In this paper, a high-dimensional statistical signal processing is revisited with the aim of introducing the concept of vector signal representation derived from the Riesz transforms, which are the natural extension and generalization of the one-dimensional Hilbert transform. Under the new concepts of vector correlations proposed recently, the statistical properties of the vector signal representation for random signal are presented and some applications to speckle metrology developed recently are reviewed to demonstrate the unique capability of Riesz transforms.
The statistical property of the polarization speckle generated from scattering from a rough surface has been investigated theoretically. Under the Gaussian assumption of the random electric fields, the joint probability density function of the Stokes parameters has been derived. These results can be regarded as a development and extension of previous works on the fully developed polarization speckle where no constant specular background intensity exists. Our result provides more concise description about the surface roughness and surface anisotropy.
The statistical properties of the spatial derivatives of the Stokes parameters for polarization speckle are investigated
theoretically and experimentally. Based on the Gaussian assumption, the six-dimensional joint probability density
function (p.d.f) for the derivatives of the Stokes parameters (S1, S2, and S3) are all derived analytically for the first time.
Subsequently, three two-dimensional p.d.f of derivatives for each Stokes parameters and the corresponding six marginal
p.d.f are also given. Based on polar-interferometry, experiments have also been conducted to demonstrate the validity of
the principle.
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