516 research outputs found
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
Extreme Value Analysis of Empirical Frame Coefficients and Implications for Denoising by Soft-Thresholding
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 a
reconstruction of the original signal. In order to properly balance the amount
of noise to be removed and the relevant signal features to be kept, a precise
understanding of the statistical properties of thresholding is important. For
that purpose we derive the asymptotic distribution of max_{\omega \in \Omega_n}
|| for a wide class of redundant frames
(\phi_\omega^n: \omega \in \Omega_n}. Based on our theoretical results we give
a rationale for universal extreme value thresholding techniques yielding
asymptotically sharp confidence regions and smoothness estimates corresponding
to prescribed significance levels. The results cover many frames used in
imaging and signal recovery applications, such as redundant wavelet systems,
curvelet frames, or unions of bases. We show that `generically' a standard
Gumbel law results as it is known from the case of orthonormal wavelet bases.
However, for specific highly redundant frames other limiting laws may occur. We
indeed verify that the translation invariant wavelet transform shows a
different asymptotic behaviour.Comment: [Content: 39 pages, 4 figures] Note that in this version 4 we have
slightely changed the title of the paper and we have rewritten parts of the
introduction. Except for corrected typos the other parts of the paper are the
same as the original versions
Combining local regularity estimation and total variation optimization for scale-free texture segmentation
Texture segmentation constitutes a standard image processing task, crucial to
many applications. The present contribution focuses on the particular subset of
scale-free textures and its originality resides in the combination of three key
ingredients: First, texture characterization relies on the concept of local
regularity ; Second, estimation of local regularity is based on new multiscale
quantities referred to as wavelet leaders ; Third, segmentation from local
regularity faces a fundamental bias variance trade-off: In nature, local
regularity estimation shows high variability that impairs the detection of
changes, while a posteriori smoothing of regularity estimates precludes from
locating correctly changes. Instead, the present contribution proposes several
variational problem formulations based on total variation and proximal
resolutions that effectively circumvent this trade-off. Estimation and
segmentation performance for the proposed procedures are quantified and
compared on synthetic as well as on real-world textures
Wavelet Analysis and Denoising: New Tools for Economists
This paper surveys the techniques of wavelets analysis and the associated methods of denoising. The Discrete Wavelet Transform and its undecimated version, the Maximum Overlapping Discrete Wavelet Transform, are described. The methods of wavelets analysis can be used to show how the frequency content of the data varies with time. This allows us to pinpoint in time such events as major structural breaks. The sparse nature of the wavelets representation also facilitates the process of noise reduction by nonlinear wavelet shrinkage , which can be used to reveal the underlying trends in economic data. An application of these techniques to the UK real GDP (1873-2001) is described. The purpose of the analysis is to reveal the true structure of the data - including its local irregularities and abrupt changes - and the results are surprising.Wavelets, Denoising, Structural breaks, Trend estimation
Variational Multiscale Nonparametric Regression: Algorithms and Implementation
Many modern statistically efficient methods come with tremendous
computational challenges, often leading to large-scale optimisation problems.
In this work, we examine such computational issues for recently developed
estimation methods in nonparametric regression with a specific view on image
denoising. We consider in particular certain variational multiscale estimators
which are statistically optimal in minimax sense, yet computationally
intensive. Such an estimator is computed as the minimiser of a smoothness
functional (e.g., TV norm) over the class of all estimators such that none of
its coefficients with respect to a given multiscale dictionary is statistically
significant. The so obtained multiscale Nemirowski-Dantzig estimator (MIND) can
incorporate any convex smoothness functional and combine it with a proper
dictionary including wavelets, curvelets and shearlets. The computation of MIND
in general requires to solve a high-dimensional constrained convex optimisation
problem with a specific structure of the constraints induced by the statistical
multiscale testing criterion. To solve this explicitly, we discuss three
different algorithmic approaches: the Chambolle-Pock, ADMM and semismooth
Newton algorithms. Algorithmic details and an explicit implementation is
presented and the solutions are then compared numerically in a simulation study
and on various test images. We thereby recommend the Chambolle-Pock algorithm
in most cases for its fast convergence. We stress that our analysis can also be
transferred to signal recovery and other denoising problems to recover more
general objects whenever it is possible to borrow statistical strength from
data patches of similar object structure.Comment: Codes are available at https://github.com/housenli/MIN
Pointwise adaptive estimation for robust and quantile regression
A nonparametric procedure for robust regression estimation and for quantile
regression is proposed which is completely data-driven and adapts locally to
the regularity of the regression function. This is achieved by considering in
each point M-estimators over different local neighbourhoods and by a local
model selection procedure based on sequential testing. Non-asymptotic risk
bounds are obtained, which yield rate-optimality for large sample asymptotics
under weak conditions. Simulations for different univariate median regression
models show good finite sample properties, also in comparison to traditional
methods. The approach is extended to image denoising and applied to CT scans in
cancer research
Shape Constrained Regularisation by Statistical Multiresolution for Inverse Problems: Asymptotic Analysis
This paper is concerned with a novel regularisation technique for solving
linear ill-posed operator equations in Hilbert spaces from data that is
corrupted by white noise. We combine convex penalty functionals with
extreme-value statistics of projections of the residuals on a given set of
sub-spaces in the image-space of the operator. We prove general consistency and
convergence rate results in the framework of Bregman-divergences which allows
for a vast range of penalty functionals. Various examples that indicate the
applicability of our approach will be discussed. We will illustrate in the
context of signal and image processing that the presented method constitutes a
locally adaptive reconstruction method
A Multiscale Approach for Statistical Characterization of Functional Images
Increasingly, scientific studies yield functional image data, in which the observed data consist of sets of curves recorded on the pixels of the image. Examples include temporal brain response intensities measured by fMRI and NMR frequency spectra measured at each pixel. This article presents a new methodology for improving the characterization of pixels in functional imaging, formulated as a spatial curve clustering problem. Our method operates on curves as a unit. It is nonparametric and involves multiple stages: (i) wavelet thresholding, aggregation, and Neyman truncation to effectively reduce dimensionality; (ii) clustering based on an extended EM algorithm; and (iii) multiscale penalized dyadic partitioning to create a spatial segmentation. We motivate the different stages with theoretical considerations and arguments, and illustrate the overall procedure on simulated and real datasets. Our method appears to offer substantial improvements over monoscale pixel-wise methods. An Appendix which gives some theoretical justifications of the methodology, computer code, documentation and dataset are available in the online supplements
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