1,497 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
Statistical Multiresolution Estimation for Variational Imaging: With an Application in Poisson-Biophotonics
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. The resulting convex
optimization problem is approached by a combination of an inexact alternating
direction method of multipliers and Dykstra's projection algorithm. We describe
a novel method for balancing data-fit and regularity that is fully automatic
and allows for a sound statistical interpretation. The performance of our
estimation approach is studied for various problems in imaging. Among others,
this includes deconvolution problems that arise in Poisson nanoscale
fluorescence microscopy
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
A Multiscale Pyramid Transform for Graph Signals
Multiscale transforms designed to process analog and discrete-time signals
and images cannot be directly applied to analyze high-dimensional data residing
on the vertices of a weighted graph, as they do not capture the intrinsic
geometric structure of the underlying graph data domain. In this paper, we
adapt the Laplacian pyramid transform for signals on Euclidean domains so that
it can be used to analyze high-dimensional data residing on the vertices of a
weighted graph. Our approach is to study existing methods and develop new
methods for the four fundamental operations of graph downsampling, graph
reduction, and filtering and interpolation of signals on graphs. Equipped with
appropriate notions of these operations, we leverage the basic multiscale
constructs and intuitions from classical signal processing to generate a
transform that yields both a multiresolution of graphs and an associated
multiresolution of a graph signal on the underlying sequence of graphs.Comment: 16 pages, 13 figure
MDL Denoising Revisited
We refine and extend an earlier MDL denoising criterion for wavelet-based
denoising. We start by showing that the denoising problem can be reformulated
as a clustering problem, where the goal is to obtain separate clusters for
informative and non-informative wavelet coefficients, respectively. This
suggests two refinements, adding a code-length for the model index, and
extending the model in order to account for subband-dependent coefficient
distributions. A third refinement is derivation of soft thresholding inspired
by predictive universal coding with weighted mixtures. We propose a practical
method incorporating all three refinements, which is shown to achieve good
performance and robustness in denoising both artificial and natural signals.Comment: Submitted to IEEE Transactions on Information Theory, June 200
Image Fusion via Sparse Regularization with Non-Convex Penalties
The L1 norm regularized least squares method is often used for finding sparse
approximate solutions and is widely used in 1-D signal restoration. Basis
pursuit denoising (BPD) performs noise reduction in this way. However, the
shortcoming of using L1 norm regularization is the underestimation of the true
solution. Recently, a class of non-convex penalties have been proposed to
improve this situation. This kind of penalty function is non-convex itself, but
preserves the convexity property of the whole cost function. This approach has
been confirmed to offer good performance in 1-D signal denoising. This paper
demonstrates the aforementioned method to 2-D signals (images) and applies it
to multisensor image fusion. The problem is posed as an inverse one and a
corresponding cost function is judiciously designed to include two data
attachment terms. The whole cost function is proved to be convex upon suitably
choosing the non-convex penalty, so that the cost function minimization can be
tackled by convex optimization approaches, which comprise simple computations.
The performance of the proposed method is benchmarked against a number of
state-of-the-art image fusion techniques and superior performance is
demonstrated both visually and in terms of various assessment measures
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