Abstract The term lambda-tomography refers to an operator that preserves the singular support of a function and, more generally, its wavefront set. We review the basics of the Radon transform (RT), microlocal analysis and lambda tomography. Then, we develop a novel analytical expression that allows us to evaluate some pseudo-differential operators (related to the RT) on a function with the shearlet system. We also develop their respective numerical implementation with the ShearLab toolbox. As a consequence, the implementation is simple, accurate and fast. A first application is on the lambda and local operators of the region-of-interest (ROI) tomography, where highly incomplete samples are measured. Taking advantage of the fact that the lambda operator accentuates boundaries and singularities, the second application is a very competitive algorithm aimed to detect edge-like features in synthetic, natural and real medical images along with their orientations. 2020 Mathematics Subject Classification: 42C40, 42-08, 65T99, 68U10

We use well-known intertwining relations of integral operators between the Euclidean space and the manifold of planes/lines to develop new analytical inverses of the Radon/John transforms. The integral operators are convolutions with shearlets as kernels. The shearlets are smooth compactly supported and away from the origin in the Fourier domain, therefore the operators are bounded. We show a series of relations in the form of shearlet coefficients, convolutions and inner products in different spaces. This framework yields potential applications since there is a natural translation from the continuous to the discrete theory of shearlets. We also develop a new biorthogonal shearlet decomposition for the 3D Radon transform inversion. This shades light on the differences between our method of integral (bounded) operators and the biorthogonal curvelet/shearlet decomposition of differential (unbounded) operators.

Having in mind some applications in operator theory, we extend the class of shearlet-based non-homogeneous Triebel–Lizorkin spaces, originally introduced in Vera (Appl Comput Harmon Anal 12:130–150, 2013). The extension consists in allowing the value $$p=\infty $$ as one of the parameters describing the spaces. We also establish some basic properties of these new spaces, including the fact that for a particular set of parameters one gets a properly larger space than the space of local bounded mean oscillation functions from Goldberg (Duke Math J 46:27–42, 1979).

We find a new and simple inversion formula for the 2D Radon transform (RT) with a straight use of the shearlet system and of well-known properties of the RT. Since the continuum theory of shearlets has a natural translation to the discrete theory, we also obtain a computable algorithm that recovers a digital image from noisy samples of the 2D Radon transform which also preserves edges. A very well-known RT inversion in the applied harmonic analysis community is the biorthogonal curvelet decomposition (BCD). The BCD uses an intertwining relation of differential (unbounded) operators between functions in Euclidean and Radon domains. Hence the BCD is ill-posed since the inverse is unstable in the presence of noise. In contrast, our inversion method makes use of an intertwining relation of integral transformations with very smooth kernels and compact support away from the origin in the Fourier domain, i.e. bounded operators. Therefore, we obtain, at least, the same asymptotic behavior of mean-square error as the BCD (and its shearlet version) for the class of cartoon-like functions. Numerical simulations show that our inverse surpasses, by far, the inverse based on the BCD. Our algorithm uses only fast transformations.

Restricted non linear approximation is a generalization of the N‐term approximation in which a measure on the index set of the approximants controls the type, instead of the number, of elements in the approximation. Thresholding is a well‐known type of non linear approximation. We relate a generalized upper and lower Temlyakov property with the decreasing rate of the thresholding approximation. This relation is in the form of a characterization through some general discrete Lorentz spaces. Thus, not only we recover some results in the literature but find new ones. As an application of these results, we compress and reduce noise of some images with wavelets and shearlets and show, at least empirically, that the L2‐norm is not necessarily the best norm to measure the approximation error.

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~$B$.

Shearlets on the cone provide Parseval frames for L-2. They also provide near-optimal approximation for the class epsilon of cartoon-like images. Moreover, there are spaces associated to them other than L-2 and there exist embeddings between these and classical spaces.We prove approximation properties of the cone-adapted shearlet system coefficients in a more general context. Namely, when the target shearlet sequence belongs to a class or space different to that obtained from a shearlet sequence of a f is an element of epsilon and when the error is not necessarily measured in the L-2-norm (or, since the shearlet system is a frame, the l(2)-norm) but in a norm of a much wider family of smoothness spaces of "high" anisotropy. We first prove democracy of shearlet frames in shear anisotropic inhomogeneous Besov and Triebel Lizorkin sequence spaces. Then, we prove embeddings between approximation spaces and discrete weighted Lorentz spaces in the framework of shearlet coefficients. Simultaneously, we also prove that these embeddings are equivalent to Jackson and Bernstein type inequalities. This allows us to find real interpolation between these highly anisotropic sequence spaces. We also describe how some of these results can be extended to other shearlet and curvelet generated spaces. Finally, we show some examples of embeddings between wavelet approximation spaces and shearlet approximation spaces and obtain a similar result stated in L-2(R-2) for the curvelet smoothness spaces. This also paves the way to the use of thresholding algorithms in compression or noise reduction. (C) 2016 Elsevier Inc. All rights reserved.

Shearlets on the cone are a multi-scale and multi-directional discrete system that have near-optimal representation of the so-called cartoon-like functions. They form Parseval frames, have better geometrical sensitivity than traditional wavelets and an implementable framework. Recently, it has been proved that some smoothness spaces can be associated to discrete systems of shearlets. Moreover, there exist embeddings between the classical isotropic dyadic spaces and the shearlet generated spaces. We prove boundedness of pseudo-differential operators (PDO’s) with non regular symbols on the shear anisotropic inhomogeneous Besov spaces and on the shear anisotropic inhomogeneous Triebel–Lizorkin spaces (which are up to now the only Triebel–Lizorkin-type spaces generated by either shearlets or curvelets and more generally by any parabolic molecule, as far as we know). The type of PDO’s that we study includes the classical Hörmander definition with x-dependent parameter \(\delta \) for a range limited by the anisotropy associated to the class. One of the advantages is that the anisotropy of the shearlet spaces is not adapted to that of the PDO.

We find a new and simple inversion formula of the Radon transform RT with the only use of the shearlet system and of well-known properties of RT. No intertwining relation of differential operators in Euclidean space and Radon domain is used. As a consequence, an additive noise is not incremented. Since the continuum theory of shearlets has a straight translation to the discrete theory, we find a fast, stable and computable algorithm that recovers a digital image from noisy samples of the Radon transform preserving edges. In the process, we find a more natural and easier-to-construct density-compensation weight functions for the ShearLab toolbox.

We define distribution spaces in ℝd via ℓq(Lp) norms of a sequence of convolutions of [Formula: see text] with smooth functions, the shearlet system. Then, we define associated sequence spaces and prove characterization with the shearlet coefficients. We also prove continuous embeddings between some shear anisotropic inhomogeneous spaces and between classical (dyadic isotropic) inhomogeneous spaces and shear anisotropic inhomogeneous spaces.

Ernesto Laclau, emeritus professor of the University of Essex, UK, visited Ecuador in March 2012. In his open conference at FLACSO-, Quito, Laclau presented his peculiar interpretation of "populism". The present article sketches the main lines of Laclau's lecture and draws a critique on some of his main theoretical and political tenets.

El presente artículo refiere a las propuestas de Ernesto Laclau que revisan la consideración tradicional acerca del populismo. En marzo de 2012, Laclau estuvo de visita en Ecuador y ofreció una conferencia magistral en FLACSO. El artículo a continuación responde en parte a lo dicho por Laclau. Se esbozan críticas a algunos de sus planteamientos.

Restricted non-linear approximation is a type of N-term approximation where a measure ν on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel–Lizorkin spaces by showing the Temlyakov property in this setting. Moreover, new interpolation results for Triebel–Lizorkin and Besov spaces are obtained.

The shearlets are a special case of the wavelets with composite dilation that, among other things, have a basis-like structure and multi-resolution analysis properties. These relatively new representation systems have encountered wide range of applications, generally surpassing the performance of their ancestors due to their directional sensitivity. Both theories of coorbit spaces and decomposition spaces provide a way of associating some kind of smoothness spaces to shearlets. However, these smoothness spaces are closer to classical Besov type spaces. Here, instead, we define a kind of highly anisotropic inhomogeneous Triebel–Lizorkin spaces and prove that it can be characterized with the so-called “shearlets on the cone” coefficients. We first prove the boundedness of the analysis and synthesis operators with the “traditional” shearlets coefficients. Then, with the development of the smooth Parseval frames of shearlets of Guo and Labate we are able to prove a reproducing identity, which was previously possible only for the L2 case. We also find some embeddings of the (classical) dyadic spaces into these highly anisotropic spaces, and vice versa, for certain ranges of parameters. In order to keep a concise document we develop our results in the “weightless” case (w=1) and give hints on how to develop the weighted case.

Some consequences of the Restricted Isometry Property (RIP) of matrices have been applied to develop a greedy algorithm called "ROMP" (Regularized Orthogonal Matching Pursuit) to recover sparse signals and to approximate non-sparse ones. These consequences were subsequently applied to other greedy and thresholding algorithms like "SThresh", "CoSaMP", "StOMP" and "SWCGP". In this paper, we find another consequence of the RIP property and use it to analyze the approximation to k-sparse signals with Stagewise Weak versions of Gradient Pursuit (SWGP), Matching Pursuit (SWMP) and Orthogonal Matching Pursuit (SWOMP). We combine the above mentioned algorithms with another selection rule similar to the ones that have appeared in the literature showing that results are obtained with less restrictions in the RIP constant, but we need a smaller threshold parameter for the coefficients. The results of some experiments are shown.

Shearlets on the cone provide Parseval frames for $L^2$. They also provide near-optimal approximation for the class $\mathcal{E}$ of cartoon-like images. Moreover, there are spaces associated to them other than $L^2$ and there exist embeddings between these and classical spaces. We prove approximation properties of the cone-adapted shearlets in a more general context, namely, when the target function belongs to a class or space different to $\mathcal{E}$ and when the error is not necessarily measured in the $L^2$-norm but in a much wider family of smoothness space of high anisotropy.

We define distribution spaces of a sequence of convolutions of a set of distributions with smooth functions, the shearlet system. Then, we define associated sequence spaces and prove characterizations. We also show a reproducing identity in the class of distributions. Finally, we prove Sobolev-type embeddings within the shear anisotropic inhomogeneous spaces and embeddings between (classical dyadic) isotropic inhomogeneous spaces and shear anisotropic inhomogeneous spaces.

En el presente artículo consigno algunas reflexiones que me ha despertado la lectura de la obra de Ernesto Laclau. En especial, encontré sumamente sugestiva su reconsideración del viejo tópico del populismo. A mi entender, Laclau no solamente ha dado con una nueva forma de considerar el tema, sino lo político tout court. Mi perspectiva de lectura y evaluación de las propuestas de Laclau se apoya en el psicoanálisis, que constituye para mí el horizonte de cualquier reconstrucción posible de las ciencias sociales.This article offers a number of reflections inspired by a reading of Ernesto Laclau’s works. I found particularly suggestive his reconsideration of the old topic of populism. As I understand it, Laclau has found a new way to approach not only this issue, but also politics tout court. My perspective on reading and evaluating Laclau’s proposals is based on psychoanalysis, which for me constitutes the horizon of all possible social science reconstructions.

This paper presents the GoalBit Starter platform. GoalBit is an open source peer-to-peer distribution system of real-time video streams over Internet. The main advantage of a P2P architecture is the possibility of using available upload bandwidth in the hosts connected. The main difficulty is that these hosts are typically highly dynamic, they continuously enter and leave the network. To deal with this problem a mesh connectivity approach is used (Bittorrent-like) where the stream is decomposed into several pieces and shared between different peers. Nowadays, the GoalBit platform is used by operators and by final users to broadcast their live contents. To illustrate its potential, we present some empirical results measured in an emulation of a GoalBit P2P streaming, with more than 300 peers concurrently connected.