Classification of edges using compactly supported shearlets

We analyze the detection and classification of singularities of functions $f={\chi }_S$, where $S?{\mathbb{R}}^d$ and $d=2,3$. It will be shown how the set ?S can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if ?S is a $d?1$ dimensional piecewise smooth manifold with $d=2$ or 3, we will classify smooth and non-smooth components of ?S. This improves previous results given for shearlet systems with a certain band-limited generator, since the estimates we derive are uniform. Moreover, we will show that our bounds are optimal. Along the way, we also obtain novel results on the characterization of wavefront sets in 3 dimensions by compactly supported shearlets. Finally, geometric properties of ?S such as curvature are described in terms of the continuous shearlet transform of f.     

Compactly supported shearlets are optimally sparse

Cartoon-like images, i.e., C² functions which are smooth apart from a C² discontinuity curve, have by now become a standard model for measuring sparse (nonlinear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit sparse approximations within this model, which are optimal up to a log-factor. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications.In this paper, we present the first complete proof of optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of (non-tight) shearlet frames with shearlet generators having compact support and satisfying some weak directional vanishing moment conditions.     

Shearlet coorbit spaces and associated Banach frames

In this paper, we study the relationships of the newly developed continuous shearlet transform with the coorbit space theory. It turns out that all the conditions that are needed to apply the coorbit space theory can indeed be satisfied for the shearlet group. Consequently, we establish new families of smoothness spaces, the shearlet coorbit spaces. Moreover, our approach yields Banach frames for these spaces in a quite natural way. We also study the approximation power of best n-term approximation schemes and present some first numerical experiments.     

Construction of Compactly Supported Shearlet Frames

Shearlet tight frames have been extensively studied in recent years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital settings. However, these studies only concerned shearlet tight frames generated by a band-limited shearlet, whereas for practical purposes compact support in spatial domain is crucial.     

Duality Principles in Frame Theory

Duality principles in Gabor theory such as the Ron–Shen duality principle and the Wexler–Raz biorthogonality relations play a fundamental role for analyzing Gabor systems. In this article we present a general approach to derive duality principles in abstract frame theory. For each sequence in a separable Hilbert space we define a corresponding sequence dependent only on two orthonormal bases. Then we characterize exactly properties of the first sequence in terms of the associated one, which yields duality relations for the abstract frame setting. In the last part we apply our results to Gabor systems.     

Quantitative Comparison of Tandem Mass Spectra Obtained on Various Instruments

The similarity between two tandem mass spectra, which were measured on different instruments, was compared quantitatively using the similarity index (SI), defined as the dot product of the square root of peak intensities in the respective spectra. This function was found to be useful for comparing energy-dependent tandem mass spectra obtained on various instruments. Spectral comparisons show the similarity index in a 2D “heat map”, indicating which collision energy combinations result in similar spectra, and how good this agreement is. The results and methodology can be used in the pharma industry to design experiments and equipment well suited for good reproducibility. We suggest that to get good long-term reproducibility, it is best to adjust the collision energy to yield a spectrum very similar to a reference spectrum. It is likely to yield better results than using the same tuning file, which, for example, does not take into account that contamination of the ion source due to extended use may influence instrument tuning. The methodology may be used to characterize energy dependence on various instrument types, to optimize instrumentation, and to study the influence or correlation between various experimental parameters.

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Graphical Abstract ?

     

A generalization of Gram–Schmidt orthogonalization generating all Parseval frames

Given an arbitrary finite sequence of vectors in a finite-dimensional Hilbert space, we describe an algorithm, which computes a Parseval frame for the subspace generated by the input vectors while preserving redundancy exactly. We further investigate several of its properties. Finally, we apply the algorithm to several numerical examples.      

A new identity for Parseval frames

In this paper we establish a surprising new identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss the derived results.

     

Formation of solvated ions in the atmospheric interface of an electrospray ionization triple-quadrupole mass spectrometer

A simple method capable of generating and investigating various solvent clusters and solvated ions was developed. The technique opens a door to studying these complexes on commercially available instruments. Formation of the desired solvated ion in the gas phase was achieved by introducing the appropriate volatile solvent vapour into the curtain gas stream. Capabilities of the technique are illustrated by generating alkali, alkaline earth and transition metal cations solvated by various volatile compounds such as water, methanol and acetonitrile. Depending on the ligands and on the experimental conditions, clusters of 2-100 molecules may be observed. Isotope labelling suggests that these are formed by a re-solvation process in the curtain gas region.