144 research outputs found

    Function spaces vs. Scaling functions: Some issues in image classification

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    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

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    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 LpL^p (1≀p≀∞1\le p\le \infty) 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

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    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 Bqα,p(R2)\mathrm{B}^{\alpha,p}_q(\mathbb{R}^2). In particular B1α,1(R2)\mathrm{B}^{\alpha,1}_1(\mathbb{R}^2) with α≈1\alpha \approx 1 is a preferred model in image analysis for describing natural images. In case of noisy Radon data the total FBP reconstruction error ∄f−fLΎ∄≀∄f−fL∄+∄fL−fLΎ∄\|f-f_L^\delta\| \le \|f-f_L\|+ \|f_L - f_L^\delta\| splits into an approximation error and a data error, where LL serves as regularization parameter. In this paper, we study the approximation error of FBP reconstructions for target functions f∈L1(R2)∩Bqα,p(R2)f \in \mathrm{L}^1(\mathbb{R}^2) \cap \mathrm{B}^{\alpha,p}_q(\mathbb{R}^2) with positive α∈̞N\alpha \not\in \mathbb{N} and 1≀p,q≀∞1 \leq p,q \leq \infty. We prove that the Lp\mathrm{L}^p-norm of the inherent FBP approximation error f−fLf-f_L 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 WW. 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

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    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

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    Analysis on the unit sphere S2\mathbb{S}^{2} 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 S2\mathbb{S}^{2},   S3\>\>\mathbb{S}^{3} and the rotation group SO(3)SO(3). 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 Sd\mathbb{S}^{d} and SO(3)SO(3).Comment: The final publication is available at http://www.springerlink.co

    Problems on averages and lacunary maximal functions

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    We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First, we obtain an H1H^1 to L1,∞L^{1,\infty} bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an LpL^p regularity bound for some p>1p>1. Secondly, we obtain a necessary and sufficient condition for L2L^2 boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an LpL^p 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

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    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 RnR^n. 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 C∞C^\infty functions.Comment: 23 page
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