, Aug. 1998, doi:10.1109/34.709572
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