1,994 research outputs found
Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging
Left-invariant PDE-evolutions on the roto-translation group (and
their resolvent equations) have been widely studied in the fields of cortical
modeling and image analysis. They include hypo-elliptic diffusion (for contour
enhancement) proposed by Citti & Sarti, and Petitot, and they include the
direction process (for contour completion) proposed by Mumford. This paper
presents a thorough study and comparison of the many numerical approaches,
which, remarkably, is missing in the literature. Existing numerical approaches
can be classified into 3 categories: Finite difference methods, Fourier based
methods (equivalent to -Fourier methods), and stochastic methods (Monte
Carlo simulations). There are also 3 types of exact solutions to the
PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in
previous works by Duits and van Almsick in 2005. Here we provide an overview of
these 3 types of exact solutions and explain how they relate to each of the 3
numerical approaches. We compute relative errors of all numerical approaches to
the exact solutions, and the Fourier based methods show us the best performance
with smallest relative errors. We also provide an improvement of Mathematica
algorithms for evaluating Mathieu-functions, crucial in implementations of the
exact solutions. Furthermore, we include an asymptotical analysis of the
singularities within the kernels and we propose a probabilistic extension of
underlying stochastic processes that overcomes the singular behavior in the
origin of time-integrated kernels. Finally, we show retinal imaging
applications of combining left-invariant PDE-evolutions with invertible
orientation scores.Comment: A final and corrected version of the manuscript is Published in
Numerical Mathematics: Theory, Methods and Applications (NM-TMA), vol. (9),
p.1-50, 201
Multiscale modelling and identification of a class of lattice dynamical systems
A new multiscale modelling framework is introduced to describe a class of lattice dynamical systems (LDS), which can be used to model natural systems involving multiphysics
and the multi-resolution facets of a single spatio-temporal dynamical system. The emphasis of the paper is on the multi-resolution facets, with respect to the spatial domain, of a single spatio-temporal dynamical system by using a Haar wavelet decomposition technique. A multiscale identification method for such systems is then proposed, which can be considered as a dual of the multigrid method. The proposed identification method involves three
steps: the system dynamics at some specific scale of interest are identified using a recursive least-squares algorithm; the residual is then projected onto coarser scales using Haar wavelets and the parameter estimation errors are minimized; and finally a coarse correction
procedure is applied to the original scale. An outstanding advantage of the proposed identification method is a saving on the computational costs. Numerical examples are provided
to demonstrate the application of the proposed new approach
Iterative algorithms for total variation-like reconstructions in seismic tomography
A qualitative comparison of total variation like penalties (total variation,
Huber variant of total variation, total generalized variation, ...) is made in
the context of global seismic tomography. Both penalized and constrained
formulations of seismic recovery problems are treated. A number of simple
iterative recovery algorithms applicable to these problems are described. The
convergence speed of these algorithms is compared numerically in this setting.
For the constrained formulation a new algorithm is proposed and its convergence
is proven.Comment: 28 pages, 8 figures. Corrected sign errors in formula (25
Adaptive multiresolution analysis based on synchronization
We propose an adaptive multiscale approach to data analysis based on synchronization. The approach is nonlinear, data driven in the sense that it does not rely on a priori chosen basis, and automatically determines the data scale. Numerical results for one- and two-dimensional cases illustrate that the method works effectively for the usual modulated signals such as chirps, etc., as well as for more complicated data with multiple scales. The method extends straightforwardly to functions defined on weighted graphs and grids in high dimensions. Connections with some other recent approaches to multiscale analysis are briefly discussed
Numerical solution of the inverse Gardner equation
In this paper, the numerical solution of the inverse Gardner equation will be considered. The Haar wavelet collocation method (HWCM) will be used to determine the unknown boundary condition which is estimated from an over-specified condition at a boundary. In this regard, we apply the HWCM for discretizing the space derivatives and then use a quasilinearization technique to linearize the nonlinear term in the equations. It is proved that the proposed method has the order of convergence O(∆x). The efficiency and robustness of the proposed approach for solving the inverse Gardner equation are demonstrated by one numerical example.Publisher's Versio
Wavelets and their use
This review paper is intended to give a useful guide for those who want to
apply discrete wavelets in their practice. The notion of wavelets and their use
in practical computing and various applications are briefly described, but
rigorous proofs of mathematical statements are omitted, and the reader is just
referred to corresponding literature. The multiresolution analysis and fast
wavelet transform became a standard procedure for dealing with discrete
wavelets. The proper choice of a wavelet and use of nonstandard matrix
multiplication are often crucial for achievement of a goal. Analysis of various
functions with the help of wavelets allows to reveal fractal structures,
singularities etc. Wavelet transform of operator expressions helps solve some
equations. In practical applications one deals often with the discretized
functions, and the problem of stability of wavelet transform and corresponding
numerical algorithms becomes important. After discussing all these topics we
turn to practical applications of the wavelet machinery. They are so numerous
that we have to limit ourselves by some examples only. The authors would be
grateful for any comments which improve this review paper and move us closer to
the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh
- …