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Paul Scheunders - One of the best experts on this subject based on the ideXlab platform.
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a multivalued image wavelet representation based on multiscale Fundamental Forms
IEEE Transactions on Image Processing, 2002Co-Authors: Paul ScheundersAbstract:A new wavelet representation for multivalued images is presented. The idea for this representation is based on the First Fundamental Form that provides a local measure for the contrast of a multivalued image. In this paper, this concept is extended toward multiscale Fundamental Forms using the dyadic wavelet transForm of Mallat (1992). The multiscale Fundamental Forms provide a local measure for the contrast of a multivalued image at different scales. The representation allows for a multiscale edge description of multivalued images. A variety of applications is presented, including multispectral image fusion, color image enhancement and multivalued image noise filtering. In an experimental section, the presented techniques are compared to single valued and/or single scale algorithms that were previously described in the literature. The techniques, based on the new representation are demonstrated to outperForm the others.
Kotaro Yamada - One of the best experts on this subject based on the ideXlab platform.
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hypersurfaces with light like points in a lorentzian manifold
Journal of Geometric Analysis, 2019Co-Authors: Masaaki Umehara, Kotaro YamadaAbstract:Consider a constant mean curvature immersion $$F:U(\subset \varvec{R}^n)\rightarrow M$$ into an arbitrary Lorentzian $$(n+1)$$-manifold M. A point $$o\in U$$ is called a light-like point if the First Fundamental Form $$\mathrm{d}s^2$$ of F degenerates at o. We denote by $$B_F$$ the determinant function of the symmetric matrix associated to $$\mathrm{d}s^2$$ with respect to a local coordinate system at o. A light-like point o is said to be degenerate if the exterior derivative of $$B_F$$ vanishes at o. We show that if o is a degenerate light-like point, then the image of F contains a light-like geodesic segment of M passing through f(o) (cf. Theorem E). This explains why several known examples of constant mean curvature hypersurface in the Lorentz–Minkowski $$(n+1)$$-space Form $$\varvec{R}^{n+1}_1$$ contain light-like lines on their sets of light-like points, under a suitable regularity condition of F. Several related results are also given.
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surfaces with light like points in lorentz minkowski 3 space with applications
arXiv: Differential Geometry, 2017Co-Authors: Masaaki Umehara, Kotaro YamadaAbstract:With several concrete examples of zero mean curvature surfaces in $\boldsymbol{R}^3_1$ containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the First Fundamental Form of a surface degenerates is said to be light-like. We also show a theorem on a property of light-like points of a surface in $\boldsymbol{R}^3_1$ whose mean curvature vector is smoothly extendable. This explains why such surfaces will contain a light-like line when they do not change causal types. Moreover, several applications of these two results are given.
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surfaces with light like points in lorentz minkowski 3 space with applications
International Meeting on Lorentzian Geometry, 2016Co-Authors: Masaaki Umehara, Kotaro YamadaAbstract:With several concrete examples of zero mean curvature surfaces in the Lorentz-Minkowski 3-space \(\varvec{R}^3_1\) containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the First Fundamental Form of a surface degenerates is said to be light-like. We also show a theorem on a property of light-like points of a surface in \(\varvec{R}^3_1\) whose mean curvature vector is smoothly extendable. This explains why such surfaces will contain a light-like line when they do not change causal types. Moreover, several applications of these two results are given.
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coherent tangent bundles and gauss bonnet Formulas for wave fronts
arXiv: Differential Geometry, 2009Co-Authors: Kentaro Saji, Masaaki Umehara, Kotaro YamadaAbstract:We give a definition of `coherent tangent bundles', which is an intrinsic Formulation of wave fronts. In our application of coherent tangent bundles for wave fronts, the First Fundamental Forms and the third Fundamental Forms are considered as induced metrics of certain homomorphisms between vector bundles. They satisfy the completely same conditions, and so can reverse roles with each other. For a given wave front of a 2-manifold, there are two Gauss-Bonnet Formulas. By exchanging the roles of the Fundamental Forms, we get two new additional Gauss-Bonnet Formulas for the third Fundamental Form. Surprisingly, these are different from those for the First Fundamental Form, and using these four Formulas, we get several new results on the topology and geometry of wave fronts.
Gerd Verbouwe - One of the best experts on this subject based on the ideXlab platform.
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divisibility of the second Fundamental Form of hypersurfaces of space Forms
Results in Mathematics, 2008Co-Authors: Franki Dillen, Gerd VerbouweAbstract:We classify the hypersurfaces of space Forms for which the cubic Form is divisible by the metric (g|C). In other words, when does the symmetric traceless part of the cubic Form vanish, where the trace is taken with respect to the First Fundamental Form?
Masaaki Umehara - One of the best experts on this subject based on the ideXlab platform.
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hypersurfaces with light like points in a lorentzian manifold
Journal of Geometric Analysis, 2019Co-Authors: Masaaki Umehara, Kotaro YamadaAbstract:Consider a constant mean curvature immersion $$F:U(\subset \varvec{R}^n)\rightarrow M$$ into an arbitrary Lorentzian $$(n+1)$$-manifold M. A point $$o\in U$$ is called a light-like point if the First Fundamental Form $$\mathrm{d}s^2$$ of F degenerates at o. We denote by $$B_F$$ the determinant function of the symmetric matrix associated to $$\mathrm{d}s^2$$ with respect to a local coordinate system at o. A light-like point o is said to be degenerate if the exterior derivative of $$B_F$$ vanishes at o. We show that if o is a degenerate light-like point, then the image of F contains a light-like geodesic segment of M passing through f(o) (cf. Theorem E). This explains why several known examples of constant mean curvature hypersurface in the Lorentz–Minkowski $$(n+1)$$-space Form $$\varvec{R}^{n+1}_1$$ contain light-like lines on their sets of light-like points, under a suitable regularity condition of F. Several related results are also given.
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surfaces with light like points in lorentz minkowski 3 space with applications
arXiv: Differential Geometry, 2017Co-Authors: Masaaki Umehara, Kotaro YamadaAbstract:With several concrete examples of zero mean curvature surfaces in $\boldsymbol{R}^3_1$ containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the First Fundamental Form of a surface degenerates is said to be light-like. We also show a theorem on a property of light-like points of a surface in $\boldsymbol{R}^3_1$ whose mean curvature vector is smoothly extendable. This explains why such surfaces will contain a light-like line when they do not change causal types. Moreover, several applications of these two results are given.
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surfaces with light like points in lorentz minkowski 3 space with applications
International Meeting on Lorentzian Geometry, 2016Co-Authors: Masaaki Umehara, Kotaro YamadaAbstract:With several concrete examples of zero mean curvature surfaces in the Lorentz-Minkowski 3-space \(\varvec{R}^3_1\) containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the First Fundamental Form of a surface degenerates is said to be light-like. We also show a theorem on a property of light-like points of a surface in \(\varvec{R}^3_1\) whose mean curvature vector is smoothly extendable. This explains why such surfaces will contain a light-like line when they do not change causal types. Moreover, several applications of these two results are given.
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coherent tangent bundles and gauss bonnet Formulas for wave fronts
arXiv: Differential Geometry, 2009Co-Authors: Kentaro Saji, Masaaki Umehara, Kotaro YamadaAbstract:We give a definition of `coherent tangent bundles', which is an intrinsic Formulation of wave fronts. In our application of coherent tangent bundles for wave fronts, the First Fundamental Forms and the third Fundamental Forms are considered as induced metrics of certain homomorphisms between vector bundles. They satisfy the completely same conditions, and so can reverse roles with each other. For a given wave front of a 2-manifold, there are two Gauss-Bonnet Formulas. By exchanging the roles of the Fundamental Forms, we get two new additional Gauss-Bonnet Formulas for the third Fundamental Form. Surprisingly, these are different from those for the First Fundamental Form, and using these four Formulas, we get several new results on the topology and geometry of wave fronts.
Dario L Ringach - One of the best experts on this subject based on the ideXlab platform.
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anisotropic diffusion of multivalued images with applications to color filtering
IEEE Transactions on Image Processing, 1996Co-Authors: Guillermo Sapiro, Dario L RingachAbstract:A general framework for anisotropic diffusion of multivalued images is presented. We propose an evolution equation where, at each point in time, the directions and magnitudes of the maximal and minimal rate of change in the vector-image are First evaluated. These are given by eigenvectors and eigenvalues of the First Fundamental Form in the given image metric. Then, the image diffuses via a system of coupled differential equations in the direction of minimal change. The diffusion "strength" is controlled by a function that measures the degree of dissimilarity between the eigenvalues. We apply the proposed framework to the filtering of color images represented in CIE-L*a*b* space.