Marin-Quintero, Maider J.

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2004 - 200912010 - 20121
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    Killing vector fields for a special class of metrics
    (2004) Marin-Quintero, Maider J.; Rózga, Krzysztof; College of Arts and Sciences - Sciences; Portnoy, Arturo; Steinberg, Lev; Department of Mathematics; Velázquez, Esov
    A one parameter local group of isometries of Riemannian or more generally pseudo-Riemannian manifolds is generated by a Killing vector field which is subjected to the commonly named Killing equations. The latter constitute an over determined system of first order partial differential equations which are even linear and homogeneous. However, in general such a system is not completely integrable. A brief presentation of the well-known results on the existence of non- trivial Killing vector fields (i.e. nontrivial solutions of Killing equations) is provided. These results also suggest a method of constructing Killing vector fields, which consists basically of studying consequences of the so called integrability conditions. That last part requires usually quite involved symbolic computations and therefore can be aided by the appropriate computer programs. The method is applied to a class of pseudo Riemannian structures that depends on two arbitrary holomorphic functions of one complex variable. Some constraints on these functions arise as a consequence of the existence of nontrivial Killing vector fields. The nature of these constraints and an explicit form of a Killing field are presented as the final result.
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    An adaptive spectrally weighted structure  tensor applied to tensor anisotropic nonlinear  diffusion for hyperspectral images  
    (2012) Marin-Quintero, Maider J.; Vélez Reyes, Miguel; College of Engineering; Hunt, Shawn; Manien, Vidya; Rivera Gallego, Wilson; Department of Electrical and Computer Engineering; Maldonado Fortunet, Francisco
    The structure tensor for vector valued images is most often defined as the average of the scalar structure tensors in each band. The problem with this definition is the assumption that all bands provide the same amount of edge information giving them the same weights. As a result non-edge pixels can be reinforced and edges can be weakened resulting in a poor performance by processes that depend on the structure tensor. Iterative processes, in particular, are vulnerable to this phenomenon. In this work, a structure tensor for Hyperspectral Images (HSI) is proposed. The initial matrix field is calculated using a weighted smoothed gradient. The weights are based on the Heat Operator. This definition is motivated by the fact that in HSI, neighboring spectral bands are highly correlated, as are the bands of its gradient. To use the heat operator, the smoothed gradient is modeled as the initial heat distribution on a compact manifold M. A Tensor Anisotropic Nonlinear Diffusion (TAND) method using the spectrally weighted structure tensor is proposed to do two kind of processing: Image regularization known as Edge Enhancing Diffusion (EED) and structure enhancement known as Coherence Enhancing Diffusion (CED). Diffusion tensor and a stopping criteria were also developed in this work. Comparisons between methods show that the structure tensor with weights based on the heat operator better discriminates edges that need to be persistent during the iterative process with EED and produces more complete edges with CED. Remotely sensed and biological HSI are used in the experiments.