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Axel Munk

  1. Extreme value analysis of frame coefficients and implications for image denoising.

    Fri, 05/11/2012 - 15:01 — SomNambul
    Authors: Axel Munk, Markus Haltmeier
    Subjects: Numerical Analysis
    Abstract

    Denoising by frame thresholding is one of the most basic and efficient
    methods for recovering a discrete signal or image from data that are corrupted
    by additive Gaussian white noise. The basic idea is to select a frame of
    analyzing elements that separates the data in few large coefficients due to the
    signal and many small coefficients mainly due to the noise $\epsilon_n$.
    Removing all data coefficients being in magnitude below a certain threshold
    yields an approximation to the original signal.

  2. Statistical Multiresolution Estimation for Variational Imaging: With an Application in Poisson-Biophotonics.

    Wed, 04/18/2012 - 15:01 — SomNambul
    Authors: Axel Munk, Philipp Marnitz, Klaus Frick
    Subjects: Applications
    Abstract

    In this paper we present a spatially-adaptive method for image reconstruction
    that is based on the concept of statistical multiresolution estimation as
    introduced in [Frick K, Marnitz P, and Munk A. "Statistical multiresolution
    Dantzig estimation in imaging: Fundamental concepts and algorithmic framework".
    Electron. J. Stat., 6:231-268, 2012]. It constitutes a variational
    regularization technique that uses an supremum-type distance measure as
    data-fidelity combined with a convex cost functional.

  3. Multiscale Methods for Shape Constraints in Deconvolution.

    Fri, 07/08/2011 - 15:01 — SomNambul
    Authors: Axel Munk, Johannes Schmidt-Hieber, Lutz Duembgen
    Subjects: Statistics
    Abstract

    We derive multiscale statistics for deconvolution in order to detect
    qualitative features of the unknown density. An important example covered
    within this framework is to test for local monotonicity on all scales
    simultaneously. The errors in the deconvolution model are restricted to a
    certain class of distributions that include Laplace, Gamma and Exponential
    random variables. Our approach relies on inversion formulas for deconvolution
    operators. For multiscale testing, we consider a calibration, motivated by the
    modulus of continuity of Brownian motion.

  4. Statistical Multiresolution Estimation in Imaging: Fundamental Concepts and Algorithmic Framework.

    Tue, 01/25/2011 - 16:01 — SomNambul
    Authors: Axel Munk, Philipp Marnitz, Klaus Frick
    Subjects: Applications
    Abstract

    In this paper we introduce a general class of statistical multiresolution
    estimators and develop an algorithmic framework for computing those. By this we
    mean estimators that are defined as solutions of convex optimization problems
    with $\ell_\infty$-type constraints. We employ a combination of an alternating
    direction augmented Lagrangian technique with Dykstra's algorithm for computing
    orthogonal projections onto intersections of convex sets. The capability of the
    proposed method is illustrated by various examples from imaging.

  5. M\"{o}bius deconvolution on the hyperbolic plane with application to impedance density estimation.

    Thu, 10/21/2010 - 15:01 — SomNambul
    Authors: Axel Munk, Peter T. Kim, Stephan F. Huckemann, Ja-Yong Koo
    Subjects: Statistics
    Abstract

    In this paper we consider a novel statistical inverse problem on the
    Poincar\'{e}, or Lobachevsky, upper (complex) half plane. Here the Riemannian
    structure is hyperbolic and a transitive group action comes from the space of
    $2\times2$ real matrices of determinant one via M\"{o}bius transformations. Our
    approach is based on a deconvolution technique which relies on the
    Helgason--Fourier calculus adapted to this hyperbolic space. This gives a
    minimax nonparametric density estimator of a hyperbolic density that is
    corrupted by a random M\"{o}bius transform.

  6. Nonparametric estimation of the volatility under microstructure noise: wavelet adaptation.

    Wed, 07/28/2010 - 15:01 — SomNambul
    Authors: Axel Munk, Johannes Schmidt-Hieber, Marc Hoffmann
    Subjects: Statistics
    Abstract

    We study nonparametric estimation of the volatility function of a diffusion
    process from discrete data, when the data are blurred by additional noise. This
    noise can be white or correlated, and serves as a model for microstructure
    effects in financial modeling, when the data are given on an intra-day scale.
    By developing pre-averaging techniques combined with wavelet thresholding, we
    construct adaptive estimators that achieve a nearly optimal rate within a large
    scale of smoothness constraints of Besov type.

  7. Locally Adaptive Regularization of Linear Statistical Inverse Problems.

    Thu, 03/18/2010 - 16:01 — SomNambul
    Authors: Axel Munk, Philipp Marnitz, Klaus Frick
    Subjects: Statistics
    Abstract

    This paper is concerned with a novel regularization technique for solving
    linear ill-posed operator equations in Hilbert spaces from data that is
    corrupted by white noise. As fit-to-data measures we employ extreme-value
    statistics of projections of residuals on a given set of sub-spaces in the
    image-space of the operator. We show that the proposed regularization technique
    exhibits local adaptive behaviour and chooses the amount of regularization in a
    data-driven way. This also leads to honest confidence-regions.

  8. Lower bounds for volatility estimation in microstructure noise models.

    Wed, 02/17/2010 - 16:01 — SomNambul
    Authors: Axel Munk, Johannes Schmidt-Hieber
    Subjects: Statistics
    Abstract

    In this paper we derive lower bounds in minimax sense for estimation of the
    instantaneous volatility if the diffusion type part cannot be observed directly
    but under some additional Gaussian noise. Three different models are
    considered. Our technique is based on a general inequality for Kullback-Leibler
    divergence of multivariate normal random variables and spectral analysis of the
    processes. The derived lower bounds are indeed optimal. Upper bounds can be
    found in Munk and Schmidt-Hieber [18].

  9. Locally adaptive image denoising by a statistical multiresolution criterion.

    Mon, 02/01/2010 - 16:01 — SomNambul
    Authors: Axel Munk, Zakhar Kabluchko, Thomas Hotz, Philipp Marnitz, Rahel Stichtenoth, Laurie Davies
    Subjects: Methodology
    Abstract

    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.

  10. Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise.

    Mon, 08/24/2009 - 13:10 — SomNambul
    Authors: Axel Munk, Johannes Schmidt-Hieber
    Subjects: Methodology
    Abstract

    We consider the models Y_{i,n}=\int_0^{i/n}
    \sigma(s)dW_s+\tau(i/n)\epsilon_{i,n}, and \tilde
    Y_{i,n}=\sigma(i/n)W_{i/n}+\tau(i/n)\epsilon_{i,n}, i=1,...,n, where W_t
    denotes a standard Brownian motion and \epsilon_{i,n} are centered i.i.d.
    random variables with E(\epsilon_{i,n}^2)=1 and finite fourth moment.
    Furthermore, \sigma and \tau are unknown deterministic functions and W_t and
    (\epsilon_{1,n},...,\epsilon_{n,n}) are assumed to be independent processes.
    Based on a spectral decomposition of the covariance structures we derive series
    estimators for \sigma^2 and \tau^2 and investigate t

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