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ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 2008
"... Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment ..."
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Cited by 43 (6 self)
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Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes.
On computing rational GaussChebyshev quadrature formulas. 2004
 In preparation
"... Abstract. We provide an algorithm to compute the nodes and weights for GaussChebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside [−1, 1]. Contrary to existing rational quadrature formulas, the computational effort is very low, even for ..."
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Cited by 28 (22 self)
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Abstract. We provide an algorithm to compute the nodes and weights for GaussChebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside [−1, 1]. Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order O(n). This method is based on the derivation of explicit expressions for Chebyshev orthogonal rational functions, which are (thus far) the only examples of explicitly known orthogonal rational functions on [−1,1] with arbitrary real poles outside this interval. 1.
Computing orthogonal rational functions with poles near the boundary
, 2006
"... Computing orthogonal rational functions is a far from trivial problem, especially for poles close to the interval of integration. In this paper we analyze some of the difficulties involved and present two different approaches to solve this problem. ..."
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Cited by 24 (18 self)
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Computing orthogonal rational functions is a far from trivial problem, especially for poles close to the interval of integration. In this paper we analyze some of the difficulties involved and present two different approaches to solve this problem.
A quadrature formula based on Chebyshev rational functions
 IMA Journal of Numerical Analysis
"... Several generalisations to the classical Gauss quadrature formulas have been made over the last few years. When the integrand has singularities near the interval of integration, formulas based on rational functions give more accurate results than the classical quadrature rules based on polynomials. ..."
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Cited by 5 (4 self)
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Several generalisations to the classical Gauss quadrature formulas have been made over the last few years. When the integrand has singularities near the interval of integration, formulas based on rational functions give more accurate results than the classical quadrature rules based on polynomials. In this paper we present one such generalisation which uses results from the theory of orthogonal rational functions. Compared to similar existing formulas, it has the advantage of improved stability and smaller quadrature weights. 1
An Analysis of Sphere Tessellations for Pose Estimation of 3D Objects Using Spherically Correlated Images ∗
"... Eigendecomposition is a common technique used for pose detection of threedimensional (3D) objects from twodimensional (2D) images. It has been shown in previous work that the eigendecomposition can be estimated using spherical sampling in conjunction with the Spherical Harmonic Transform. The iss ..."
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Cited by 5 (5 self)
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Eigendecomposition is a common technique used for pose detection of threedimensional (3D) objects from twodimensional (2D) images. It has been shown in previous work that the eigendecomposition can be estimated using spherical sampling in conjunction with the Spherical Harmonic Transform. The issue then becomes deciding on the best tessellation of the sphere to define the sampling pattern. In this paper we evaluate three popular tessellations and compare and contrast their computational performance, as well as their estimation accuracy for the eigendecomposition of this spherical data set. 1.
Fast Eigenspace Decomposition of Images of Objects With Variation in Illumination and Pose
"... Abstract—Many appearancebased classification problems such as principal component analysis, linear discriminant analysis, and locally preserving projections involve computing the principal components (eigenspace) of a large set of images. Although the online expense associated with appearancebased ..."
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Abstract—Many appearancebased classification problems such as principal component analysis, linear discriminant analysis, and locally preserving projections involve computing the principal components (eigenspace) of a large set of images. Although the online expense associated with appearancebased techniques is small, the offline computational burden becomes prohibitive for practical applications. This paper presents a method to reduce the expense of computing the eigenspace decomposition of a set of images when variations in both illumination and pose are present. In particular, it is shown that the set of images of an object under a wide range of illumination conditions and a fixed pose can be significantly reduced by projecting these data onto a few lowfrequency spherical harmonics, producing a set of “harmonic images. ” It is then shown that the dimensionality of the set of harmonic images at different poses can be further reduced by utilizing the fast Fourier transform. An eigenspace decomposition is then applied in the spectral domain at a much lower dimension, thereby significantly reducing the computational expense. An analysis is also provided, showing that the principal eigenimages computed assuming a single illumination source are capable of recovering a significant amount of information from images of objects when multiple illumination sources exist. Index Terms—Eigenspace decomposition, Fourier transform, illumination variation, pose estimation, spherical harmonics.
Eigendecomposition of Images Correlated on S¹, S², and SO(3) Using Spectral Theory
, 2009
"... Eigendecomposition represents one computationally efficient approach for dealing with object detection and pose estimation, as well as other visionbased problems, and has been applied to sets of correlated images for this purpose. The major drawback in using eigendecomposition is the off line comp ..."
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Eigendecomposition represents one computationally efficient approach for dealing with object detection and pose estimation, as well as other visionbased problems, and has been applied to sets of correlated images for this purpose. The major drawback in using eigendecomposition is the off line computational expense incurred by computing the desired subspace. This off line expense increases drastically as the number of correlated images becomes large (which is the case when doing fully general 3D pose estimation). Previous work has shown that for data correlated on S¹, Fourier analysis can help reduce the computational burden of this off line expense. This paper presents a method for extending this technique to data correlated on S² as well as SO(3) by sampling the sphere appropriately. An algorithm is then developed for reducing the off line computational burden associated with computing the eigenspace by exploiting the spectral information of this spherical data set using spherical harmonics and WignerD functions. Experimental results are presented to compare the proposed algorithm to the true eigendecomposition, as well as assess the computational savings.