A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators

This chapter offers a detailed survey on intrinsically localized frames and the corresponding matrix representation of operators. We re-investigate the properties of localized frames and the associated Banach spaces in full detail. We investigate the representation of operators using localized frames in a Galerkin-type scheme. We show how the boundedness and the invertibility of matrices and operators are linked and give some sufficient and necessary conditions for the boundedness of operators between the associated Banach spaces.Comment: 32 page

Generating binary partial Hadamard matrices

This paper deals with partial binary Hadamard matrices. Although there is a fast simple way to generate about a half (which is the best asymptotic bound known so far, see de Launey (2000) and de Launey and Gordon (2001)) of a full Hadamard matrix, it cannot provide larger partial Hadamard matrices beyond this bound. In order to overcome such a limitation, we introduce a particular subgraph Gt of Ito’s Hadamard Graph Δ(4t) (Ito, 1985), and study some of its properties,which facilitates that a procedure may be designed for constructing large partial Hadamard matrices. The key idea is translating the problem of extending a given clique in Gt into a Constraint Satisfaction Problem, to be solved by Minion (Gent et al., 2006). Actually, iteration of this process ends with large partial Hadamard matrices, usually beyond the bound of half a full Hadamard matrix, at least as our computation capabilities have led us thus far

Depth-resolved two-dimensional Stokes vectors of backscattered light and Mueller matrices of biological tissue measured with optical coherence tomography

Mueller matrices provide a complete characterization of the optical polarization properties of biological tissue. A polarization-sensitive optical coherence tomography (OCT) system was built and used to investigate the optical polarization properties of biological tissues and other turbid media. The apparent degree of polarization (DOP) of the backscattered light was measured with both liquid and solid scattering samples. The DOP maintains the value of unity within the detectable depth for the solid sample, whereas the DOP decreases with the optical depth for the liquid sample. Two-dimensional depth-resolved images of both the Stokes vectors of the backscattered light and the full Mueller matrices of biological tissue were measured with this system. These polarization measurements revealed some tissue structures that are not perceptible with standard OCT