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Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian
, 2007
"... We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz ..."
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Cited by 20 (1 self)
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We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the KarhunenLoève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.
MODELING OF WAVE RESONANCES IN LOWCONTRAST PHOTONIC CRYSTALS ∗
"... Abstract. Coupledmode equations are derived from Maxwell equations for modeling of lowcontrast cubiclattice photonic crystals in three spatial dimensions. Coupledmode equations describe resonantly interacting Bloch waves in stop bands of the photonic crystal. We study the linear boundaryvalue pr ..."
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Cited by 15 (8 self)
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Abstract. Coupledmode equations are derived from Maxwell equations for modeling of lowcontrast cubiclattice photonic crystals in three spatial dimensions. Coupledmode equations describe resonantly interacting Bloch waves in stop bands of the photonic crystal. We study the linear boundaryvalue problem for stationary transmission of four counterpropagating and two oblique waves on the plane. Wellposedness of the boundaryvalue problem is proved by using the method of separation of variables and generalized Fourier series. For applications in photonic optics, we compute integral invariants for transmission, reflection, and diffraction of resonant waves.
On convergence rates equivalency and sampling strategies in in functional deconvolution models
 THE ANNALS OF STATISTICS
, 2010
"... Using the asymptotical minimax framework, we examine convergence rates equivalency between a continuous functional deconvolution model and its reallife discrete counterpart over a wide range of Besov balls and for the L 2risk. For this purpose, all possible models are divided into three groups. Fo ..."
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Cited by 9 (4 self)
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Using the asymptotical minimax framework, we examine convergence rates equivalency between a continuous functional deconvolution model and its reallife discrete counterpart over a wide range of Besov balls and for the L 2risk. For this purpose, all possible models are divided into three groups. For the models in the first group, which we call uniform, the convergence rates in the discrete and the continuous models coincide no matter what the sampling scheme is chosen, and hence the replacement of the discrete model by its continuous counterpart is legitimate. For the models in the second group, to which we refer as regular, one can point out the best sampling strategy in the discrete model, but not every sampling scheme leads to the same convergence rates; there are at least two sampling schemes which deliver different convergence rates in the discrete model (i.e., at least one of the discrete models leads to convergence rates that are different from the convergence rates in the continuous model). The third group consists of models for which, in general, it is impossible to devise the best sampling strategy; we call these
Soap film smoothing
 Journal of the Royal Statistical Society B
, 2008
"... Conventional smoothing methods sometimes perform badly when used to smooth data over complex domains, by smoothing inappropriately across boundary features, such as peninsulas. Solutions to this smoothing problem tend to be computationally complex, and not to provide model smooth functions which are ..."
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Cited by 8 (4 self)
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Conventional smoothing methods sometimes perform badly when used to smooth data over complex domains, by smoothing inappropriately across boundary features, such as peninsulas. Solutions to this smoothing problem tend to be computationally complex, and not to provide model smooth functions which are appropriate for incorporating as components of other models, such as generalized additive models, or mixed additive models. In this paper we propose a class of smoothers appropriate for smoothing over difficult regions of R 2, which can be represented in terms of a low rank basis and one or two quadratic penalties. The key features of these smoothers are (i) that they do not ‘smooth across ’ boundary features; (ii) that their representation in terms of a basis and penalties allows straightforward incorporation as components of GAMs, mixed models and other nonstandard models; (iii) that smoothness selection for these model components is straightforward to accomplish in a computationally efficient manner via GCV, AIC or REML, for example; (iv) that their low rank means that their use is computationally efficient.
CAMERA RESPONSE FUNCTION SIGNATURE FOR DIGITAL FORENSICS – PART II: SIGNATURE EXTRACTION
"... Part I of this twopart paper proposed a robust way to detect local points on linearisophote surface in an image. Only a subset of these points corresponds to linear surface in image irradiance and provides useful information about the camera response function (CRF). In Part II, we show that, for s ..."
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Cited by 6 (2 self)
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Part I of this twopart paper proposed a robust way to detect local points on linearisophote surface in an image. Only a subset of these points corresponds to linear surface in image irradiance and provides useful information about the camera response function (CRF). In Part II, we show that, for some images, this subset of linear points could constitute a very small portion of the candidate set and the remaining points are often considered as noise. Our previous approach was to eliminate the noise using a learningbased method. The learningbased method could only reduce the noise but not eliminate it completely. Hence, it fails when the proportion of linear points is too small. As a main contribution in Part II, we introduce the concept of edge profile and consider the candidate points as discrete samples of an edge profile. Instead of eliminating the unwanted candidate points as noise, we use them to instantiate the edge profiles. Assuming that every edge profile has a linear component in image irradiance, the interactions of the edge profiles in the space of linear geometric invariants may correspond to the linear part which is CRFindicating. Such a model is shown to be sound and effective in both simulation images and real camera images. Index Terms — Camera response function, image forensics, geometric invariants, edge profiles 1.
HighOrder NonReflecting Boundary Conditions for the Linearized 2D Euler Equations: No Mean Flow Case,” submitted to Wave Motion
, 2008
"... Approved for public release; distribution is unlimited. ..."
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Cited by 6 (1 self)
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Approved for public release; distribution is unlimited.
COMPARISON OF THE CLASSICAL BMO WITH THE BMO SPACES ASSOCIATED WITH OPERATORS AND APPLICATIONS
, 2006
"... Abstract. Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMOL space associated with L which was introduced recently by Duong and Yan. We discuss applications of the new BMOL spaces in the theory of singular integration such as BMOL est ..."
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Cited by 6 (0 self)
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Abstract. Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMOL space associated with L which was introduced recently by Duong and Yan. We discuss applications of the new BMOL spaces in the theory of singular integration such as BMOL estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space BMOL might coincide with or might be essentially different from the classical BMO space. 1.
A (2006) Manifold learning and representations for image analysis and visualization
, 2006
"... i A manifold is a mathematical concept which generalizes surfaces to higher dimensions. The values of signals and data are sometimes naturally described as points in manifolds – they are manifoldvalued. In this thesis some recently proposed spectral methods for manifold learning are applied to a vi ..."
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Cited by 4 (0 self)
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i A manifold is a mathematical concept which generalizes surfaces to higher dimensions. The values of signals and data are sometimes naturally described as points in manifolds – they are manifoldvalued. In this thesis some recently proposed spectral methods for manifold learning are applied to a visualization problem in medical imaging. 3D volume data of the human brain, acquired using Diffusion Tensor MRI, is post processed in a novel way in order to represent and visualize the shape and connectivity of white matter fiber bundles. In addition to this realworld application of manifold learning, the contributions to a generic framework for processing of manifoldvalued signals and data consist of the following. 1) The idea of the diffusion mean, which is a preliminary result related to the extrinsic and intrinsic means in certain manifolds. 2) A representation for extrinsic manifoldvalued signal processing in SO(3), Q, which is useful
STOCHASTIC REDUCTIONS FOR INERTIAL FLUIDSTRUCTURE INTERACTIONS SUBJECT TO THERMAL FLUCTUATIONS
"... Abstract. We investigate the dynamics of elastic microstructures that interact with a fluid flow when subject to thermal fluctuations. We perform analysis to obtain systematically simplified descriptions of the mechanics in the limiting regimes when (i) the coupling forces that transfer momentum bet ..."
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Abstract. We investigate the dynamics of elastic microstructures that interact with a fluid flow when subject to thermal fluctuations. We perform analysis to obtain systematically simplified descriptions of the mechanics in the limiting regimes when (i) the coupling forces that transfer momentum between the fluid and microstructures is strong, (ii) the mass of the microstructures is small relative to the displaced mass of the fluid, and (iii) the response to stresses results in hydrodynamics that relax rapidly to a quasisteadystate relative to the motions of the microstructure. We derive effective equations using a singular perturbation analysis of the Backward Kolmogorov equations of the stochastic process. Our continuum mechanics description is based on the Stochastic Eulerian Lagrangian Method (SELM) which provides a framework for approximation of the fluidstructure interactions when subject to thermal fluctuations. We perform a dimension analysis of the SELM equations to identify key nondimensional groups and to characterize precisely each of the limiting physical regimes. The reduced equations offer insights into the physical accuracy of SELM descriptions in comparison with classical results. The reduced equations also elimintate rapid timescales from the dynamics and provide possible approaches for the development of more efficient computational methods for simulations of fluidstructure interactions when subject to thermal fluctuations.