arXiv subjects

Daniel Vera

Publications and source records attributed to Daniel Vera.

7 recordsLinked to original sources

arXiv2018

A simple shearlet-based reconstruction for computer tomography

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.

math.FA↗

arXiv2016

Region-of-Interest reconstruction from truncated cone-beam projections

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 $Γ$ 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 $ε>0$, we prove that if $C$ is sufficiently large our iterative reconstruction algorithm converges uniformly to an $ε$-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 $ε$-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~$B$.

math-ph↗

arXiv2014

Democracy of shearlet bases with applications to approximation and interpolation

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.

math.FA↗

arXiv2013

Greedy type algorithms for RIP matrices. A study of two selection rules

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.

cs.IT↗

arXiv2012

Shear Anisotropic Inhomogeneous Besov And Triebel-Lizorkin Spaces In $R^d$

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.

math.FA↗

arXiv2012

Triebel-Lizorkin Spaces and Shearlets on the Cone in $\mathbb{R}^2$

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. However, little is known about their relation with spaces other than $L^2$. Here, we find a characterization of a kind of anisotropic inhomogeneous Triebel-Lizorkin spaces (to be defined) 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 $L^2$ case. We also find some embeddings of the (classical) dyadic spaces into these highly anisotropic spaces, and viceversa, 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.

math.FA↗

arXiv2011

Restricted non-linear approximation in sequence spaces and applications to wavelet bases and interpolation

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 Temlyakow property in this setting. Moreover, new interpolation results for Triebel-Lizorkin and Besov spaces are obtained.

math.CA↗