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

Optimally sparse approximations of 3D functions by compactly supported shearlet frames

We study efficient and reliable methods of capturing and sparsely representing anisotropic structures in 3D data. As a model class for multidimensional data with anisotropic features, we introduce generalized three-dimensional cartoon-like images. This function class will have two smoothness parameters: one parameter \beta controlling classical smoothness and one parameter \alpha controlling anisotropic smoothness. The class then consists of piecewise C^\beta-smooth functions with discontinuities on a piecewise C^\alpha-smooth surface. We introduce a pyramid-adapted, hybrid shearlet system for the three-dimensional setting and construct frames for L^2(R^3) with this particular shearlet structure. For the smoothness range 1<\alpha =< \beta =< 2 we show that pyramid-adapted shearlet systems provide a nearly optimally sparse approximation rate within the generalized cartoon-like image model class measured by means of non-linear N-term approximations.Comment: 56 pages, 6 figure

Globular clusters versus dark matter haloes in strong lensing observations

Small distortions in the images of Einstein rings or giant arcs offer the exciting prospect of detecting low mass dark matter haloes or subhaloes of mass below 109 M⊙ (for independent haloes, the mass refers to M200, and for subhaloes, the mass refers to the mass within tidal radius), most of which are too small to have made a visible galaxy. A very large number of such haloes are predicted to exist in the cold dark matter model of cosmogony; in contrast, other models, such as warm dark matter, predict no haloes below a mass of this order, which depends on the properties of the warm dark matter particle. Attempting to detect these small perturbers could therefore discriminate between different kinds of dark matter particles, and even rule out the cold dark matter model altogether. Globular clusters in the lens galaxy also induce distortions in the image, which could, in principle, contaminate the test. Here, we investigate the population of globular clusters in six early-type galaxies in the Virgo cluster. We find that the number density of globular clusters of mass MGC ∼ 106 M⊙ is comparable to that of the dark matter perturbers (subhaloes in the lenses and haloes along the line of sight of comparable mass). We show that the very different degrees of mass concentration in globular clusters and dark matter haloes result in different lensing distortions. These are detectable with milli-arcsecond resolution imaging, which can distinguish between globular cluster and dark matter halo signals

Potassium channel dysfunction in human neuronal models of Angelman syndrome

Disruptions in the ubiquitin protein ligase E3A (UBE3A) gene cause Angelman syndrome (AS). Whereas AS model mice have associated synaptic dysfunction and altered plasticity with abnormal behavior, whether similar or other mechanisms contribute to network hyperactivity and epilepsy susceptibility in AS patients remains unclear. Using human neurons and brain organoids, we demonstrate that UBE3A suppresses neuronal hyperexcitability via ubiquitin-mediated degradation of calcium- and voltage-dependent big potassium (BK) channels. We provide evidence that augmented BK channel activity manifests as increased intrinsic excitability in individual neurons and subsequent network synchronization. BK antagonists normalized neuronal excitability in both human and mouse neurons and ameliorated seizure susceptibility in an AS mouse model. Our findings suggest that BK channelopathy underlies epilepsy in AS and support the use of human cells to model human developmental diseases

Simulating quantum statistics with entangled photons: a continuous transition from bosons to fermions

In contrast to classical physics, quantum mechanics divides particles into two classes-bosons and fermions-whose exchange statistics dictate the dynamics of systems at a fundamental level. In two dimensions quasi-particles known as 'anyons' exhibit fractional exchange statistics intermediate between these two classes. The ability to simulate and observe behaviour associated to fundamentally different quantum particles is important for simulating complex quantum systems. Here we use the symmetry and quantum correlations of entangled photons subjected to multiple copies of a quantum process to directly simulate quantum interference of fermions, bosons and a continuum of fractional behaviour exhibited by anyons. We observe an average similarity of 93.6\pm0.2% between an ideal model and experimental observation. The approach generalises to an arbitrary number of particles and is independent of the statistics of the particles used, indicating application with other quantum systems and large scale application.Comment: 10 pages, 5 figure