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The axon initial segment (AIS) is the first 20- to 60-μm segment of the axon proximal to the soma of a neuron. This highly specialized subcellular domain is the initiation site of the action potential and contains a high concentration of voltage-gated ion channels held in place by a complex nexus of scaffolding and regulatory proteins that ensure proper electrical activity of the neuron. Studies have shown that dysfunction of many AIS channels and scaffolding proteins occurs in a variety of neuropsychiatric and neurodegenerative diseases, raising the need to develop accurate methods for visualization and quantification of the AIS and its protein content in models of normal and disease conditions. In this article, we describe methods for immunolabeling AIS proteins in cultured neurons and brain slices as well as methods for quantifying protein expression and pattern distribution using fluorescent labeling of these proteins. © 2019 by John WileySons, Inc.

This paper presents the results of a study aimed at investigating the potential of Compressive Sensing (CS) technologies for optical space instruments. Besides assessing the pros and cons for a wide set of proposed instrumental concepts for space applications, the study analyzed in further detail two CS-based instrument concepts, each targeting a specific application: an UV-VIS hyperspectral imager on orbiter for stellar spectro-photometry and a MIR camera for sky observation and real-time detection of Near Earth Objects (NEO). The proposed UV-VIS hyperspectral imager relies on a classical CS approach and addresses the CS reconstruction of the full image in order to implement slitless spectrophotometry of stars. The CS-based MIR camera for NEO detection instead explores a novel approach aiming at information extraction without a prior full reconstruction of the image. Besides outlining the optical design of the instruments, its key elements and a pros and cons analysis of the architecture, this paper presents the performance assessment of these instruments for typical application scenarios by means of simulated data. The results showed that, from the point of view of data reconstruction quality, a good performance can be achieved by the designed instruments in terms of compression ratio (CR) and image reconstruction. In terms of system budgets, the CS architecture offered only some marginal benefits with respect to their traditional counterparts, mainly due to the lack of a compression board. Most advantages are instead provided in terms of downlink requirements and memory buffer.

When it comes to computed tomography (CT), the possibility to reconstruct a small region-of-interest (ROI) using truncated projection data is particularly appealing due to its potential to lower radiation exposure and reduce the scanning time. However, ROI reconstruction from truncated projections is an ill-posed inverse problem, with the ill-posedness becoming more severe when the ROI size is getting smaller. To address this problem, both ad hoc analytic formulas and iterative numerical schemes have been proposed in the literature. In this paper, we introduce a novel approach for ROI CT reconstruction, formulated as a convex optimization problem with a regularized term based on shearlets. Our numerical implementation consists of an iterative scheme based on the scaled gradient projection (SGP) method and is tested in the context of fan beam CT. Our results show that this approach is essentially insensitive to the location of the ROI and remains very stable also when the ROI size is rather small., 13 pages, 30 figures

Region-of-Interest (ROI) tomography aims at reconstructing a region of interest $C$ inside a body using only x-ray projections intersecting $C$ with the goal to reduce overall radiation exposure when only a small specific region of the body needs to be examined. We consider x-ray acquisition from sources located on a smooth curve $\Gamma$ in $\mathbb{R}^3$ verifying classical Tuy's condition. In this situation, the {\it non-trucated} cone-beam transform $D f$ of smooth densities $f$ admits an explicit inverse $Z$; however $Z$ cannot directly reconstruct $f$ from ROI-truncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities $f$ in $L^{\infty}(B)$ where $B$ is a bounded ball in $\mathbb{R}^3$, our method iterates an operator $U$ combining ROI-truncated projections, inversion by the operator $Z$ and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI $C \subset B$, given $\epsilon >0$, we prove that if $C$ is sufficiently large our iterative reconstruction algorithm converges uniformly to an $\epsilon$-accurate approximation of $f$, where the accuracy depends on the regularity of $f$ quantified in the Sobolev norm $W^5(B)$. This result shows the existence of a critical ROI radius ensuring the convergence of the ROI reconstruction algorithm to $\epsilon$-accurate approximations of $f$. We numerically verified these theoretical results using simulated acquisition of ROI-truncated cone-beam projection data for multiple acquisition geometries. Numerical experiments indicate that the critical ROI radius is fairly small with respect to the support region~$B$.

We set up a new general coorbit space theory for reproducing representations of a locally compact second countable group $G$ that are not necessarily irreducible nor integrable. Our basic assumption is that the kernel associated with the voice transform belongs to a Fr\'echet space $\mathcal T$ of functions on $G$, which generalizes the classical choice $\mathcal T=L_w^1(G)$. Our basic example is $ \mathcal T=\bigcap_{p\in(1,+\infty)} L^p(G)$, or a weighted versions of it. By means of this choice it is possible to treat, for instance, Paley-Wiener spaces and coorbit spaces related to Shannon wavelets and Schr\"odingerlets., Comment: 52 pages, 1 figures

We set up a new general coorbit space theory for reproducing representations of a locally compact second countable group $G$ that are not necessarily irreducible nor integrable. Our basic assumption is that the kernel associated with the voice transform belongs to a Fr��chet space $\mathcal T$ of functions on $G$, which generalizes the classical choice $\mathcal T=L_w^1(G)$. Our basic example is $ \mathcal T=\bigcap_{p\in(1,+\infty)} L^p(G)$, or a weighted versions of it. By means of this choice it is possible to treat, for instance, Paley-Wiener spaces and coorbit spaces related to Shannon wavelets and Schr��dingerlets., 52 pages, 1 figures

The continuous curvelet and shearlet transforms have recently been shown to be much more effective than the traditional wavelet transform in dealing with the set of discontinuities of functions and distributions. In particular, the continuous shearlet transform has the ability to provide a very precise geometrical characterization of general discontinuity curves occurring in images. In this paper, we show that these properties are useful to design improved algorithms for the analysis and detection of edges.

The affine systems generated by $\Psi \subset L^2(\mathbb{R}^n)$ are the systems $\mathcal{A}_A(\Psi) = \{D^j_A T_k\Psi : j \in \mathbb{Z},k \in \mathbb{Z}^n$ where $T_k$ are the translations, and $D_A$ the dilations with respect to an invertible matrix A. As shown in [5], there is a simple characterization for those affine systems that are a Parseval frame for $L^2(\mathbb{R}^n)$. In this paper, we correct an error in the proof of the characterization result from [5], by redefining the class of not-necessarily expanding dilation matrices for which this characterization result holds. In addition, we examine the connection between the eigenvalues of the dilation matrix A and the characterization equations of the affine system $\mathcal{A}_A(\Psi)$ that are Parseval frames. Our observations go in the same directions as other recent results in the literature that show that, when A is not expanding, the information about the eigenvalues alone is not sufficient to characterize or to determine existence of those affine systems that are Parseval frames.

In this thesis we discuss and extend the theory of shearlet systems. These systems were introduced by Guo, Kutyniok, Labate, Lim and Weiss, and have found a multitude of applications in signal- and image processing and related fields since then. The results of this thesis are split into two different but connected parts. In the first part we present a construction of a novel shearlet system on a bounded domain. This system can be used to construct adaptive discretizations of partial differential equations. The second aspect of this thesis is to analyze the shearlet transform and its ability to detect and classify edges in images. To employ shearlet systems for the discretization of partial differential equations a number of properties must be fulfilled by the shearlet system. Among others, one necessity is that the shearlet system constitutes a frame for the space of square integrable functions defined on a bounded domain. Furthermore, the system needs to be able to characterize Sobolev spaces in a suitable way. This requirement is, for example, fulfilled if the system constitutes a so-called Gelfand frame. We show that a novel construction, termed boundary shearlet system, satisfies all those requirements. We prove the Gelfand frame property partly analytically and partly numerically. Additionally, we show that boundary shearlet systems can characterize Sobolev norms by weighted norms of square summable sequence spaces. Lastly, we also demonstrate that boundary shearlet systems yield almost optimal approximation rates for functions which are smooth except for a curvilinear singularity. Along the way we find new improved approximation rates by shearlets on the two-dimensional real plane for functions whose first- or higher-order derivatives are cartoon-like functions. Furthermore, we will see from numerical experiments that boundary shearlet systems are, indeed, capable and very well suited for the discretization of elliptic partial differential equations and they lead to adaptive algorithms admitting optimal convergence rates. In the second part of this thesis we study the analysis of singularities by the shearlet transform. We describe the asymptotic behavior of the shearlet transform and provide a possibility to make a classification of different types of singularities based on that behavior. This procedure is based on works of Guo and Labate where a similar classification method was proposed for a band-limited shearlet generator. We extend these results for compactly supported shearlet systems in this thesis. Moreover, we observe that in many cases the use of compactly supported shearlets systems leads to improved results. These improvements include the existence of uniform estimates for the asymptotic behavior of the shearlet transform. Furthermore, we obtain stronger separation of the decay rates of the shearlet transform, leading to clearer classification of different types of singularities. In addition, we accurately describe the influence of the curvature of a singularity curve on the asymptotic behavior of the shearlet transform. Die vorliegende Arbeit untersucht und erweitert die Theorie der Shearletsysteme. Diese Systeme wurden seit ihrer Einführung in einer Vielzahl von Anwendungen in der Signal- und Bildverarbeitung und weiteren verwandten Gebieten eingesetzt. Die Resultate dieser Arbeit lassen sich zwei Bereichen zuordnen. Der erste Bereich beschäftigt sich mit der Konstruktion eines neuartigen Shearletsystems auf einem beschränkten Gebiet, das zur Lösung partieller Differentialgleichungen verwendet werden kann. Der zweite Bereich untersucht das Verhalten der Shearlettransformation und die Möglichkeit damit Kanten eines Bildes zu erkennen und zu klassifizieren. Um Shearletsysteme für die Diskretisierung von partiellen Differentialgleichungen verwenden zu können, müssen eine Reihe von Bedingungen erfüllt sein. Unter anderem ist es notwendig, dass, wenn ein beschränktes Gebiet gegeben ist, die Systeme in der Lage sind, Frames für den Hilbertraum der quadratintegirerbaren Funktionen definiert auf diesem Gebiet zu bilden und Sobolevräume auf beschränkten Gebieten in einer geeigneten Form zu beschreiben. Diese Eigenschaft ist erfüllt, wenn das zugrunde liegende System einen sogenannten Gelfandframe bildet. Wir werden zeigen, dass eine neue Konstruktion eines Shearletsystems auf einem beschränkten Gebiet, welche wir boundary Shearletsystem nennen werden, alle oben genannten Eigenschaften erfüllt. Die Gelfandframeeigenschaft weisen wir teils analytisch und teils numerisch nach. Darüber hinaus werden wir auch beweisen, dass boundary Shearletsysteme, Sobolevräume anhand ihrer Analysiskoeffizienten beschreiben. Des Weiteren wird gezeigt, dass boundary Shearletsysteme optimale Approximationsraten für Funktionen, die abgesehen von einer kurvenförmigen Singularität glatt sind, aufweisen. Auf dem Weg zu diesen Resultaten beweisen wir neue, verbesserte Approximationsraten durch Shearlets auf der gesamten reelen Ebene für Funktionen, deren Ableitungen kurvenförmige Singularitäten aufweisen. Außerdem werden wir anhand einer numerischen Analyse sehen, dass boundary Shearletsysteme in der Tat geeignet sind, um elliptische partielle Differentialgleichungen mittels adaptiver numerischer Algorithmen zu lösen, sodass die numerischen Prozeduren optimale Konvergenzraten aufweisen. Im zweiten Teil dieser Arbeit beschäftigen wir uns mit der Analyse von Singularitäten anhand der Shearlettransformation. Dieser Teil der Arbeit beschreibt das asymptotische Verhalten der Shearlettransformation und die Möglichkeit, anhand dieses Verhaltens eine Klassifizierung von Kurvensingularitäten vorzunehmen. Diese Analyse basiert auf den Arbeiten von Guo und Labate. Dort wurde die Klassifizierung anhand eines bandbeschränkten Shearlets vorgenommen. Die Resultate dieser Arbeiten werden für Shearlets mit kompaktem Träger erweitert. Die neuen Resultate belegen, dass in vielen Fällen eine Analyse mithilfe von Shearlets mit kompaktem Träger bessere Ergebnisse liefert als die klassischen Resultate. Die Verbesserungen beinhalten den Nachweis gleichmäßiger Abfallraten der Shearlettransformation, eine bessere Unterscheidbarkeit der Abfallraten unterschiedlich zu klassifizierender Singularitätskurven, sowie eine genaue Beschreibung des Einflusses der Krümmung einer Singularitätskurve auf das asymptotische Verhalten der Shearlettransformation.