13,575 research outputs found
Wavelets and Fast Numerical Algorithms
Wavelet based algorithms in numerical analysis are similar to other transform
methods in that vectors and operators are expanded into a basis and the
computations take place in this new system of coordinates. However, due to the
recursive definition of wavelets, their controllable localization in both space
and wave number (time and frequency) domains, and the vanishing moments
property, wavelet based algorithms exhibit new and important properties.
For example, the multiresolution structure of the wavelet expansions brings
about an efficient organization of transformations on a given scale and of
interactions between different neighbouring scales. Moreover, wide classes of
operators which naively would require a full (dense) matrix for their numerical
description, have sparse representations in wavelet bases. For these operators
sparse representations lead to fast numerical algorithms, and thus address a
critical numerical issue.
We note that wavelet based algorithms provide a systematic generalization of
the Fast Multipole Method (FMM) and its descendents.
These topics will be the subject of the lecture. Starting from the notion of
multiresolution analysis, we will consider the so-called non-standard form
(which achieves decoupling among the scales) and the associated fast numerical
algorithms. Examples of non-standard forms of several basic operators (e.g.
derivatives) will be computed explicitly.Comment: 32 pages, uuencoded tar-compressed LaTeX file. Uses epsf.sty (see
`macros'
MAGMA: Multi-level accelerated gradient mirror descent algorithm for large-scale convex composite minimization
Composite convex optimization models arise in several applications, and are
especially prevalent in inverse problems with a sparsity inducing norm and in
general convex optimization with simple constraints. The most widely used
algorithms for convex composite models are accelerated first order methods,
however they can take a large number of iterations to compute an acceptable
solution for large-scale problems. In this paper we propose to speed up first
order methods by taking advantage of the structure present in many applications
and in image processing in particular. Our method is based on multi-level
optimization methods and exploits the fact that many applications that give
rise to large scale models can be modelled using varying degrees of fidelity.
We use Nesterov's acceleration techniques together with the multi-level
approach to achieve convergence rate, where
denotes the desired accuracy. The proposed method has a better
convergence rate than any other existing multi-level method for convex
problems, and in addition has the same rate as accelerated methods, which is
known to be optimal for first-order methods. Moreover, as our numerical
experiments show, on large-scale face recognition problems our algorithm is
several times faster than the state of the art
Current-mode piecewise-linear function generators
We present a systematic design technique for current-mode piecewise-linear (PWL) function generators. It uses two building blocks: a high-resolution current rectifier, and a programmable current amplifier. We show how to arrange these blocks to obtain basic non-linearities from which generic characteristics are built through aggregations. Measurements from a 1.0 /spl mu/m CMOS prototype chip show 10 pA resolution in the rectification operation and 0.6% non-linearity errors in the programmable scaling operation for 2 /spl mu/A input current range
ShearLab 3D: Faithful Digital Shearlet Transforms based on Compactly Supported Shearlets
Wavelets and their associated transforms are highly efficient when
approximating and analyzing one-dimensional signals. However, multivariate
signals such as images or videos typically exhibit curvilinear singularities,
which wavelets are provably deficient of sparsely approximating and also of
analyzing in the sense of, for instance, detecting their direction. Shearlets
are a directional representation system extending the wavelet framework, which
overcomes those deficiencies. Similar to wavelets, shearlets allow a faithful
implementation and fast associated transforms. In this paper, we will introduce
a comprehensive carefully documented software package coined ShearLab 3D
(www.ShearLab.org) and discuss its algorithmic details. This package provides
MATLAB code for a novel faithful algorithmic realization of the 2D and 3D
shearlet transform (and their inverses) associated with compactly supported
universal shearlet systems incorporating the option of using CUDA. We will
present extensive numerical experiments in 2D and 3D concerning denoising,
inpainting, and feature extraction, comparing the performance of ShearLab 3D
with similar transform-based algorithms such as curvelets, contourlets, or
surfacelets. In the spirit of reproducible reseaerch, all scripts are
accessible on www.ShearLab.org.Comment: There is another shearlet software package
(http://www.mathematik.uni-kl.de/imagepro/members/haeuser/ffst/) by S.
H\"auser and G. Steidl. We will include this in a revisio
Making Maps Of The Cosmic Microwave Background: The MAXIMA Example
This work describes Cosmic Microwave Background (CMB) data analysis
algorithms and their implementations, developed to produce a pixelized map of
the sky and a corresponding pixel-pixel noise correlation matrix from time
ordered data for a CMB mapping experiment. We discuss in turn algorithms for
estimating noise properties from the time ordered data, techniques for
manipulating the time ordered data, and a number of variants of the maximum
likelihood map-making procedure. We pay particular attention to issues
pertinent to real CMB data, and present ways of incorporating them within the
framework of maximum likelihood map-making. Making a map of the sky is shown to
be not only an intermediate step rendering an image of the sky, but also an
important diagnostic stage, when tests for and/or removal of systematic effects
can efficiently be performed. The case under study is the MAXIMA data set.
However, the methods discussed are expected to be applicable to the analysis of
other current and forthcoming CMB experiments.Comment: Replaced to match the published version, only minor change
Image interpolation using Shearlet based iterative refinement
This paper proposes an image interpolation algorithm exploiting sparse
representation for natural images. It involves three main steps: (a) obtaining
an initial estimate of the high resolution image using linear methods like FIR
filtering, (b) promoting sparsity in a selected dictionary through iterative
thresholding, and (c) extracting high frequency information from the
approximation to refine the initial estimate. For the sparse modeling, a
shearlet dictionary is chosen to yield a multiscale directional representation.
The proposed algorithm is compared to several state-of-the-art methods to
assess its objective as well as subjective performance. Compared to the cubic
spline interpolation method, an average PSNR gain of around 0.8 dB is observed
over a dataset of 200 images
Data-Driven Time-Frequency Analysis
In this paper, we introduce a new adaptive data analysis method to study
trend and instantaneous frequency of nonlinear and non-stationary data. This
method is inspired by the Empirical Mode Decomposition method (EMD) and the
recently developed compressed (compressive) sensing theory. The main idea is to
look for the sparsest representation of multiscale data within the largest
possible dictionary consisting of intrinsic mode functions of the form , where , consists of the
functions smoother than and . This problem can
be formulated as a nonlinear optimization problem. In order to solve this
optimization problem, we propose a nonlinear matching pursuit method by
generalizing the classical matching pursuit for the optimization problem.
One important advantage of this nonlinear matching pursuit method is it can be
implemented very efficiently and is very stable to noise. Further, we provide a
convergence analysis of our nonlinear matching pursuit method under certain
scale separation assumptions. Extensive numerical examples will be given to
demonstrate the robustness of our method and comparison will be made with the
EMD/EEMD method. We also apply our method to study data without scale
separation, data with intra-wave frequency modulation, and data with incomplete
or under-sampled data
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