144 research outputs found
Function spaces vs. Scaling functions: Some issues in image classification
Criteria based on the computation of fractal dimensions have been used in order to perform image analysis and classification; we show that such criteria often amount to deter- mine the regularity of the image in some classes of function spaces, and that looking for richer criteria naturally leads to the introduction of new classes of function spaces. We will investigate the properties of some of these classes, and show which type of additional information they yield for the initial image
Inversion of noisy Radon transform by SVD based needlet
A linear method for inverting noisy observations of the Radon transform is
developed based on decomposition systems (needlets) with rapidly decaying
elements induced by the Radon transform SVD basis. Upper bounds of the risk of
the estimator are established in () norms for functions
with Besov space smoothness. A practical implementation of the method is given
and several examples are discussed
Error analysis for filtered back projection reconstructions in Besov spaces
Filtered back projection (FBP) methods are the most widely used
reconstruction algorithms in computerized tomography (CT). The ill-posedness of
this inverse problem allows only an approximate reconstruction for given noisy
data. Studying the resulting reconstruction error has been a most active field
of research in the 1990s and has recently been revived in terms of optimal
filter design and estimating the FBP approximation errors in general Sobolev
spaces.
However, the choice of Sobolev spaces is suboptimal for characterizing
typical CT reconstructions. A widely used model are sums of characteristic
functions, which are better modelled in terms of Besov spaces
. In particular
with is a preferred
model in image analysis for describing natural images.
In case of noisy Radon data the total FBP reconstruction error
splits into an
approximation error and a data error, where serves as regularization
parameter. In this paper, we study the approximation error of FBP
reconstructions for target functions with positive and . We prove that the -norm
of the inherent FBP approximation error can be bounded above by
\begin{equation*} \|f - f_L\|_{\mathrm{L}^p(\mathbb{R}^2)} \leq c_{\alpha,q,W}
\, L^{-\alpha} \, |f|_{\mathrm{B}^{\alpha,p}_q(\mathbb{R}^2)} \end{equation*}
under suitable assumptions on the utilized low-pass filter's window function
. This then extends by classical methods to estimates for the total
reconstruction error.Comment: 32 pages, 8 figure
Locally adaptive image denoising by a statistical multiresolution criterion
We demonstrate how one can choose the smoothing parameter in image denoising
by a statistical multiresolution criterion, both globally and locally. Using
inhomogeneous diffusion and total variation regularization as examples for
localized regularization schemes, we present an efficient method for locally
adaptive image denoising. As expected, the smoothing parameter serves as an
edge detector in this framework. Numerical examples illustrate the usefulness
of our approach. We also present an application in confocal microscopy
Splines and Wavelets on Geophysically Relevant Manifolds
Analysis on the unit sphere found many applications in
seismology, weather prediction, astrophysics, signal analysis, crystallography,
computer vision, computerized tomography, neuroscience, and statistics.
In the last two decades, the importance of these and other applications
triggered the development of various tools such as splines and wavelet bases
suitable for the unit spheres , and the
rotation group . Present paper is a summary of some of results of the
author and his collaborators on generalized (average) variational splines and
localized frames (wavelets) on compact Riemannian manifolds. The results are
illustrated by applications to Radon-type transforms on and
.Comment: The final publication is available at http://www.springerlink.co
Problems on averages and lacunary maximal functions
We prove three results concerning convolution operators and lacunary maximal
functions associated to dilates of measures. First, we obtain an to
bound for lacunary maximal operators under a dimensional
assumption on the underlying measure and an assumption on an regularity
bound for some . Secondly, we obtain a necessary and sufficient condition
for boundedness of lacunary maximal operator associated to averages over
convex curves in the plane. Finally we prove an regularity result for
such averages. We formulate various open problems.Comment: To appear in the Marcinkiewicz Centenary Volume (Banach Center
Publications 95
On Bogovski\u{\i} and regularized Poincar\'e integral operators for de Rham complexes on Lipschitz domains
We study integral operators related to a regularized version of the classical
Poincar\'e path integral and the adjoint class generalizing Bogovski\u{\i}'s
integral operator, acting on differential forms in . We prove that these
operators are pseudodifferential operators of order -1. The Poincar\'e-type
operators map polynomials to polynomials and can have applications in finite
element analysis. For a domain starlike with respect to a ball, the special
support properties of the operators imply regularity for the de Rham complex
without boundary conditions (using Poincar\'e-type operators) and with full
Dirichlet boundary conditions (using Bogovski\u{\i}-type operators). For
bounded Lipschitz domains, the same regularity results hold, and in addition we
show that the cohomology spaces can always be represented by
functions.Comment: 23 page
- âŠ