The Experts below are selected from a list of 195 Experts worldwide ranked by ideXlab platform
Anne Gelb - One of the best experts on this subject based on the ideXlab platform.
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Fourier reconstruction of univariate Piecewise-Smooth Functions from non-uniform spectral data with exponential convergence rates
Applied and Computational Harmonic Analysis, 2015Co-Authors: Rodrigo B. Platte, Alexander Gutierrez, Anne GelbAbstract:Abstract Reconstruction of Piecewise Smooth Functions from non-uniform Fourier data arises in sensing applications such as magnetic resonance imaging (MRI). This paper presents a new method that uses edge information to recover the Fourier transform of a Piecewise Smooth Function from data that is sparsely sampled at high frequencies. The approximation is based on a combination of polynomials multiplied by complex exponentials. We obtain super-algebraic convergence rates for a large class of Functions with one jump discontinuity, and geometric convergence rates for Functions that decay exponentially fast in the physical domain when the derivatives satisfy a certain bound. Exponential convergence is also proved for Piecewise analytic Functions of compact support. Our method can also be used to improve initial jump location estimates, which are calculated from the available Fourier data, through an iterative process. Finally, if the Fourier transform is approximated at integer values, then the IFFT can be used to reconstruct the underlying Function. Post-processing techniques, such as spectral reprojection, can then be used to reduce Gibbs oscillations.
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Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data
Journal of Scientific Computing, 2011Co-Authors: Wolfgang Stefan, Anne Gelb, Aditya Viswanathan, Rosemary A. RenautAbstract:We present a new method for estimating the edges in a Piecewise Smooth Function from blurred and noisy Fourier data. The proposed method is constructed by combining the so called concentration factor edge detection method, which uses a finite number of Fourier coefficients to approximate the jump Function of a Piecewise Smooth Function, with compressed sensing ideas. Due to the global nature of the concentration factor method, Gibbs oscillations feature prominently near the jump discontinuities. This can cause the misidentification of edges when simple thresholding techniques are used. In fact, the true jump Function is sparse, i.e. zero almost everywhere with non-zero values only at the edge locations. Hence we adopt an idea from compressed sensing and propose a method that uses a regularized deconvolution to remove the artifacts. Our new method is fast, in the sense that it only needs the solution of a single l 1 minimization. Numerical examples demonstrate the accuracy and robustness of the method in the presence of noise and blur.
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Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data
Journal of Scientific Computing, 2007Co-Authors: Anne GelbAbstract:Spectral series expansions of Piecewise Smooth Functions are known to yield poor results, with spurious oscillations forming near the jump discontinuities and reduced convergence throughout the interval of approximation. The spectral reprojection method, most notably the Gegenbauer reconstruction method, can restore exponential convergence to Piecewise Smooth Function approximations from their (pseudo-)spectral coefficients. Difficulties may arise due to numerical robustness and ill-conditioning of the reprojection basis polynomials, however. This paper considers non-classical orthogonal polynomials as reprojection bases for a general order (finite or spectral) reconstruction of Piecewise Smooth Functions. Furthermore, when the given data are discrete grid point values, the reprojection polynomials are constructed to be orthogonal in the discrete sense, rather than by the usual continuous inner product. No calculation of optimal quadrature points is therefore needed. This adaptation suggests a method to approximate Piecewise Smooth Functions from discrete non-uniform data, and results in a one-dimensional approximation that is accurate and numerically robust.
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A Hybrid Approach to Spectral Reconstruction of Piecewise Smooth Functions
Journal of Scientific Computing, 2000Co-Authors: Anne GelbAbstract:Consider a Piecewise Smooth Function for which the (pseudo-)spectral coefficients are given. It is well known that while spectral partial sums yield exponentially convergent approximations for Smooth Functions, the results for Piecewise Smooth Functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks in the application of spectral methods. Various types of reconstruction methods developed for the recovery of Piecewise Smooth Functions have met with varying degrees of success. The Gegenbauer reconstruction method, originally proposed by Gottlieb et al. has the particularly impressive ability to reconstruct Piecewise analytic Functions with exponential convergence up to the points of discontinuity. However, it has been sharply criticized for its high cost and susceptibility to round-off error. In this paper, a new approach to Gegenbauer reconstruction is considered, resulting in a reconstruction method that is less computationally intensive and costly, yet still enjoys superior convergence. The idea is to create a procedure that combines the well known exponential filtering method in Smooth regions away from the discontinuities with the Gegenbauer reconstruction method in regions close to the discontinuities. This hybrid approach benefits from both the simplicity of exponential filtering and the high resolution properties of the Gegenbauer reconstruction method. Additionally, a new way of computing the Gegenbauer coefficients from Jacobian polynomial expansions is introduced that is both more cost effective and less prone to round-off errors.
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the resolution of the gibbs phenomenon for spherical harmonics
Mathematics of Computation, 1997Co-Authors: Anne GelbAbstract:Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to Functions with discontinuities, causing undesirable oscillations over the entire sphere. Recently, methods for removing the Gibbs phenomenon for one-dimensional discontinuous Functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first N expansion coefficients (either Fourier or Gegenbauer) of a Piecewise analytic Function f(x) is enough to recover an exponentially convergent approximation to the point values of f(x) in any subinterval in which the Function is analytic. Here we take a similar approach, proving that knowledge of the first N spherical harmonic coefficients yield an exponentially convergent approximation to a spherical Piecewise Smooth Function f(θ, Φ) in any subinterval [θ 1 , θ 2 ], Φ ∈ [0,2π], where the Function is analytic. Thus we entirely overcome the Gibbs phenomenon.
Carlos Fernandezgranda - One of the best experts on this subject based on the ideXlab platform.
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towards a mathematical theory of super resolution
Communications on Pure and Applied Mathematics, 2014Co-Authors: Emmanuel J Candes, Carlos FernandezgrandaAbstract:This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object—the high end of its spectrum—from coarse scale information only—from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0,1] and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up to a frequency cutoff fc. We show that one can super-resolve these point sources with infinite precision—i.e., recover the exact locations and amplitudes—by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2/fc. This result extends to higher dimensions and other models. In one dimension, for instance, it is possible to recover a Piecewise Smooth Function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary. © 2014 Wiley Periodicals, Inc.
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towards a mathematical theory of super resolution
arXiv: Information Theory, 2012Co-Authors: Emmanuel J Candes, Carlos FernandezgrandaAbstract:This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object---the high end of its spectrum---from coarse scale information only---from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in $[0,1]$ and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up until a frequency cut-off $f_c$. We show that one can super-resolve these point sources with infinite precision---i.e. recover the exact locations and amplitudes---by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least $2/f_c$. This result extends to higher dimensions and other models. In one dimension for instance, it is possible to recover a Piecewise Smooth Function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the {\em super-resolution factor} vary.
Emmanuel J Candes - One of the best experts on this subject based on the ideXlab platform.
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towards a mathematical theory of super resolution
Communications on Pure and Applied Mathematics, 2014Co-Authors: Emmanuel J Candes, Carlos FernandezgrandaAbstract:This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object—the high end of its spectrum—from coarse scale information only—from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0,1] and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up to a frequency cutoff fc. We show that one can super-resolve these point sources with infinite precision—i.e., recover the exact locations and amplitudes—by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least 2/fc. This result extends to higher dimensions and other models. In one dimension, for instance, it is possible to recover a Piecewise Smooth Function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary. © 2014 Wiley Periodicals, Inc.
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towards a mathematical theory of super resolution
arXiv: Information Theory, 2012Co-Authors: Emmanuel J Candes, Carlos FernandezgrandaAbstract:This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object---the high end of its spectrum---from coarse scale information only---from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in $[0,1]$ and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up until a frequency cut-off $f_c$. We show that one can super-resolve these point sources with infinite precision---i.e. recover the exact locations and amplitudes---by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least $2/f_c$. This result extends to higher dimensions and other models. In one dimension for instance, it is possible to recover a Piecewise Smooth Function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the {\em super-resolution factor} vary.
Pier Luigi Dragotti - One of the best experts on this subject based on the ideXlab platform.
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Centralized and Distributed Semiparametric Compression of Piecewise Smooth Functions
IEEE Transactions on Signal Processing, 2011Co-Authors: Varit Chaisinthop, Pier Luigi DragottiAbstract:This paper introduces novel wavelet-based semiparametric centralized and distributed compression methods for a class of 1-D Piecewise Smooth Functions. Classical centralized compression schemes are based on a relatively complex, nonlinear encoder and a simple, linear decoder. Recently, a new paradigm in compression called distributed source coding has emerged. This setup involves multiple encoders, where each one partially observes the source, and a centralized decoder. First, we focus on the dual situation of the centralized compression with a simple encoder and a complex decoder. We show that, by incorporating parametric estimation into the decoding procedure, it is possible to achieve the same rate-distortion performance as that of a conventional wavelet-based compression scheme. Second, we consider the distributed compression scenario, where each independent encoder partially observes the 1-D Piecewise Smooth Function. We propose a new wavelet-based distributed compression scheme that uses parametric estimation to perform joint decoding. Our analysis shows that it is possible for the proposed scheme to achieve the same compression performance as that of a joint encoding scheme.
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EUSIPCO - Semi-parametric compression of Piecewise Smooth Functions
2009Co-Authors: Varit Chaisinthop, Pier Luigi DragottiAbstract:This paper introduces a new wavelet-based compression scheme that combines the use of linear approximation and parametric estimation. Our proposed scheme differs from the conventional wavelet-based schemes in two ways: first, the encoder uses linear approximation and second, the decoding process is non-linear as it is combined with parametric estimation. We consider a simple model of one-dimensional (1-D) Piecewise Smooth Function and show that, with our scheme, it is possible to achieve the same decay in the distortion-rate bound as conventional wavelet-based schemes that employ non-linear approximation. A practical compression algorithm that achieves the distortion bound and uses the new concept of sampling of signal with finite rate of innovation is also presented together with the simulation results.
Rodrigo B. Platte - One of the best experts on this subject based on the ideXlab platform.
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Fourier reconstruction of univariate Piecewise-Smooth Functions from non-uniform spectral data with exponential convergence rates
Applied and Computational Harmonic Analysis, 2015Co-Authors: Rodrigo B. Platte, Alexander Gutierrez, Anne GelbAbstract:Abstract Reconstruction of Piecewise Smooth Functions from non-uniform Fourier data arises in sensing applications such as magnetic resonance imaging (MRI). This paper presents a new method that uses edge information to recover the Fourier transform of a Piecewise Smooth Function from data that is sparsely sampled at high frequencies. The approximation is based on a combination of polynomials multiplied by complex exponentials. We obtain super-algebraic convergence rates for a large class of Functions with one jump discontinuity, and geometric convergence rates for Functions that decay exponentially fast in the physical domain when the derivatives satisfy a certain bound. Exponential convergence is also proved for Piecewise analytic Functions of compact support. Our method can also be used to improve initial jump location estimates, which are calculated from the available Fourier data, through an iterative process. Finally, if the Fourier transform is approximated at integer values, then the IFFT can be used to reconstruct the underlying Function. Post-processing techniques, such as spectral reprojection, can then be used to reduce Gibbs oscillations.