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

Intermediately Executed Code is the Key to Find Refactorings that Improve Temporal Data Locality

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Solving ptychography with a convex relaxation

Ptychography is a powerful computational imaging technique that transforms a collection of low-resolution images into a high-resolution sample reconstruction. Unfortunately, algorithms that are currently used to solve this reconstruction problem lack stability, robustness, and theoretical guarantees. Recently, convex optimization algorithms have improved the accuracy and reliability of several related reconstruction efforts. This paper proposes a convex formulation of the ptychography problem. This formulation has no local minima, it can be solved using a wide range of algorithms, it can incorporate appropriate noise models, and it can include multiple a priori constraints. The paper considers a specific algorithm, based on low-rank factorization, whose runtime and memory usage are near-linear in the size of the output image. Experiments demonstrate that this approach offers a 25% lower background variance on average than alternating projections, the current standard algorithm for ptychographic reconstruction.Comment: 8 pages, 8 figure

A fast patch-dictionary method for whole image recovery

Various algorithms have been proposed for dictionary learning. Among those for image processing, many use image patches to form dictionaries. This paper focuses on whole-image recovery from corrupted linear measurements. We address the open issue of representing an image by overlapping patches: the overlapping leads to an excessive number of dictionary coefficients to determine. With very few exceptions, this issue has limited the applications of image-patch methods to the local kind of tasks such as denoising, inpainting, cartoon-texture decomposition, super-resolution, and image deblurring, for which one can process a few patches at a time. Our focus is global imaging tasks such as compressive sensing and medical image recovery, where the whole image is encoded together, making it either impossible or very ineffective to update a few patches at a time. Our strategy is to divide the sparse recovery into multiple subproblems, each of which handles a subset of non-overlapping patches, and then the results of the subproblems are averaged to yield the final recovery. This simple strategy is surprisingly effective in terms of both quality and speed. In addition, we accelerate computation of the learned dictionary by applying a recent block proximal-gradient method, which not only has a lower per-iteration complexity but also takes fewer iterations to converge, compared to the current state-of-the-art. We also establish that our algorithm globally converges to a stationary point. Numerical results on synthetic data demonstrate that our algorithm can recover a more faithful dictionary than two state-of-the-art methods. Combining our whole-image recovery and dictionary-learning methods, we numerically simulate image inpainting, compressive sensing recovery, and deblurring. Our recovery is more faithful than those of a total variation method and a method based on overlapping patches

An optimal adaptive wavelet method for First Order System Least Squares

In this paper, it is shown that any well-posed 2nd order PDE can be reformulated as a well-posed first order least squares system. This system will be solved by an adaptive wavelet solver in optimal computational complexity. The applications that are considered are second order elliptic PDEs with general inhomogeneous boundary conditions, and the stationary Navier-Stokes equations.Comment: 40 page

An optimally concentrated Gabor transform for localized time-frequency components

Gabor analysis is one of the most common instances of time-frequency signal analysis. Choosing a suitable window for the Gabor transform of a signal is often a challenge for practical applications, in particular in audio signal processing. Many time-frequency (TF) patterns of different shapes may be present in a signal and they can not all be sparsely represented in the same spectrogram. We propose several algorithms, which provide optimal windows for a user-selected TF pattern with respect to different concentration criteria. We base our optimization algorithm on $l^p$-norms as measure of TF spreading. For a given number of sampling points in the TF plane we also propose optimal lattices to be used with the obtained windows. We illustrate the potentiality of the method on selected numerical examples

A jigsaw puzzle framework for homogenization of high porosity foams

An approach to homogenization of high porosity metallic foams is explored. The emphasis is on the \Alporas{} foam and its representation by means of two-dimensional wire-frame models. The guaranteed upper and lower bounds on the effective properties are derived by the first-order homogenization with the uniform and minimal kinematic boundary conditions at heart. This is combined with the method of Wang tilings to generate sufficiently large material samples along with their finite element discretization. The obtained results are compared to experimental and numerical data available in literature and the suitability of the two-dimensional setting itself is discussed.Comment: 11 pages, 7 figures, 3 table