Optimal Estimation and Detection in Homogeneous Spaces

This paper presents estimation and detection techniques in homogeneous spaces that are optimal under the squared error loss function. The data is collected on a manifold which forms a homogeneous space under the transitive action of a compact Lie group. Signal estimation problems are addressed by formulating Wiener-Hopf equations for homogeneous spaces. The coefficient functions of these equations are the signal correlations which are assumed to be known. The resulting coupled integral equations on the manifold are converted to Wiener-Hopf convolutional integral equations on the group. These are solved using the Peter-Weyl theory of Fourier transform on compact Lie groups. The computational complexity of this algorithm is reduced using the bi-invariance of the correlations with respect to a stabilizer subgroup. The theory of matched filtering for isotropic signal fields is developed for signal classification where given a set of template signals on the manifold and a noisy test signal, the objective is to optimally detect the template buried in the test signal. This is accomplished by designing a filter on the manifold that maximizes the signal-to-noise-ratio (SNR) of the filtered output. An expression for the SNR is obtained as a ratio of quadratic forms expressed as Haar integrals over the transformation group. These integrals are expressed in the Fourier domain as infinite sums over the irreducible representations. Simplification of these sums is achieved by invariance properties of the signal function and the noise correlation function. The Wiener filter and matched filter are developed for an abstract homogeneous space and then specialized to the case of spherical signals under the action of the rotation group. Applications of these algorithms to denoising of 3D surface data, visual navigation with omnidirectional camera and detection of compact embedded objects in the stochastic background are discussed with experimental results.