1,463 research outputs found
Kernel-based Image Reconstruction from Scattered Radon Data
Computerized tomography requires suitable numerical methods for the approximation of a bivariate
function f from a finite set of discrete Radon data, each of whose data samples represents one line
integral of f . In standard reconstruction methods, specific assumptions concerning the geometry
of the Radon lines are usually made. In relevant applications of image reconstruction, however,
such assumptions are often too restrictive. In this case, one would rather prefer to work with
reconstruction methods allowing for arbitrary distributions of scattered Radon lines.
This paper proposes a novel image reconstruction method for scattered Radon data, which combines
kernel-based scattered data approximation with a well-adapted regularization of the Radon transform.
This results in a very flexible numerical algorithm for image reconstruction, which works for arbitrary
distributions of Radon lines. This is in contrast to the classical filtered back projection, which
essentially relies on a regular distribution of the Radon lines, e.g. parallel beam geometry. The good
performance of the kernel-based image reconstruction method is illustrated by numerical examples
and comparisons
A rescaled method for RBF approximation
In the recent paper [8], a new method to compute stable kernel-based
interpolants has been presented. This \textit{rescaled interpolation} method
combines the standard kernel interpolation with a properly defined rescaling
operation, which smooths the oscillations of the interpolant. Although
promising, this procedure lacks a systematic theoretical investigation. Through
our analysis, this novel method can be understood as standard kernel
interpolation by means of a properly rescaled kernel. This point of view allow
us to consider its error and stability properties
Approximation Theory XV: San Antonio 2016
These proceedings are based on papers presented at the international conference Approximation Theory XV, which was held May 22\u201325, 2016 in San Antonio, Texas. The conference was the fifteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 146 participants. The book contains longer survey papers by some of the invited speakers covering topics such as compressive sensing, isogeometric analysis, and scaling limits of polynomials and entire functions of exponential type.
The book also includes papers on a variety of current topics in Approximation Theory drawn from areas such as advances in kernel approximation with applications, approximation theory and algebraic geometry, multivariate splines for applications, practical function approximation, approximation of PDEs, wavelets and framelets with applications, approximation theory in signal processing, compressive sensing, rational interpolation, spline approximation in isogeometric analysis, approximation of fractional differential equations, numerical integration formulas, and trigonometric polynomial approximation
Partition of unity interpolation using stable kernel-based techniques
In this paper we propose a new stable and accurate approximation technique
which is extremely effective for interpolating large scattered data sets. The
Partition of Unity (PU) method is performed considering Radial Basis Functions
(RBFs) as local approximants and using locally supported weights. In
particular, the approach consists in computing, for each PU subdomain, a stable
basis. Such technique, taking advantage of the local scheme, leads to a
significant benefit in terms of stability, especially for flat kernels.
Furthermore, an optimized searching procedure is applied to build the local
stable bases, thus rendering the method more efficient
On the spectral distribution of kernel matrices related to\ud radial basis functions
This paper focuses on the spectral distribution of kernel matrices related to radial basis functions. The asymptotic behaviour of eigenvalues of kernel matrices related to radial basis functions with different smoothness are studied. These results are obtained by estimated the coefficients of an orthogonal expansion of the underlying kernel function. Beside many other results, we prove that there are exactly (k+d−1/d-1) eigenvalues in the same order for analytic separable kernel functions like the Gaussian in Rd. This gives theoretical support for how to choose the diagonal scaling matrix in the RBF-QR method (Fornberg et al, SIAM J. Sci. Comput. (33), 2011) which can stably compute Gaussian radial basis function interpolants
On kernel engineering via Paley–Wiener
A radial basis function approximation takes the form
where the coefficients a 1,…,a n are real numbers, the centres b 1,…,b n are distinct points in ℝ d , and the function φ:ℝ d →ℝ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which φ is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure μ for which the convolution ψ=μ φ is a function of compact support, and when φ is polyharmonic. The novelty of this construction is its use of the Paley–Wiener theorem to identify compact support via analysis of the Fourier transform of the new kernel ψ, so providing a new form of kernel engineering
Cholesky-factorized sparse Kernel in support vector machines
Support Vector Machine (SVM) is one of the most powerful machine learning algorithms due to its convex optimization formulation and handling non-linear classification. However, one of its main drawbacks is the long time it takes to train large data sets. This limitation is often aroused when applying non-linear kernels (e.g. RBF Kernel) which are usually required to obtain better separation for linearly inseparable data sets. In this thesis, we study an approach that aims to speed-up the training time by combining both the better performance of RBF kernels and fast training by a linear solver, LIBLINEAR. The approach uses an RBF kernel with a sparse matrix which is factorized using Cholesky decomposition. The method is tested on large artificial and real data sets and compared to the standard RBF and linear kernels where both the accuracy and training time are reported. For most data sets, the result shows a huge training time reduction, over 90\%, whilst maintaining the accuracy
A simple method for finite range decomposition of quadratic forms and Gaussian fields
We present a simple method to decompose the Green forms corresponding to a
large class of interesting symmetric Dirichlet forms into integrals over
symmetric positive semi-definite and finite range (properly supported) forms
that are smoother than the original Green form. This result gives rise to
multiscale decompositions of the associated Gaussian free fields into sums of
independent smoother Gaussian fields with spatially localized correlations. Our
method makes use of the finite propagation speed of the wave equation and
Chebyshev polynomials. It improves several existing results and also gives
simpler proofs.Comment: minor correction for t<
One-Step Recurrences for Stationary Random Fields on the Sphere
Recurrences for positive definite functions in terms of the space dimension
have been used in several fields of applications. Such recurrences typically
relate to properties of the system of special functions characterizing the
geometry of the underlying space. In the case of the sphere the (strict) positive definiteness of the zonal function
is determined by the signs of the coefficients in the
expansion of in terms of the Gegenbauer polynomials , with
. Recent results show that classical differentiation and
integration applied to have positive definiteness preserving properties in
this context. However, in these results the space dimension changes in steps of
two. This paper develops operators for zonal functions on the sphere which
preserve (strict) positive definiteness while moving up and down in the ladder
of dimensions by steps of one. These fractional operators are constructed to
act appropriately on the Gegenbauer polynomials
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